Advanced Quantum Mechanics: Materials and Photons [3 ed.] 9783030578695

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Graduate Texts in Physics

Rainer Dick

Advanced Quantum Mechanics Materials and Photons Third Edition

Graduate Texts in Physics Series Editors Kurt H. Becker, NYU Polytechnic School of Engineering, Brooklyn, NY, USA ` Jean-Marc Di Meglio, Matie`re et Systemes Complexes, Bâtiment Condorcet, Université Paris Diderot, Paris, France Morten Hjorth-Jensen, Department of Physics, Blindern, University of Oslo, Oslo, Norway Bill Munro, NTT Basic Research Laboratories, Atsugi, Japan William T. Rhodes, Department of Computer and Electrical Engineering and Computer Science, Florida Atlantic University, Boca Raton, FL, USA Susan Scott, Australian National University, Acton, Australia H. Eugene Stanley, Center for Polymer Studies, Physics Department, Boston University, Boston, MA, USA Martin Stutzmann, Walter Schottky Institute, Technical University of Munich, Garching, Germany Andreas Wipf, Institute of Theoretical Physics, Friedrich-Schiller-University Jena, Jena, Germany

Graduate Texts in Physics publishes core learning/teaching material for graduateand advanced-level undergraduate courses on topics of current and emerging fields within physics, both pure and applied. These textbooks serve students at the MS- or PhD-level and their instructors as comprehensive sources of principles, definitions, derivations, experiments and applications (as relevant) for their mastery and teaching, respectively. International in scope and relevance, the textbooks correspond to course syllabi sufficiently to serve as required reading. Their didactic style, comprehensiveness and coverage of fundamental material also make them suitable as introductions or references for scientists entering, or requiring timely knowledge of, a research field.

More information about this series at http://www.springer.com/series/8431

Rainer Dick

Advanced Quantum Mechanics Materials and Photons Third Edition

Rainer Dick Department of Physics University of Saskatchewan Saskatoon, SK, Canada

ISSN 1868-4513 ISSN 1868-4521 (electronic) Graduate Texts in Physics ISBN 978-3-030-57869-5 ISBN 978-3-030-57870-1 (eBook) https://doi.org/10.1007/978-3-030-57870-1 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2016, 2020 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Cover illustration: © Rost-9D / Getty Images / iStock This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

Quantum mechanics was invented in an era of intense and seminal scientific research between 1900 and 1928 (and in many regards continues to be developed and expanded) because neither the properties of atoms and electrons nor the spectrum of radiation from heat sources could be explained by the classical theories of mechanics, electrodynamics, and thermodynamics. It was a major intellectual achievement and a breakthrough of curiosity-driven fundamental research which formed quantum theory into one of the pillars of our present understanding of the fundamental laws of nature. The properties and behavior of every elementary particle are governed by the laws of quantum theory. However, the rule of quantum mechanics is not limited to atomic and subatomic scales, but also affects macroscopic systems in a direct and profound manner. The electric and thermal conductivity properties of materials are determined by quantum effects, and the electromagnetic spectrum emitted by a star is primarily determined by the quantum properties of photons. It is therefore not surprising that quantum mechanics permeates all areas of research in advanced modern physics and materials science, and training in quantum mechanics plays a prominent role in the curriculum of every major physics or chemistry department. The ubiquity of quantum effects in materials implies that quantum mechanics also evolved into a major tool for advanced technological research. The construction of the first nuclear reactor in Chicago in 1942 and the development of nuclear technology could not have happened without a proper understanding of the quantum properties of particles and nuclei. However, the real breakthrough for a wide recognition of the relevance of quantum effects in technology occurred with the invention of the transistor in 1948 and the ensuing rapid development of semiconductor electronics. This proved once and for all the importance of quantum mechanics for the applied sciences and engineering, only 22 years after the publication of the Schrödinger equation! Electronic devices like transistors rely heavily on the quantum mechanical emergence of energy bands in materials, which can be considered as a consequence of combination of many atomic orbitals or as a consequence of delocalized electron states probing a lattice structure. Today the rapid developments of spintronics, photonics, and nanotechnology provide continuing testimony to the technological relevance of quantum mechanics. v

vi

Preface

As a consequence, every physicist, chemist, and electrical engineer nowadays has to learn aspects of quantum mechanics, and we are witnessing a time when also mechanical and aerospace engineers are advised to take at least a second-year course, due to the importance of quantum mechanics for elasticity and stability properties of materials. Furthermore, quantum information appears to become increasingly relevant for computer science and information technology, and a whole new area of quantum technology will likely follow in the wake of this development. Therefore, it seems safe to posit that within the next two generations, second- and third-year quantum mechanics courses will become as abundant and important in the curricula of science and engineering colleges as first- and second-year calculus courses. Quantum mechanics continues to play a dominant role in particle physics and atomic physics—after all, the standard model of particle physics is a quantum theory, and the spectra and stability of atoms cannot be explained without quantum mechanics. However, most scientists and engineers use quantum mechanics in advanced materials research. Furthermore, the dominant interaction mechanisms in materials (beyond the nuclear level) are electromagnetic, and many experimental techniques in materials science are based on photon probes. The introduction to quantum mechanics in the present book takes this into account by including aspects of condensed matter theory and the theory of photons at earlier stages and to a larger extent than other quantum mechanics texts. Quantum properties of materials provide neat and very interesting illustrations of time-independent and timedependent perturbation theory, and many students are better motivated to master the concepts of quantum mechanics when they are aware of the direct relevance for modern technology. A focus on the quantum mechanics of photons and materials is also perfectly suited to prepare students for future developments in quantum information technology, where entanglement of photons or spins, decoherence, and time evolution operators will be key concepts. Indeed, the rapid advancement of experimental quantum physics, nanoscience, and quantum technology warrants regular updates of our courses on quantum theory. Therefore, besides containing more than 50 additional end of chapter problems, the third edition also features a discussion of chiral spin-momentum locking through Rashba spin–orbit coupling and the resulting Edelstein effects in Problem 22.31, as well as the new Chap. 19 on epistemic and ontic interpretations of quantum states. Other special features of the discussion of quantum mechanics in this book concern attention to relevant mathematical aspects which otherwise can only be found in journal articles or mathematical monographs. Special appendices include a mathematically rigorous discussion of the completeness of Sturm–Liouville eigenfunctions in one spatial dimension, an evaluation of the Baker–Campbell–Hausdorff formula to higher orders, and a discussion of logarithms of matrices. Quantum mechanics has an extremely rich and beautiful mathematical structure. The growing prominence of quantum mechanics in the applied sciences and engineering has already reinvigorated increased research efforts on its mathematical aspects. Both students who study quantum mechanics for the sake of its numerous applications

Preface

vii

and mathematically inclined students with a primary interest in the formal structure of the theory should therefore find this book interesting. This book emerged from a quantum mechanics course which I had introduced at the University of Saskatchewan in 2001. It should be suitable for both advanced undergraduate and introductory graduate courses on the subject. To make advanced quantum mechanics accessible to wider audiences which might not have been exposed to standard second- and third-year courses on atomic physics, analytical mechanics, and electrodynamics, important aspects of these topics are briefly, but concisely introduced in special chapters and appendices. The success and relevance of quantum mechanics has reached far beyond the realms of physics research, and physicists have a duty to disseminate the knowledge of quantum mechanics as widely as possible. Saskatoon, SK, Canada

Rainer Dick

Contents

1

The Need for Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Electromagnetic Spectra and Discrete Energy Levels. . . . . . . . . . . . . 1.2 Blackbody Radiation and Planck’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Blackbody Spectra and Photon Fluxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 The Photoelectric Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Wave-Particle Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Why Schrödinger’s Equation?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Interpretation of Schrödinger’s Wave Function . . . . . . . . . . . . . . . . . . . 1.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 3 9 15 15 17 19 23

2

Self-Adjoint Operators and Eigenfunction Expansions . . . . . . . . . . . . . . . . 2.1 The δ Function and Fourier Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Self-Adjoint Operators and Completeness of Eigenstates . . . . . . . . 2.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25 25 30 35

3

Simple Model Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Barriers in Quantum Mechanics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Box Approximations for Quantum Wells, Quantum Wires and Quantum Dots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 The Attractive δ Function Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Evolution of Free Schrödinger Wave Packets . . . . . . . . . . . . . . . . . . . . . 3.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37 37 44 48 51 57

4

Notions from Linear Algebra and Bra-Ket Notation . . . . . . . . . . . . . . . . . . . 4.1 Notions from Linear Algebra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Bra-ket Notation in Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 The Adjoint Schrödinger Equation and the Virial Theorem . . . . . . 4.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63 64 74 79 82

5

Formal Developments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Uncertainty Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Frequency Representation of States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Dimensions of States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

87 87 92 95 ix

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5.4 5.5 5.6

Gradients and Laplace Operators in General Coordinate Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 Separation of Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

6

Harmonic Oscillators and Coherent States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Basic Aspects of Harmonic Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Solution of the Harmonic Oscillator by the Operator Method . . . . 6.3 Construction of the x-Representation of the Eigenstates . . . . . . . . . 6.4 Lemmata for Exponentials of Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Coherent States. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

105 105 106 109 112 115 123

7

Central Forces in Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Separation of Center of Mass Motion and Relative Motion . . . . . . 7.2 The Concept of Symmetry Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Operators for Kinetic Energy and Angular Momentum. . . . . . . . . . . 7.4 Matrix Representations of the Rotation Group . . . . . . . . . . . . . . . . . . . . 7.5 Construction of the Spherical Harmonic Functions . . . . . . . . . . . . . . . 7.6 Basic Features of Motion in Central Potentials. . . . . . . . . . . . . . . . . . . . 7.7 Free Spherical Waves: The Free Particle with Sharp Mz , M 2 . . . . 7.8 Bound Energy Eigenstates of the Hydrogen Atom . . . . . . . . . . . . . . . . 7.9 Spherical Coulomb Waves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.10 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

129 129 132 134 136 141 146 147 152 162 166

8

Spin and Addition of Angular Momentum Type Operators . . . . . . . . . . . 8.1 Spin and Magnetic Dipole Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Transformation of Scalar, Spinor, and Vector Wave Functions Under Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Addition of Angular Momentum Like Quantities . . . . . . . . . . . . . . . . . 8.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

175 176

9

10

179 181 187

Stationary Perturbations in Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . 9.1 Time-Independent Perturbation Theory Without Degeneracies . . 9.2 Time-Independent Perturbation Theory With Degenerate Energy Levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

189 189

Quantum Aspects of Materials I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Bloch’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Wannier States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Time-Dependent Wannier States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 The Kronig-Penney Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5 kp Perturbation Theory and Effective Mass . . . . . . . . . . . . . . . . . . . . . . . 10.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

203 203 207 210 212 217 218

195 200

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11

Scattering Off Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 The Free Energy-Dependent Green’s Function . . . . . . . . . . . . . . . . . . . . 11.2 Potential Scattering in the Born Approximation . . . . . . . . . . . . . . . . . . 11.3 Scattering Off a Hard Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Rutherford Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

225 227 231 237 241 246

12

The Density of States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 Counting of Oscillation Modes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 The Continuum Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3 The Density of States in the Energy Scale . . . . . . . . . . . . . . . . . . . . . . . . . 12.4 Density of States for Free Non-relativistic Particles and for Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5 The Density of States for Other Quantum Systems . . . . . . . . . . . . . . . 12.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

249 250 253 255

13

257 258 260

Time-Dependent Perturbations in Quantum Mechanics . . . . . . . . . . . . . . . 13.1 Pictures of Quantum Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 The Dirac Picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3 Transitions Between Discrete States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4 Transitions from Discrete States into Continuous States: Ionization or Decay Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.5 Transitions from Continuous States into Discrete States: Capture Cross Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.6 Transitions Between Continuous States: Scattering . . . . . . . . . . . . . . . 13.7 Expansion of the Scattering Matrix to Higher Orders. . . . . . . . . . . . . 13.8 Energy-Time Uncertainty. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

265 266 272 276

14

Path Integrals in Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1 Correlation and Green’s Functions for Free Particles . . . . . . . . . . . . . 14.2 Time Evolution in the Path Integral Formulation. . . . . . . . . . . . . . . . . . 14.3 Path Integrals in Scattering Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

311 312 315 321 327

15

Coupling to Electromagnetic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.1 Electromagnetic Couplings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2 Stark Effect and Static Polarizability Tensors . . . . . . . . . . . . . . . . . . . . . 15.3 Dynamical Polarizability Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

331 331 339 341 349

16

Principles of Lagrangian Field Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.1 Lagrangian Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2 Symmetries and Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.3 Applications to Schrödinger Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . 16.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

353 353 356 360 362

281 290 293 298 300 301

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17

Non-relativistic Quantum Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.1 Quantization of the Schrödinger Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.2 Time Evolution for Time-Dependent Hamiltonians . . . . . . . . . . . . . . . 17.3 The Connection Between First and Second Quantized Theory . . . 17.4 The Dirac Picture in Quantum Field Theory . . . . . . . . . . . . . . . . . . . . . . 17.5 Inclusion of Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.6 Two-Particle Interaction Potentials and Equations of Motion . . . . 17.7 Expectation Values and Exchange Terms . . . . . . . . . . . . . . . . . . . . . . . . . . 17.8 From Many Particle Theory to Second Quantization . . . . . . . . . . . . . 17.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

367 368 377 379 384 389 397 405 408 409

18

Quantization of the Maxwell Field: Photons . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.1 Lagrange Density and Mode Expansion for the Maxwell Field . . 18.2 Photons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.3 Coherent States of the Electromagnetic Field . . . . . . . . . . . . . . . . . . . . . 18.4 Photon Coupling to Relative Motion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.5 Energy-Momentum Densities and Time Evolution in Quantum Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.6 Photon Emission Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.7 Photon Absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.8 Stimulated Emission of Photons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.9 Photon Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.10 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

431 431 438 441 443

19

446 450 459 467 469 479

Epistemic and Ontic Quantum States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.1 Stern-Gerlach Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.2 Non-locality from Entanglement?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.3 Quantum Jumps and the Continuous Evolution of Quantum States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.4 Photon Emission Revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.5 Particle Location . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

491 494 497

20

Quantum Aspects of Materials II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.1 The Born-Oppenheimer Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.2 Covalent Bonding: The Dihydrogen Cation . . . . . . . . . . . . . . . . . . . . . . . 20.3 Bloch and Wannier Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.4 The Hubbard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.5 Vibrations in Molecules and Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.6 Quantized Lattice Vibrations: Phonons . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.7 Electron-Phonon Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

515 516 520 530 534 536 548 554 558

21

Dimensional Effects in Low-Dimensional Systems. . . . . . . . . . . . . . . . . . . . . . 21.1 Quantum Mechanics in d Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Inter-Dimensional Effects in Interfaces and Thin Layers . . . . . . . . . 21.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

563 563 569 575

500 504 505 510

Contents

xiii

22

Relativistic Quantum Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.1 The Klein-Gordon Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 Klein’s Paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.3 The Dirac Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.4 The Energy-Momentum Tensor for Quantum Electrodynamics . . 22.5 The Non-relativistic Limit of the Dirac Equation . . . . . . . . . . . . . . . . . 22.6 Covariant Quantization of the Maxwell Field . . . . . . . . . . . . . . . . . . . . . 22.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

583 583 592 595 605 610 619 624

23

Applications of Spinor QED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.1 Two-Particle Scattering Cross Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.2 Electron Scattering off an Atomic Nucleus . . . . . . . . . . . . . . . . . . . . . . . . 23.3 Photon Scattering by Free Electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.4 Møller Scattering. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

643 643 649 654 666 674

A

Lagrangian Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 677

B

The Covariant Formulation of Electrodynamics . . . . . . . . . . . . . . . . . . . . . . . . 689

C

Completeness of Sturm–Liouville Eigenfunctions . . . . . . . . . . . . . . . . . . . . . . 711

D

Properties of Hermite Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 729

E

The Baker–Campbell–Hausdorff Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733

F

The Logarithm of a Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 737

G

Dirac γ Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 743

H

Spinor Representations of the Lorentz Group . . . . . . . . . . . . . . . . . . . . . . . . . . 755

I

Transformation of Fields Under Reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 767

J

Green’s Functions in d Dimensions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 773

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 799 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805

To the Students

Congratulations! You have reached a stage in your studies where the topics of your inquiry become ever more interesting and more relevant for modern research in basic science and technology. Together with your professors, I will have the privilege to accompany you along the exciting road of your own discovery of the bizarre and beautiful world of quantum mechanics. I will aspire to share my own excitement that I continue to feel for the subject and for science in general. You will be introduced to many analytical and technical skills that are used in everyday applications of quantum mechanics. These skills are essential in virtually every aspect of modern research. A proper understanding of a materials science measurement at a synchrotron requires a proper understanding of photons and quantum mechanical scattering, just like manipulation of qubits in quantum information research requires a proper understanding of spin and photons and entangled quantum states. Quantum mechanics is ubiquitous in modern research. It governs the formation of microfractures in materials, the conversion of light into chemical energy in chlorophyll or into electric impulses in our eyes, and the creation of particles at the Large Hadron Collider. Technical mastery of the subject is of utmost importance for understanding quantum mechanics. Trying to decipher or apply quantum mechanics without knowing how it really works in the calculation of wave functions, energy levels, and cross sections is just idle talk and always prone to misconceptions. Therefore, we will go through a great many technicalities and calculations, because you and I (and your professor!) have a common goal: You should become an expert in quantum mechanics. However, there is also another message in this book. The apparently exotic world of quantum mechanics is our world. Our bodies and all the world around us are built on quantum effects and ruled by quantum mechanics. It is not apparent and only visible to the cognoscenti. Therefore, we have developed a mode of thought

xv

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To the Students

and explanation of the world that is based on classical pictures—mostly waves and particles in mechanical interaction. This mode of thought was amended by the notions of gravitational and electromagnetic forces, thus culminating in a powerful tool called classical physics. However, by 1900 those who were paying attention had caught enough glimpses of the underlying non-classical world to embark on the exciting journey of discovering quantum mechanics. The discoveries of the early quantum scientists paved the way for many surprising revelations and insights. For example, every single atom in your body is ruled by the laws of quantum mechanics and could not even exist as a classical particle. The electrons that provide the light for your long nights of studying generate this light in stochastic quantum jumps from a state of a single electron to a state of an electron and a photon. And maybe the most striking example of all: There is absolutely nothing classical in the sunlight that provides the energy for all life on Earth. Indeed, the shape of the continuous solar spectrum is determined by a single quantum effect, viz. the parsing of light into photons. Furthermore, the nuclear reactions which produce those photons are entirely ruled by quantum mechanics. And after billions of years, when our sun has exhausted its supply of nuclear fuel, its burnt-out core will be stabilized against gravitational collapse again by a single quantum effect, viz. the fact that no two electrons can exist in the same quantum state. Quantum mechanics stabilizes both the smallest structures that we know, including atoms and atomic nuclei, and the largest structures that we know, including neutron stars, white dwarfs, and hydrogen-burning main sequence stars. Quantum theory is not a young theory any more. The scientific foundations of the subject were developed over half a century between 1900 and 1949, and many of the mathematical foundations were even developed in the nineteenth century. The steepest ascent in the development of quantum theory appeared between 1924 and 1928, when matrix mechanics, Schrödinger’s equation, the Dirac equation, and field quantization were invented. I have included numerous references to original papers from this period, not to ask you to read all those papers—after all, the primary purpose of a textbook is to put major achievements into context, provide an introductory overview at an appropriate level, and replace often indirect and circuitous original derivations with simpler explanations—but to honor the people who brought the then-nascent theory to maturity. Quantum theory is an extremely well-established and developed theory now, which has proven itself on numerous occasions. However, we still continue to improve our collective understanding of the theory and its wide-ranging applications, and we test its predictions and its probabilistic interpretation with ever-increasing accuracy. The implications and applications of quantum mechanics are limitless, and we are witnessing a time when many technologies have reached their “quantum limit,” which is a misnomer for the fact that any methods of classical physics are just useless in trying to describe or predict the behavior of atomic scale devices. It is a “limit” for those who do not want to learn quantum physics. For you, it holds the promise of excitement and opportunity if you are prepared to work hard and if you can understand the calculations.

To the Students

xvii

Quantum mechanics combines power and beauty in a way that even supersedes advanced analytical mechanics and electrodynamics. Quantum mechanics is universal and therefore incredibly versatile, and if you have a sense for mathematical beauty: The structure of quantum mechanics is breathtaking, indeed. I sincerely hope that reading this book will be an enjoyable and exciting experience for you.

To the Instructor

Dear Colleague, As professors of quantum mechanics courses, we enjoy the privilege of teaching one of the most exciting subjects in the world. However, we often have to do this with fewer lecture hours than were available for the subject in the past, when at the same time we should include more material to prepare students for research or modern applications of quantum mechanics. Furthermore, students have become more mobile between universities (which is good) and between academic programs (which can have positive and negative implications). Therefore, we are facing the task to teach an advanced subject to an increasingly heterogeneous student body with very different levels of preparation. Nowadays, the audience in a fourth-year undergraduate or beginning graduate course often includes students who have not gone through a course on Lagrangian mechanics or have not seen the covariant formulation of electrodynamics in their electromagnetism courses. I deal with this problem by including one special lecture on each topic in my quantum mechanics course, and this is what Appendices A and B are for. I have also tried to be as inclusive as possible without sacrificing content or level of understanding by starting at a level that would correspond to an advanced second-year modern physics or quantum chemistry course and then follow a steeply ascending route that takes the students all the way from Planck’s law to the photon scattering tensor. The selection and arrangement of topics in this book are determined by the desire to develop an advanced undergraduate and introductory graduate-level course that is useful to as many students as possible, in the sense of giving them a head start into major current research areas or modern applications of quantum mechanics without neglecting the necessary foundational training. There is a core of knowledge that every student is expected to know by heart after having taken a course in quantum mechanics. Students must know the Schrödinger equation. They must know how to solve the harmonic oscillator and the Coulomb problem, and they must know how to extract information from the wave function. They should also be able to apply basic perturbation theory, and they should

xix

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To the Instructor

understand that a wave function x|ψ(t) is only one particular representation of a quantum state |ψ(t). In a North American physics program, students would traditionally learn all these subjects in a 300-level Quantum Mechanics course. Here these subjects are discussed in Chaps. 1–7 and 9. This allows the instructor to use this book also in 300-level courses or introduce those chapters in a 400-level or graduate course if needed. Depending on their specialization, there will be an increasing number of students from many different science and engineering programs who will have to learn these subjects at M.Sc. or beginning Ph.D. level before they can learn about photon scattering or quantum effects in materials, and catering to these students will also become an increasingly important part of the mandate of physics departments. Including Chaps. 1–7 and 9 with the book is part of the philosophy of being as inclusive as possible to disseminate knowledge in advanced quantum mechanics as widely as possible. Additional training in quantum mechanics in the past traditionally focused on atomic and nuclear physics applications, and these are still very important topics in fundamental and applied science. However, a vast number of our current students in quantum mechanics will apply the subject in materials science in a broad sense encompassing condensed matter physics, chemistry, and engineering. For these students, it is beneficial to see Bloch’s theorem, Wannier states, and basics of the theory of covalent bonding embedded with their quantum mechanics course. Another important topic for these students is quantization of the Schrödinger field. Indeed, it is also useful for students in nuclear and particle physics to learn quantization of the Schrödinger field because it makes quantization of gauge fields and relativistic matter fields so much easier if they know quantum field theory in the non-relativistic setting. Furthermore, many of our current students will use or manipulate photon probes in their future graduate and professional work. A proper discussion of photon– matter interactions is therefore also important for a modern quantum mechanics course. This should include minimal coupling, quantization of the Maxwell field, and applications of time-dependent perturbation theory for photon absorption, emission, and scattering. Students should also know the Klein–Gordon and Dirac equations after completion of their course, not only to understand that Schrödinger’s equation is not the final answer in terms of wave equations for matter particles, but also to understand the nature of relativistic corrections like the Pauli term or spin–orbit coupling. The scattering matrix is introduced as early as possible in terms of matrix elements of the time evolution operator on states in the interaction picture, Sf i (t, t  ) = f |UD (t, t  )|i, cf. Eq. (13.59). This representation of the scattering matrix appears so naturally in ordinary time-dependent perturbation theory that it makes no sense to defer the notion of an S-matrix to the discussion of scattering in quantum field theory with two or more particles in the initial state. It actually mystifies the scattering matrix to defer its discussion until field quantization has been introduced. Conversely, introducing the scattering matrix even earlier in the framework of scattering off static potentials is counterproductive, because its natural

To the Instructor

xxi

and useful definition as matrix elements of a time evolution operator cannot properly be introduced at that level, and the notion of the scattering matrix does not really help with the calculation of cross sections for scattering off static potentials. I have also emphasized the discussion of the various roles of transition matrix elements depending on whether the initial or final states are discrete or continuous. It helps students to understand transition probabilities, decay rates, absorption cross sections, and scattering cross sections if the discussion of these concepts is integrated into one chapter, cf. Chap. 13. Furthermore, I have put an emphasis on canonical field quantization. Path integrals provide a very elegant description for free–free scattering, but bound states and energy levels and basic many-particle quantum phenomena like exchange holes are very efficiently described in the canonical formalism. Feynman rules also appear more intuitive in the canonical formalism of explicit particle creation and annihilation. The core advanced topics in quantum mechanics that an instructor might want to cover in a traditional 400-level or introductory graduate course are included with Chaps. 8, 11–13, 15–18, and 22. However, instructors of a more inclusive course for general science and engineering students should include materials from Chaps. 1–7 and 9, as appropriate. The direct integration of training in quantum mechanics with the foundations of condensed matter physics, field quantization, and quantum optics is very important for the advancement of science and technology. I hope that this book will help to achieve that goal. I would greatly appreciate your comments and criticism. Please send them to [email protected].

Chapter 1

The Need for Quantum Mechanics

Quantum mechanics was initially invented because classical mechanics, thermodynamics and electrodynamics provided no means to explain the properties of atoms, electrons, and electromagnetic radiation. Furthermore, it became clear after the introduction of Schrödinger’s equation and the quantization of Maxwell’s equations that we cannot explain any physical property of matter and radiation without the use of quantum theory. We will see a lot of evidence for this in the following chapters. However, in the present chapter we will briefly and selectively review the early experimental observations and discoveries which led to the development of quantum mechanics over a period of intense research between 1900 and 1928.

1.1 Electromagnetic Spectra and Evidence for Discrete Energy Levels The first evidence that classical physics was incomplete appeared in unexpected properties of electromagnetic spectra. Thin gases of atoms or molecules emit line spectra which contradict the fact that a classical system of electric charges can oscillate at any frequency, and therefore can emit radiation of any frequency. This was a major scientific puzzle from the 1850s until the inception of the Schrödinger equation in 1926. Contrary to a thin gas, a hot body does emit a continuous spectrum, but even those spectra were still puzzling because the shape of heat radiation spectra could not be explained by classical thermodynamics and electrodynamics. In fact, classical physics provided no means at all to predict any sensible shape for the spectrum of a heat source! But at last, hot bodies do emit a continuous spectrum and therefore, from a classical point of view, their spectra are not quite as strange and unexpected as line spectra. It is therefore not surprising that the first real clues for a solution © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 R. Dick, Advanced Quantum Mechanics, Graduate Texts in Physics, https://doi.org/10.1007/978-3-030-57870-1_1

1

2

1 The Need for Quantum Mechanics

to the puzzles of electromagnetic spectra emerged when Max Planck figured out a way to calculate the spectra of heat sources under the simple, but classically extremely counterintuitive assumption that the energy in heat radiation of frequency f is quantized in integer multiples of a minimal energy quantum hf , E = nhf,

n ∈ N.

(1.1)

The constant h that Planck had introduced to formulate this equation became known as Planck’s constant and it could be measured from the shape of heat radiation spectra. A modern value is h = 6.626 × 10−34 J · s = 4.136 × 10−15 eV · s. We will review the puzzle of heat radiation and Planck’s solution in the next section, because Planck’s calculation is instructive and important for the understanding of incandescent light sources and it illustrates in a simple way how quantization of energy levels yields results which are radically different from predictions of classical physics. Albert Einstein then pointed out that Eq. (1.1) also explains the photoelectric effect. He also proposed that Planck’s quantization condition is not a property of any particular mechanism for generation of electromagnetic waves, but an intrinsic property of electromagnetic waves. However, once Eq. (1.1) is accepted as an intrinsic property of electromagnetic waves, it is a small step to make the connection with line spectra of atoms and molecules and conclude that these line spectra imply existence of discrete energy levels in atoms and molecules. Somehow atoms and molecules seem to be able to emit radiation only by jumping from one discrete energy state into a lower discrete energy state. This line of reasoning, combined with classical dynamics between electrons and nuclei in atoms then naturally leads to the Bohr-Sommerfeld theory of atomic structure. This became known as old quantum theory. Apparently, the property which underlies both the heat radiation puzzle and the puzzle of line spectra is discreteness of energy levels in atoms, molecules, and electromagnetic radiation. Therefore, one major motivation for the development of quantum mechanics was to explain discrete energy levels in atoms, molecules, and electromagnetic radiation. It was Schrödinger’s merit to find an explanation for the discreteness of energy levels in atoms and molecules through his wave equation [151] (h¯ ≡ h/2π ) ih¯

∂ h¯ 2 ψ(x, t) = − ψ(x, t) + V (x)ψ(x, t). ∂t 2m

(1.2)

A large part of this book will be dedicated to the discussion of Schrödinger’s equation. An intuitive motivation for this equation will be given in Sect. 1.6. Ironically, the fundamental energy quantization condition (1.1) for electromagnetic waves, which precedes the realization of discrete energy levels in atoms and molecules, cannot be derived by solving a wave equation, but emerges from the quantization of Maxwell’s equations. This is at the heart of understanding photons and the quantum theory of electromagnetic waves. We will revisit this issue in

1.2 Blackbody Radiation and Planck’s Law

3

Chap. 18. However, we can and will discuss already now the early quantum theory of the photon and what it means for the interpretation of spectra from incandescent sources.

1.2 Blackbody Radiation and Planck’s Law Historically, Planck’s deciphering of the spectra of incandescent heat and light sources played a key role for the development of quantum mechanics, because it included the first proposal of energy quanta, and it implied that line spectra are a manifestation of energy quantization in atoms and molecules. Planck’s radiation law is also extremely important in astrophysics and in the technology of heat and light sources. Generically, the heat radiation from an incandescent source is contaminated with radiation reflected from the source. Pure heat radiation can therefore only be observed from a non-reflecting, i.e. perfectly black body. Hence the name blackbody radiation for pure heat radiation. Physicists in the late nineteenth century recognized that the best experimental realization of a black body is a hole in a cavity wall. If the cavity is kept at temperature T , the hole will emit perfect heat radiation without contamination from any reflected radiation. Suppose we have a heat radiation source (or thermal emitter) at temperature T . The power per area radiated from a thermal emitter at temperature T is denoted as its exitance (or emittance) e(T ). In the blackbody experiments e(T ) · A is the energy per time leaking through a hole of area A in a cavity wall. To calculate e(T ) as a function of the temperature T , as a first step we need to find out how it is related to the density u(T ) of energy stored in the heat radiation. One half of the radiation will have a velocity component towards the hole, because all the radiation which moves under an angle ϑ ≤ π/2 relative to the axis going through the hole will have a velocity component v(ϑ) = c cos ϑ in the direction of the hole. To find out the average speed v of the radiation in the direction of the hole, we have to average c cos ϑ over the solid angle  = 2π sr of the forward direction 0 ≤ ϕ ≤ 2π , 0 ≤ ϑ ≤ π/2: v=

c 2π





2π

π/2

dϕ 0

dϑ sin ϑ cos ϑ =

0

c . 2

(1.3)

The effective energy current density towards the hole is energy density moving in forward direction × average speed in forward direction: u(T ) c c = u(T ) , 2 2 4 and during the time t an amount of energy

(1.4)

4

1 The Need for Quantum Mechanics

c E = u(T ) tA 4

(1.5)

will escape through the hole. Therefore the emitted power per area E/(tA) = e(T ) is c e(T ) = u(T ) . 4

(1.6)

However, Planck’s radiation law is concerned with the spectral exitance e(f, T ), which is defined in such a way that  e[f1 ,f2 ] (T ) =

f2

df e(f, T )

(1.7)

f1

is the power per area emitted in radiation with frequencies f1 ≤ f ≤ f2 . In particular, the total exitance is  e(T ) = e[0,∞] (T ) =

∞

df e(f, T ).

(1.8)

0

Operationally, the spectral exitance is the power per area emitted with frequencies f ≤ f  ≤ f + f , and normalized by the width f of the frequency interval, e(f, T ) = lim

f →0

e[f,f +f ] (T ) e[0,f +f ] − e[0,f ] (T ) ∂ = lim = e[0,f ] (T ). f →0 f f ∂f

The spectral exitance e(f, T ) can also be denoted as the emitted power per area and per unit of frequency or as the spectral exitance in the frequency scale. The spectral energy density u(f, T ) is defined in the same way. If we measure the energy density u[f,f +f ] (T ) in radiation with frequency between f and f + f , then the energy per volume and per unit of frequency (i.e. the spectral energy density in the frequency scale) is u[f,f +f ] (T ) ∂ = u[0,f ] (T ), f →0 f ∂f

u(f, T ) = lim

(1.9)

and the total energy density in radiation is  u(T ) =

∞

df u(f, T ).

(1.10)

0

The equation e(T ) = u(T )c/4 also applies separately in each frequency interval [f, f + f ], and therefore must also hold for the corresponding spectral densities, c e(f, T ) = u(f, T ) . 4

(1.11)

1.2 Blackbody Radiation and Planck’s Law

5

The following facts were known before Planck’s work in 1900. • The prediction from classical thermodynamics for the spectral exitance e(f, T ) (Rayleigh-Jeans law) was wrong, and actually non-sensible! • The exitance e(T ) satisfies Stefan’s law1 (Stefan, 1879; Boltzmann, 1884) e(T ) = σ T 4 ,

(1.12)

with the Stefan–Boltzmann constant σ = 5.6704 × 10−8   • The spectral exitance e(λ, T ) = e(f, T )

f =c/λ

W . m2 K4

(1.13)

· c/λ2 per unit of wavelength (i.e.

the spectral exitance in the wavelength scale) has a maximum at a wavelength λmax · T = 2.898 × 10−3 m · K = 2898 μm · K.

(1.14)

This is Wien’s displacement law (Wien, 1893). The puzzle was to explain the observed curves e(f, T ) and to explain why classical thermodynamics had failed. We will explore these questions through a calculation of the spectral energy density u(f, T ). Equation (1.11) then also yields e(f, T ). The key observation for the calculation of u(f, T ) is to realize that u(f, T ) can be split into two factors. If we want to know the radiation energy density u[f,f +df ] = u(f, T )df in the small frequency interval [f, f +df ], then we can first ask ourselves how many different electromagnetic oscillation modes per volume, (f )df , exist in that frequency interval. Each oscillation mode will then contribute an energy E(f, T ) to the radiation energy density, where E(f, T ) is the expectation value of energy in an electromagnetic oscillation mode of frequency f at temperature T , u(f, T )df = (f )df E(f, T ).

(1.15)

The spectal energy density u(f, T ) can therefore be calculated in two steps: 1. Calculate the number (f ) of oscillation modes per volume and per unit of frequency (“counting of oscillation modes”). 2. Calculate the mean energy E(f, T ) in an oscillation of frequency f at temperature T . The results can then be combined to yield the spectral energy density u(f, T ) = (f )E(f, T ).

1 References

are enclosed in square brackets, e.g. [1]. For historical context, I have also included parenthetical remarks with the names of scientists and years referring to events preceding the development of quantum mechanics and for the emergence of particular applications.

6

1 The Need for Quantum Mechanics

The number of electromagnetic oscillation modes per volume and per unit of frequency is an important quantity in quantum mechanics and will be calculated explicitly in Chap. 12, with the result 8πf 2 . c3

(f ) =

The corresponding density of oscillation modes in the wavelength scale is  c 8π  · 2 = 4. (λ) = (f ) f =c/λ λ λ

(1.16)

(1.17)

Statistical physics predicts that the probability PT (E) to find an oscillation of energy E in a system at temperature T should be exponentially suppressed,   E 1 exp − . PT (E) = kB T kB T

(1.18)

The possible values of E are not restricted in classical physics, but can vary continuously between 0 ≤ E < ∞. For example, for any classical oscillation with fixed frequency f , continuously increasing the amplitude yields a continuous increase in energy. The mean energy of an oscillation at temperature T according to classical thermodynamics is therefore   E

 classical

∞

=



∞

dE EPT (E) =

0

0

  E E exp − = kB T . dE kB T kB T

Therefore the spectral energy density in blackbody radiation and the corresponding spectral exitance according to classical thermodynamics should be 2πf 2 c = kB T , 4 c2 (1.19) but this is obviously nonsensical: it would predict that every heat source should emit a diverging amount of energy at high frequencies/short wavelengths! This is the ultraviolet catastrophe of the Rayleigh-Jeans law. Max Planck observed in 1900 that he could derive an equation which matches the spectra of heat sources perfectly if he assumes that the energy in electromagnetic waves of frequency f is quantized in multiples of the frequency, u(f, t) = (f )kB T =

8πf 2 kB T , c3

E = nhf = n

e(f, T ) = u(f, T )

hc , λ

n ∈ N.

(1.20)

The exponential suppression of high energy oscillations then reads   nhf PT (E) = PT (n) ∝ exp − , kB T

(1.21)

1.2 Blackbody Radiation and Planck’s Law

7

but due to the discreteness of the energy quanta hf , the normalized probabilities are now      nhf hf exp − PT (E) = PT (n) = 1 − exp − kB T kB T     hf hf − exp −(n + 1) , (1.22) = exp −n kB T kB T  such that ∞ n=0 PT (n) = 1. The resulting mean energy per oscillation mode is E =

∞

nhf PT (n)

n=0

=

∞ n=0

=

∞ n=0

    ∞ hf hf − nhf exp −n nhf exp −(n + 1) kB T kB T n=0

    ∞ hf hf − nhf exp −n (n + 1)hf exp −(n + 1) kB T kB T

+ hf

n=0

∞ n=0

  hf . exp −(n + 1) kB T

(1.23)

The first two sums cancel, and the last term yields the mean energy in an electromagnetic wave of frequency f at temperature T as

 exp − khf T hf B = 



E(f, T ) = hf . hf hf 1 − exp − kB T exp kB T − 1

(1.24)

Combination with (f ) from Eq. (1.16) yields Planck’s formulas for the spectral energy density and spectral exitance in heat radiation, u(f, T ) =

8π hf 3 1 

, 3 hf c exp kB T − 1

e(f, T ) =

2π hf 3 1 

. 2 hf c exp kB T − 1

(1.25)

These functions fitted the observed spectra perfectly! The spectrum e(f, T ) and the emitted power e[0,f ] (T ) with maximal frequency f are displayed for T = 5780 K in Figs. 1.1 and 1.2.

8

1 The Need for Quantum Mechanics

Fig. 1.1 The spectral emittance e(f, T ) for a heat source of temperature T = 5780 K

f Fig. 1.2 The emittance e[0,f ] (T ) = 0 df  e(f  , T ) (i.e. emitted power per area in radiation with maximal frequency f ) for a heat source of temperature T = 5780 K. The asymptote for f → ∞ is e[0,∞] (T ) ≡ e(T ) = σ T 4 = 6.33 × 107 W/m2 for the temperature T = 5780 K

1.3 Blackbody Spectra and Photon Fluxes

9

1.3 Blackbody Spectra and Photon Fluxes Their technical relevance for the quantitative analysis of incandescent light sources makes it worthwhile to take a closer look at blackbody spectra. Blackbody spectra are also helpful to elucidate the notion of spectra more closely, and to explain that a maximum in a spectrum strongly depends on the choice of independent variable (e.g. wavelength or frequency) and dependent variable (e.g. energy flux or photon flux). In particular, it is sometimes claimed that our sun has maximal radiation output at a wavelength λmax  500 nm. This statement is actually very misleading if the notion of “radiation output” is not clearly defined, and if no explanation is included that different perfectly suitable notions of radiation output yield very different wavelengths or frequencies of maximal emission. We will see below that the statement above only applies to maximal power output per unit of wavelength, i.e. if we use a monochromator which slices the wavelength axis into intervals of equal length dλ = c|df |/f 2 , then we find maximal power output in an interval around λmax  500 nm. However, we will also see that if we use a monochromator which slices the frequency axis into intervals of equal length df = c|dλ|/λ2 , then we find maximal power output in an interval around fmax  340 THz, corresponding to a wavelength c/fmax  880 nm. If we ask for maximal photon counts instead of maximal power output, we find yet other values for peaks in the spectra. Since Planck’s radiation law (1.25) yields perfect matches to blackbody spectra, it must also imply Stefan’s law and Wien’s law. Stefan’s law is readily derived in the following way. The emitted power per area is 

∞

e(T ) =



∞

df e(f, T ) =

0

dλ e(λ, T ) = 2π

0

kB4 T 4 h3 c 2



∞

dx 0

x3 . exp(x) − 1

Evaluation of the integral 

∞

dx 0

x3 = exp(x) − 1



∞

dx x 3

0

∞

exp[−(n + 1)x]

n=0

 ∞ ∞ ∞ d3 d3 1 =− dx exp(−nx) = − dn3 0 dn3 n n=1

=

∞ n=1

n=1

π4

6 = 6ζ (4) = 15 n4

(1.26)

implies e(T ) =

2π 5 kB4 4 T , 15h3 c2

(1.27)

10

1 The Need for Quantum Mechanics

i.e. Planck’s law implied a prediction for the Stefan–Boltzmann constant in terms of the Planck constant h, which could be determined previously from a fit to the spectra, σ =

2π 5 kB4 . 15h3 c2

(1.28)

An energy flux e(T ) = 6.33 × 107 W/m2 from the Sun yields a remnant energy flux at Earth’s orbit of magnitude e(T ) × (R /r⊕ )2 = 1.37 kW/m2 . Here R = 6.955 × 108 m is the radius of the Sun and r⊕ = 1.496 × 1011 m is the radius of Earth’s orbit. For the derivation of Wien’s law, we set x=

hf hc = . λkB T kB T

(1.29)

Then we have with e(λ, T ) = e(f, T )|f =c/λ c/λ2 , 

⎛ ⎞ exp λkhc hc ∂ 5 2π hc2 1 BT ⎝  



− ⎠ e(λ, T ) = ∂λ λ5 exp hc − 1 λ2 kB T exp hc − 1 λ λkB T λkB T   exp(x) 1 2π hc2 x −5 , = exp(x) − 1 λ6 exp(x) − 1

(1.30)

which implies that ∂e(λ, T )/∂λ = 0 is satisfied if and only if exp(x) =

5 . 5−x

(1.31)

This condition yields x  4.965. The wavelength of maximal spectral emittance e(λ, T ) therefore satisfies λmax · T 

hc = 2898 μm · K. 4.965kB

(1.32)

For a heat source of temperature T = 5780 K, like the surface of our sun, this yields λmax = 501 nm,

c λmax

= 598 THz,

(1.33)

see Fig. 1.3. One can also derive an analogue of Wien’s law for the frequency fmax of maximal spectral emittance e(f, T ). We have

1.3 Blackbody Spectra and Photon Fluxes

11

Fig. 1.3 The spectral emittance e(λ, T ) for a heat source of temperature T = 5780 K

 ⎞

⎛ exp λkhc T 2π hf 2 ∂ 1 hf B ⎝3 − ⎠  



e(f, T ) = ∂f kB T exp hc − 1 c2 exp hf − 1 kB T λkB T =

  exp(x) 1 2π hf 2 3 − x , exp(x) − 1 c2 exp(x) − 1

(1.34)

which implies that ∂e(f, T )/∂f = 0 is satisfied if and only if exp(x) =

3 , 3−x

(1.35)

with solution x  2.821. The frequency of maximal spectral emittance e(f, T ) therefore satisfies fmax kB GHz  2.821 = 58.79 . T h K

(1.36)

This yields for a heat source of temperature T = 5780 K, as in Fig. 1.1, fmax = 340 THz,

c = 882 nm. fmax

(1.37)

The photon fluxes in the wavelength scale and in the frequency scale, j (λ, T ) and j (f, T ), are defined below. The spectral emittance per unit of frequency, e(f, T ),

12

1 The Need for Quantum Mechanics

is directly related to the photon flux per fractional wavelength or frequency interval d ln f = df/f = −d ln λ = −dλ/λ. We have with the notations used in (1.9) for spectral densities and integrated fluxes the relations e(f, T ) = hfj (f, T ) = hf

∂ ∂ j[0,f ] (T ) = h j[0,f ] (T ) ∂f ∂ ln(f/f0 )

= hj (ln(f/f0 ), T ) = hλj (λ, T ) = hj (ln(λ/λ0 ), T ).

(1.38)

Optimization of the energy flux of a light source for given frequency bandwidth df is therefore equivalent to optimization of photon flux for fixed fractional bandwidth df/f = |dλ/λ|. The number of photons per area, per second, and per unit of wavelength emitted from a heat source of temperature T is j (λ, T ) =

2π c λ 1 

. e(λ, T ) = 4 hc λ exp hc − 1 λkB T

(1.39)

This satisfies   ∂ j (λ, T ) exp(x) j (λ, T ) = x −4 =0 ∂λ λ exp(x) − 1

(1.40)

if exp(x) =

4 . 4−x

(1.41)

This has the solution x  3.921. The wavelength of maximal spectral photon flux j (λ, T ) therefore satisfies λmax · T 

hc = 3670 μm · K. 3.921kB

(1.42)

This yields for a heat source of temperature T = 5780 K λmax = 635 nm,

c = 472 THz, λmax

(1.43)

see Fig. 1.4. The photon flux in the wavelength scale, j (λ, T ), is also related to the energy fluxes per fractional wavelength or frequency interval d ln λ = dλ/λ = −d ln f = −df/f , j (λ, T ) =

λ 1 1 f e(λ, T ) = e(ln(λ/λ0 ), T ) = e(f, T ) = e(ln(f/f0 ), T ). hc hc hc hc

1.3 Blackbody Spectra and Photon Fluxes

13

Fig. 1.4 The spectral photon flux j (λ, T ) for a heat source of temperature T = 5780 K

Therefore optimization of photon flux for fixed wavelength bandwidth dλ is equivalent to optimization of energy flux for fixed fractional bandwidth dλ/λ = |df/f |. Finally, the number of photons per area, per second, and per unit of frequency emitted from a heat source of temperature T is j (f, T ) =

2πf 2 1 e(f, T ) 

. = 2 hf hf c exp kB T − 1

(1.44)

This satisfies   ∂ j (f, T ) exp(x) j (f, T ) = 2−x =0 ∂f f exp(x) − 1

(1.45)

if exp(x) =

2 . 2−x

(1.46)

This condition is solved by x  1.594. Therefore the frequency of maximal spectral photon flux j (f, T ) in the frequency scale satisfies kB GHz fmax  1.594 = 33.21 . T h K

(1.47)

14

1 The Need for Quantum Mechanics

Fig. 1.5 The spectral photon flux j (f, T ) for a heat source of temperature T = 5780 K

This yields for a heat source of temperature T = 5780 K fmax = 192 THz,

c fmax

= 1.56 μm,

(1.48)

see Fig. 1.5. The flux of emitted photons is 

∞

j (T ) =

df j (f, T ) = 2π

0

kB3 T 3 h3 c 2



∞

dx 0

x2 . exp(x) − 1

(1.49)

Evaluation of the integral 

∞

dx 0

x2 = exp(x) − 1 =



∞

0

dx x 2

∞

exp[−(n + 1)x]

n=0

 ∞ ∞ ∞ d2 d2 1 dx exp(−nx) = dn2 0 dn2 n n=1

=

∞ 2 = 2ζ (3) n3 n=1

n=1

(1.50)

1.5 Wave-Particle Duality

15

yields j (T ) =

4π ζ (3)kB3 3 T3 15 T = 1.5205 × 10 . h3 c 2 m2 · s · K3

(1.51)

A surface temperature T = 5780 K for our sun yields a photon flux at the solar surface 2.94 × 1026 m−2 s−1 and a resulting photon flux at Earth’s orbit of 6.35 × 1021 m−2 s−1 . The average photon energy e(T )/j (T ) = 1.35 eV is in the infrared.

1.4 The Photoelectric Effect The notion of energy quanta in radiation was so revolutionary in 1900 that Planck himself speculated that this must somehow be related to the emission mechanism of radiation from the material of the source. In 1905 Albert Einstein pointed out that hitherto unexplained properties of the photoelectric effect can also be explained through energy quanta hf in ultraviolet light, and proposed that this energy quantization is likely an intrinsic property of electromagnetic waves irrespective of how they are generated. In short, the photoelectric effect observations by J.J. Thomson and Lenard revealed the following key properties: • An ultraviolet light source of frequency f will generate photoelectrons of maximal kinetic energy hf − hf0 if f > f0 , where hf0 = φ is the minimal energy to liberate photoelectrons from the photocathode. • Increasing the intensity of the incident ultraviolet light without changing its frequency will increase the photocurrent, but not change the maximal kinetic energy of the photoelectrons. Increasing the intensity must therefore liberate more photoelectrons from the photocathode, but does not impart more energy on single electrons. Einstein realized that this behavior can be explained if the incident ultraviolet light of frequency f comes in energy parcels of magnitude hf , and if the electrons in the metal can (predominantly) only absorb a single of these energy parcels.

1.5 Wave-Particle Duality When X-rays of wavelength λ0 are scattered off atoms, one observes scattered X-rays of the same wavelength λ0 in all directions. However, in the years 1921– 1923 Arthur H. Compton observed that under every scattering angle ϑ against the direction of incidence, there is also a component of scattered X-rays with a longer wavelength λ = λ0 + λC (1 − cos ϑ).

(1.52)

16

1 The Need for Quantum Mechanics

The constant λC = 2.426 pm has the same value for every atom. Compton (and also Debye) recognized that this longer wavelength component in the scattered radiation can be explained as a consequence of particle like collisions of Planck’s and Einstein’s energy parcels hf with weakly bound electrons if the energy parcels also carry momentum h/λ. Energy conservation during the collision of the electromagnetics energy parcels (meanwhile called photons) with weakly bound electrons (pe is the momentum of the recoiling electron), me c +

h = λ0



pe2 + m2e c2 +

h , λ

(1.53)

yields pe2

  h2 h2 h2 1 1 , = 2 + 2 −2 + 2me hc − λλ0 λ0 λ λ λ0

(1.54)

while momentum conservation implies pe2 =

h2 h2 h2 + 2 −2 cos ϑ. 2 λλ0 λ λ0

(1.55)

This yields for the wavelength of the scattered photon λ = λ0 +

h (1 − cos ϑ), me c

(1.56)

with excellent numerical agreement between h/me c and the measured value of λC . λC = h/me c is therefore known as the Compton wavelength. From the experimental findings on blackbody radiation, the photoelectric effect, and Compton scattering, and the ideas of Planck, Einstein, and Compton, an electromagnetic wave of frequency f = c/λ appears like a current of particles with energy hf and momentum h/λ. However, electromagnetic waves also show wavelike properties like diffraction and interference. The findings of Planck, Einstein, and Compton combined with the wavelike properties of electromagnetic waves (observed for the first time by Heinrich Hertz) constitute the first observation of wave-particle duality. Depending on the experimental setup, a physical system can sometimes behave like a wave and sometimes behave like a particle. However, the puzzle did not end there. Louis de Broglie recognized in 1923 that the orbits of the old Bohr model could be explained through closed circular electron waves if the electrons are assigned a wavelength λ = h/p, like photons. Soon thereafter, wavelike behavior of electrons was observed by Clinton Davisson and Lester Germer in 1927, when they observed interference of non-relativistic electrons scattered off the surface of Nickel crystals. At the same time, George Thomson was sending high energy electron beams (with kinetic energies between 20 keV and 60 keV) through thin metal foils and observed interference of the transmitted

1.6 Why Schrödinger’s Equation?

17

electrons, thus also confirming the wave nature of electrons. We can therefore also conclude that another major motivation for the development of quantum mechanics was to explain wave-particle duality.

1.6 Why Schrödinger’s Equation? The foundations of quantum mechanics were developed between 1900 and 1950 by some of the greatest minds of the twentieth century, from Max Planck and Albert Einstein to Richard Feynman and Freeman Dyson. The inner circle of geniuses who brought the nascent theory to maturity were Heisenberg, Born, Jordan, Schrödinger, Pauli, Dirac, and Wigner. Among all the outstanding contributions of these scientists, Schrödinger’s invention of his wave equation (1.2) was likely the most important single step in the development of quantum mechanics. Understanding this step, albeit in a simplified pedagogical way, is important for learning and understanding quantum mechanics. Ultimately, basic equations in physics have to prove themselves in comparison with experiments, and the Schrödinger equation was extremely successful in that regard. However, this does not explain how to come up with such an equation. Basic equations in physics cannot be derived from any rigorous theoretical or mathematical framewok. There is no algorithm which could have told Newton how to come up with Newton’s equation, or would have told Schrödinger how to come up with his equation (or could tell us how to come up with a fundamental theory of quantum gravity). Basic equations in physics have to be invented in an act of creative ingenuity, which certainly requires a lot of brainstorming and diligent review of pertinent experimental facts and solutions of related problems (where known). It is much easier to accept an equation and start to explore its consequences if the equation makes intuitive sense—if we can start our discussion of Schrödinger’s equation with the premise “yes, the hypothesis that Schrödinger’s equation solves the problems of energy quantization and wave-particle duality seems intuitively promising and is worth pursuing”. Therefore I will point out how Schrödinger could have invented the Schrödinger equation (although his actual thought process was much more involved and was motivated by the connection of the quantization rules of old quantum mechanics with the Hamilton-Jacobi equation of classical mechanics [162]). The problem is to come up with an equation for the motion of particles, which explains both quantization of energy levels and wave-particle duality. As a starting point, we recall that the motion of a non-relativistic particle under the influence of a conservative force F (x) = −∇V (x) is classically described by Newton’s equation m

d 2 x(t) = − ∇V (x(t)), dt 2

(1.57)

18

1 The Need for Quantum Mechanics

and this equation also implies energy conservation, E=

p2 + V (x). 2m

(1.58)

However, this cannot be the whole story, because Davisson and Germer, and G.P. Thomson had shown that at least electrons sometimes also behave like waves with wavelength λ = h/p, as predicted by de Broglie. Furthermore, Compton has demonstrated that photons of energy E = hf satisfy the relation λ = h/p between wavelength and momentum. This motivates the hypothesis that a non-relativistic particle might also satisfy the relation E = hf . A monochromatic plane wave of frequency f , wavelength λ, and direction of motion kˆ can be described by a wave function    kˆ · x − ft . (1.59) ψ(x, t) = A exp 2π i λ Substitution of the relations λ=

h , p

E = hf =

p2 2m

(1.60)

yields with h¯ ≡ h/2π   p·x p2 ψ(x, t) = A exp i − t . h¯ 2mh¯

(1.61)

Under the supposition of wave-particle duality, we have to assume that this wave function must somehow be related to the wave properties of free particles as observed in the electron diffraction experiments. However, this wave function satisfies a differential equation ih¯

h¯ 2 p2 ∂ ψ(x, t) = Eψ(x, t) = ψ(x, t) = − ψ(x, t), ∂t 2m 2m

(1.62)

because under the assumption of wave-particle duality we had to replace f with E/ h in the exponent, and we used E = p2 /2m for a free particle. This does not yet tell us how to calculate the wave function which would describe motion of particles in a potential V (x). However, comparison of the differential equation (1.62) with the classical energy equation (1.58) can give us the idea to try ih¯

∂ h¯ 2 ψ(x, t) = − ψ(x, t) + V (x)ψ(x, t) ∂t 2m

(1.63)

1.7 Interpretation of Schrödinger’s Wave Function

19

as a starting point for the calculation of wave functions for particles moving in a potential V (x). Schrödinger actually found this equation after he had found the time-independent Schrödinger equation (3.4) below, and he had demonstrated that these equations yield the correct spectrum for hydrogen atoms, where V (x) = −

e2 . 4π 0 |x|

(1.64)

Schrödinger’s solution of the hydrogen atom will be discussed in Chap. 7.

1.7 Interpretation of Schrödinger’s Wave Function The Schrödinger equation was a spectacular success right from the start, but it was not immediately clear what the physical meaning of the complex wave function ψ(x, t) is. A natural first guess would be to assume that |ψ(x, t)|2 corresponds to a physical density of the particle described by the wave function ψ(x, t). In this interpretation, an electron in a quantum state ψ(x, t) would have a spatial mass density m |ψ(x, t)|2 and a charge density −e |ψ(x, t)|2 . This interpretation would imply that waves would have prevailed over particles in wave-particle duality. However, quantum jumps are difficult to reconcile with a physical density interpretation for |ψ(x, t)|2 , and Born, Bohr and Heisenberg developed a statistical interpretation of the wave function which is still the leading paradigm for quantum mechanics. Already in June 1926, the view began to emerge that the wave function ψ(x, t) should be interpreted as a probability density amplitude2 in the sense that

2 Schrödinger

[151, paragraph on pp. 134–135, sentences 2–4]: “ψψ is a kind of weight function in the configuration space of the system. The wave mechanical configuration of the system is a superposition of many, strictly speaking of all, kinematically possible point mechanical configurations. Thereby each point mechanical configuration contributes with a certain weight to the true wave mechanical configuration, where the weight is just given by ψψ.” Of course, a weakness of this early hint at the probability interpretation is the vague reference to a “true wave mechanical configuration”. A clearer formulation of this point was offered by Born, see the reference to Born’s work below. While there may have been early agreement on the importance of a probabilistic interpretation, the question of the concept which underlies those probabilities was a contentious point between Schrödinger, who at that time may have preferred to advance a de Broglie type pilot wave interpretation, and Bohr and Born and their particle-wave complementarity interpretation. Indeed, Schrödinger himself was intrigued by the possibility of the wave function describing continuous electronic oscillations in atoms without quantum jumps, see pp. 121 and 129–130 in loc. cit. and Schrödinger’s papers in the British Journal for the Philosophy of Science 3, 109 (1952); ibid. 233 (1952)). This as well as concerns about probabilistic interpretations of superpositions of states ultimately made him sceptic regarding the probabilistic interpretation of wave functions and the concept of elementary particles. Nevertheless, in the end the probabilistic complementarity picture prevailed: There are fundamental degrees of freedom with certain quantum numbers. These degrees of freedom are quantal excitations of the vacuum, and mathematically they are described by quantum fields. Depending on the way they are probed,

20

1 The Need for Quantum Mechanics

 d 3 x |ψ(x, t)|2

PV (t) =

(1.65)

V

is the probability to find a particle (or rather, an excitation of the vacuum with minimal energy mc2 and certain other quantum numbers) in the volume V at time t. This equation implies that |ψ(x, t)|2 is the probability density to find the particle in the location x at time t. The expectation value for the location of the particle at time t is then  x(t) = d 3 x x |ψ(x, t)|2 , (1.66) where integrals without explicit limits are taken over the full range of the integration variable, i.e. here over all of R3 . Many individual particle measurements will yield the location x with a frequency proportionally to |ψ(x, t)|2 , and averaging over the observations will yield the expectation value (1.66) with a variance e.g. for the x coordinate x 2 (t) = (x − x)2 (t) = x 2 (t) − x2 (t) 2   3 2 2 3 2 . = d x x |ψ(x, t)| − d x x |ψ(x, t)|

(1.67)

This interpretation of the relation between the wave function and particle properties was essentially proposed by Max Born in the paper where he invented quantum mechanical scattering theory [17]. The Schrödinger equation (1.2) implies a local conservation law for probability ∂ |ψ(x, t)|2 + ∇ · j (x, t) = 0 ∂t

(1.68)

with the probability current density j (x, t) =

 h¯ + ψ (x, t) · ∇ψ(x, t) − ∇ψ + (x, t) · ψ(x, t) . 2im

(1.69)

The conservation law (1.68) is important for consistency of the probability interpretation of Schrödinger theory. We assume that the integral  P (t) =

d 3 x |ψ(x, t)|2

(1.70)

they exhibit wavelike or corpuscular properties, and quantum states represent probability densities for the observation of physical properties of these excitations. Whether or not to denote these excitations as particles is a matter of convenience and tradition.

1.7 Interpretation of Schrödinger’s Wave Function

21

over R3 converges. A priori this should yield a time-dependent function P (t). However, Eq. (1.68) implies d P (t) = 0, dt

(1.71)

whence P √(t) ≡ P is a positive constant. This allows for rescaling ψ(x, t) → ψ(x, t)/ P such that the new wave function still satisfies equation (1.2) and yields a normalized integral  d 3 x |ψ(x, t)|2 = 1.

(1.72)

This means that the probability to find the particle anywhere at time t is 1, as it should be. Equations (1.65) and (1.66) make sense only in conjunction with the normalization condition (1.72) We can also substitute the Schrödinger equation or the local conservation law (1.68) into  d ∂ p(t) = m x(t) = m d 3 x x |ψ(x, t)|2 (1.73) dt ∂t to find  p(t) =

h¯ d 3 x ψ + (x, t) ∇ψ(x, t). i

(1.74)

Equations (1.66) and (1.74) tell us how to extract particle like properties from the wave function ψ(x, t). At first sight, Eq. (1.74) does not seem to make a lot of intuitive sense. Why should the momentum of a particle be related to the gradient of its wave function? However, recall the Compton-de Broglie relation p = h/λ. Wave packets which are composed of shorter wavelength components oscillate more rapidly as a function of x, and therefore have a larger average gradient. Equation (1.74) is therefore in agreement with a basic relation of wave-particle duality. A related argument in favor of Eq. (1.74) arises from substitution of the Fourier transforms3  1 ψ(x, t) = √ 3 d 3 k exp(ik · x)ψ(k, t), (1.75) 2π  1 + ψ (x, t) = √ 3 d 3 k exp(− ik · x)ψ + (k, t) (1.76) 2π

3 Fourier

transformation is reviewed in Sect. 2.1.

22

1 The Need for Quantum Mechanics

in Eqs. (1.72) and (1.74). This yields  d 3 k |ψ(k, t)|2 = 1

(1.77)

and  p(t) =

d 3 k h¯ k |ψ(k, t)|2 ,

(1.78)

in perfect agreement with the Compton-de Broglie relation p = h¯ k. Apparently |ψ(k, t)|2 is a probability density in k space in the sense that  PV˜ (t) =

V˜

d 3 k |ψ(k, t)|2

(1.79)

is the probability to find the particle with a wave vector k contained in a volume V˜ in k space. We can also identify an expression for the energy of a particle which is described by a wave function ψ(x, t). The Schrödinger equation (1.2) implies the conservation law    d h¯ 2 3 +  + V (x) ψ(x, t) = 0. (1.80) d x ψ (x, t) − dt 2m Here it plays a role that we assumed time-independent potential.4 In classical mechanics, the conservation law which appears for motion in a time-independent potential is energy conservation. Therefore, we expect that the expectation value for energy is given by  E =

 h¯ 2  + V (x) ψ(x, t). d x ψ (x, t) − 2m 

3

+

(1.81)

We will also rederive this at a more advanced level in Chap. 17. From the classical relation (1.58) between energy and momentum of a particle, we should also have E =

p2  + V (x). 2m

(1.82)

Comparison of Eqs. (1.74) and (1.81) yields

4 Examples

of the Schrödinger equation with time-dependent potentials will be discussed in Chap. 13 and following chapters.

1.8 Problems

23

 p2 (t) =

2 d 3 x ψ + (x, t)(− ih∇) ψ(x, t), ¯

(1.83)

such that calculation of expectation values of powers of momentum apparently amounts to corresponding powers of the differential operator − ih∇ ¯ acting on the wave function ψ(x, t). Maybe one of the most direct observational confirmations of the statistical interpretation of the wave function is the direct observation of the build-up of electron interference patterns [136]. In the early experiments [112, 170], electrons were passing through an electric version of a double slit with a time difference that makes it extremely unlikely that two electrons interfere during their passages through the slit. The double slit is realized through an electron biprism [117], i.e. two grounded parallel plates with a fine positively charged wire running parallel to the plates, but perpendicular to the incoming electrons. Behind the biprism the electrons are observed with a scintillation screen or a camera. Each individual electron is observed to generate only a single dot on the screen. This is the behavior expected from a pointlike particle which is not spread over a physical density distribution. The first few electrons seem to generate a random pattern of dots. However, when more and more electrons hit the screen, their dots generate a collective pattern which corresponds to a distribution |ψ(x, t)|2 for double slit interference. This indicates that |ψ(x, y, z0 , t)|2 is indeed the probability density for an electron to hit the point {x, y} on the screen which is located at z0 , but it is not the physical density of a spatially extended electron.5 The standard Born interpretation of the wave function was also confirmed in a recent three-slit experiment by proving that the interference patterns from many sequential single-particle paths agree with the probability density interpretation of |ψ(x, t)|2 for single slit diffraction, double-slit interference, and triple-slit interference [158].

1.8 Problems 1.1 Plot the emittance e[0,λ] (T ) of our sun. 1.2 Suppose that the resolution of a particular monochromator scales with 1/f , i.e. if the monochromator is set to a particular frequency f the product f df = df 2 /2 of frequency and bandwidth is constant. Furthermore, assume that the monochromator is coupled to a device which produces a signal proportional to the energy of the incident radiation. In the limit df → 0, is the signal curve from this apparatus proportional to e(f, T ), e(λ, T ), j (f, T ) or j (λ, T )?

5 It

has been argued that Bohmian mechanics can also explain the biprism experiments through a pilot wave interpretation of the wave function. However, Bohmian mechanics has other problems. We will briefly discuss Bohmian mechanics in Problem 7.21.

24

1 The Need for Quantum Mechanics

1.3 Suppose that the resolution of a particular monochromator scales with f , i.e. if the monochromator is set to a particlular frequency f the fractional bandwidth df/f is constant. The monochromator is coupled to a device which produces a signal proportional to the energy of the incident radiation. The device is used for observation of a Planck spectrum. For which relation between frequency and temperature does this device yield maximal signal? 1.4 Derive the probability conservation law (1.68) from the Schrödinger equation. Hint: Multiply the Schrödinger equation with ψ + (x, t) and use also the complex conjugate equation. 1.5 We will often deal with quantum mechanics in d spatial dimensions. There are many motivations to go beyond the standard case d = 3. E.g. d = 0 is the number of spatial dimensions for an idealized quantum dot, d = 1 is often used for pedagogical purposes and also for idealized quantum wires or nanowires, and d = 2 is used for physics on surfaces and interfaces. We consider a normalized wave function ψ(x, t) in d dimensions. What are the SI units of the wave function? What are the SI units of the d-dimensional current density j for the wave function ψ(x, t)? 1.6 Derive equation (1.74) from (1.73). 1.7 Show that the Schrödinger equation (1.63) implies the conservation laws d dt





h¯ 2  + V (x) d x ψ (x, t) − 2m 3

+

n ψ(x, t) = 0,

n ∈ N0 .

(1.84)

Two particular cases of this equation appeared in Sect. 1.7. Which are those cases and what are the related conserved quantities? Why is there usually not much interest in the infinitely many higher order conservation laws (1.84) for n > 1? Hint: Think about the classical interpretation of these conservation laws. Why do the higher order conservation laws nevertheless matter in quantum mechanics? Hint: Equation (1.84) is generically different from the “similar” conservation law d(En )/dt = 0. Is there an interesting implication of the two conservation laws for n = 2? 1.8 Equation (1.73) implies that the equation p(t) = mdx(t)/dt from nonrelativistic classical mechanics is realized as an equation between expectation values in non-relativistic quantum mechanics. Show that Newton’s law holds in the following sense in non-relativistic quantum mechanics (Ehrenfest’s theorem), d p(t) = − ∇V (x)(t). dt

(1.85)

Chapter 2

Self-Adjoint Operators and Eigenfunction Expansions

The relevance of waves in quantum mechanics naturally implies that the decomposition of arbitrary wave packets in terms of monochromatic waves, commonly known as Fourier decomposition after Jean-Baptiste Fourier’s Théorie analytique de la Chaleur (1822), plays an important role in applications of the theory. Dirac’s δ function, on the other hand, gained prominence primarily through its use in quantum mechanics, although today it is also commonly used in mechanics and electrodynamics to describe sudden impulses, mass points, or point charges. Both concepts are intimately connected to the completeness of eigenfunctions of self-adjoint operators. From the quantum mechanics perspective, the problem of completeness of sets of functions concerns the problem of enumeration of all possible states of a quantum system.

2.1 The δ Function and Fourier Transforms Let f (x) be a continuous function in the interval [a, b]. Dirichlet’s equation [28, 29]  lim

κ→∞ a

b

dx 

sin[κ(x − x  )] f (x  ) = π(x − x  )



0, x ∈ / [a, b], f (x), x ∈ (a, b),

(2.1)

motivates the formal definition sin(κx) 1 = lim δ(x) = lim κ→∞ κ→∞ πx 2π  ∞ 1 = dk exp(ikx), 2π −∞



κ

−κ

dk exp(ikx)

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 R. Dick, Advanced Quantum Mechanics, Graduate Texts in Physics, https://doi.org/10.1007/978-3-030-57870-1_2

(2.2)

25

26

2 Self-Adjoint Operators and Eigenfunction Expansions

such that Eq. (2.1) can (in)formally be written as 

b









dx δ(x − x )f (x ) =

a

0, x ∈ / [a, b], f (x), x ∈ (a, b).

(2.3)

A justification for Dirichlet’s equation is given below in the derivation of Eq. (2.19). The generalization to three dimensions follows immediately from Dirichlet’s formula in a three-dimensional cube, and exhaustion of an arbitrary three-dimensional volume V by increasingly finer cubes. This yields δ(x) =

3  i=1



sin(κi xi ) 1 lim = κi →∞ π xi (2π )3 

d 3 x  δ(x − x  )f (x  ) = V

 d 3 k exp(ik · x), 0, x ∈ / V, f (x), x inside V .

(2.4)

(2.5)

The case x ∈ ∂V (x on the boundary of V ) must be analyzed on a case-by-case basis. Equation (2.4) implies  ψ(x, t) = =

d 3 x  δ(x − x  )ψ(x  , t)

1 (2π )3



 d 3 k exp(ik · x)

d 3 x  exp(− ik · x  )ψ(x  , t).

(2.6)

This can be used to introduce Fourier transforms by splitting the previous equation into two equations, 1 ψ(x, t) = √ 3 2π

 d 3 k exp(ik · x)ψ(k, t),

(2.7)

d 3 x exp(− ik · x)ψ(x, t).

(2.8)

with 1 ψ(k, t) = √ 3 2π



Use of ψ(x, t) corresponds to the x-representation of quantum mechanics. Use of ψ(k, t) corresponds to the k-representation or momentum-representation of quantum mechanics. The notation above for Fourier transforms is a little sloppy, but convenient and common in quantum mechanics. From a mathematical perspective, the Fourier ˜ transformed function ψ(k, t) should actually be denoted by ψ(k, t) to make it clear that it is not the same function as ψ(x, t) with different symbols for the first three variables. The physics notation is motivated by the observation that ψ(x, t)

2.1 The δ Function and Fourier Transforms

27

and ψ(k, t) are just different representations of the same quantum mechanical state ψ. Another often used convention for Fourier transforms is to split the factor (2π )−3 asymmetrically, or equivalently replace it with a factor 2π in the exponents,  1 d 3 k exp(ik · x)ψ(k, t), (2π )3  ψ(k, t) = d 3 x exp(− ik · x)ψ(x, t),

ψ(x, t) =

(2.9)

(2.10)

or equivalently  d 3 ν˜ exp(2π i˜ν · x)ψ(˜ν , t),

(2.11)

d 3 x exp(−2π i˜ν · x)ψ(x, t),

(2.12)

ψ(x, t) =  ψ(˜ν , t) = with the vector of wave numbers

ν˜ =

k . 2π

(2.13)

The conventions (2.7) and (2.8) are used throughout this book. The following is an argument for Eq. (2.1) and its generalizations to other representations of the δ function. The idea is to first construct a limit for the Heaviside step function or  function  (x) =

1, 0,

x > 0, x < 0,

(2.14)

and go from there. The value of (0) is often chosen to suite the needs of the problem at hands. The choice (0) = 1/2 seems intuitive and is also mathematically natural in the sense that any decomposition of a discontinuous functions in a complete set of functions (e.g. Fourier decomposition) will approximate the mean value between the left and right limit for a finite discontinuity, but in many applications other values of (0) are preferred. The  function helps us to explain Dirichlet’s equation (2.1) through the following construction. Suppose d(x) is a normalized function, 

∞

−∞

dx d(x) = 1.

(2.15)

28

2 Self-Adjoint Operators and Eigenfunction Expansions

The integral  D(x) =

x

−∞

(2.16)

dξ d(ξ )

satisfies lim D(κ · x) = (x),

(2.17)

κ→∞

0 where we apparently defined (0) as (0) = −∞ dξ d(ξ ), but this plays no role for the following reasoning. Equation (2.17) yields for every function f (x) which is differentiable in the interval [a, b] the equation 

b a

b  b  dx κ d(κ · x)f (x) = D(κ · x)f (x) − dx D(κ · x)f  (x), a

(2.18)

a

and therefore  lim

κ→∞ a

b

 dx κ d(κ · x)f (x) = (b)f (b) − (a)f (a) −

b

dx (x)f  (x)

a

= (b)f (b) − (a)f (a) − (b)[f (b) − f (0)] + (a)[f (a) − f (0)] = [(b) − (a)]f (0),

(2.19)

where we simply split the integral according to 

b

dx (x)f  (x) =



b

dx (x)f  (x) −

0

a



a

dx (x)f  (x)

(2.20)

0

to arrive at the final result. Equation (2.19) confirms lim κ d(κx) = δ(x),

(2.21)

lim κ d[κ(x − x0 )] = δ(x − x0 ).

(2.22)

κ→∞

or after shifting the argument, κ→∞

From a mathematical perspective, equations like (2.21) mean that the action of the δ distribution on a smooth funtion corresponds to integration with a kernel κ d(κx) and then taking the limit κ → ∞. Equation (2.2) is an important particular realization of Eq. (2.21) with the normalized sinc function d(x) = sinc(x)/π = sin(x)/π x. Another important realization uses the function d(x) = (π + π x 2 )−1 ,

2.1 The δ Function and Fourier Transforms

29

κ a 1 1 = lim a→0 π a 2 + x 2 π 1 + κ 2x2  ∞ 1 = lim dk exp(ikx − a|k|). a→0 2π −∞

δ(x) = lim

κ→∞

(2.23)

Note that we did not require d(x) to have a maximum at x = 0 to derive (2.21), and indeed we do not need this requirement. Consider the following example, 1 d(x) = 2



α 1 exp[−α(x − a)2 ] + π 2



β exp[−β(x − b)2 ]. π

(2.24)

This function has two maxima if α · β = 0 and if a and b are sufficiently far apart, and it even has a minimum at x = 0 if α = β and a = −b. Yet we still have   κ α lim κ d(κ · x) = lim exp[−α(κx − a)2 ] κ→∞ κ→∞ 2 π   κ β 2 + exp[−β(κx − b) ] = δ(x), 2 π

(2.25)

because the scaling with κ scales the initial maxima near a and b to a/κ → 0 and b/κ → 0.

Sokhotsky–Plemelj Relations The Sokhotsky–Plemelj relations are very useful relations involving a δ distribution,1 1 1 = P + iπ δ(x), x − i x

1 1 = P − iπ δ(x). x + i x

(2.26)

Indeed, for the practical evaluation of integrals involving singular denominators, we virtually never use these relations but evaluate the integrals with the left-hand sides directly using the Cauchy and residue theorems. The primary use of the Sokhotsky–Plemelj relations in physics and technology is to establish relations between different physical quantities. The relation between retarded Green’s functions and local densities of states is an example for this and will be derived in Sect. 21.1.

1 Sokhotsky

[161], Plemelj [135]. The “physics” version (2.26) of the Sokhotsky–Plemelj relations is of course more recent than the original references because the δ distribution was only introduced much later.

30

2 Self-Adjoint Operators and Eigenfunction Expansions

I will give a brief justification for the Sokhotsky–Plemelj relations. The relations 1 1 = x + i i



∞

0



1 dk exp[ik(x + i)] = i

0

−∞

dk exp[− ik(x + i)]

(2.27)

imply 

1 1 =− x + i 2



∞ −∞

dk cos(kx) = − π δ(x).

(2.28)

On the other hand, the real part is 

1 1 1 x . = + = 2 x + i 2(x + i) 2(x − i) x + 2

(2.29)

This implies for integration with a bounded function f (x) in [a, b] 

b a

f (x) = dx x + i



b

dx a

xf (x) − iπ [(b) − (a)]f (0). x2 + 2

(2.30)

However, the weight factor K (x) =

x x2 + 2

(2.31)

esentially cuts the region −3 < x < 3 symmetrically from the integral b a dx f (x)/x (the value 3 is chosen because xK (x) = 0.9 for x = ±3), see Fig. 2.1. Therefore we can use this factor as one possible definition of a principal value integral,  P

b

dx a

f (x) = lim →0 x



b

dx K (x)f (x).

(2.32)

a

2.2 Self-Adjoint Operators and Completeness of Eigenstates The statistical interpretation of the wave function ψ(x, t) implies that the wave functions of single stable particles should be normalized,  d 3 x |ψ(x, t)|2 = 1.

(2.33)

Time-dependence plays no role and will be suppressed in the following investigations.

2.2 Self-Adjoint Operators and Completeness of Eigenstates

31

Fig. 2.1 Comparison of 1/x with the weight factor K(x)

Indeed, we have to require a little more than just normalizability of the wave function ψ(x) itself, because the functions ∇ψ(x), ψ(x), and V (x)ψ(x) for admissible potentials V (x) should also be square integrable. We will therefore also encounter functions f (x) which may not be normalized, although they are square integrable,  d 3 x |f (x)|2 < ∞.

(2.34)

Let ψ(x) and φ(x) be two square integrable functions. The identity  d 3 x |ψ(x) − λφ(x)|2 ≥ 0

(2.35)

yields with the choice λ=

d 3 x φ + (x)ψ(x) d 3 x |φ(x)|2

(2.36)

32

2 Self-Adjoint Operators and Eigenfunction Expansions

the Schwarz inequality  2        d 3 x φ + (x)ψ(x) ≤ d 3 x |ψ(x)|2 d 3 x  φ(x  )2 .  

(2.37)

The differential operators − ih∇ ¯ and − (h¯ 2 /2m), which we associated with momentum and kinetic energy, and the potential energy V (x) all have the following properties, 

h¯ d 3 x φ + (x) ∇ψ(x) = i 



+

h¯ d 3 x ψ + (x) ∇φ(x) i



d x φ (x)ψ(x) = 3

3

+

+ ,

(2.38)

,

(2.39)

+

d x ψ (x)φ(x)

and 

d 3 x φ + (x)V (x)ψ(x) =



+

d 3 x ψ + (x)V (x)φ(x)

.

(2.40)

Eq. (2.40) is a consequence of the fact that V (x) is a real function. Equations (2.38) and (2.39) are a direct consequence of partial integrations and the fact that boundary terms at |x| → ∞ vanish under the assumptions that we had imposed on the wave functions. If two operators Ax and Bx have the property 

+

d x φ (x)Ax ψ(x) = 3

 3

+

d x ψ (x)Bx φ(x)

+ ,

(2.41)

for all wave functions of interest, then Bx is denoted as adjoint to the operator Ax . The mathematical notation for the adjoint operator to Ax is A+ x, Bx = A+ x.

(2.42)

Complex conjugation of (2.41) then immediately tells us Bx+ = Ax . An operator with the property A+ x = Ax is denoted as a self-adjoint or hermitian operator.2 Self-adjoint operators are important in quantum mechanics because they yield real expectation values, 2 We

are not addressing matters of definition of domains of operators in function spaces, see e.g. [94] or Problem 2.6. If the operators A+ x and Ax can be defined on different classes of functions, and A+ x = Ax holds on the intersections of their domains, then Ax is usually denoted as a symmetric operator. The notion of self-adjoint operator requires identical domains for both Ax and A+ x such that the domain of neither operator can be extended. If the conditions on the domains are violated, we can e.g. have a situation where Ax has no eigenfunctions at all, or where the eigenvalues of Ax are complex and the set of eigenfunctions is overcomplete. Hermiticity is

2.2 Self-Adjoint Operators and Completeness of Eigenstates

(Aψ )+ =

 

=

d 3 x ψ + (x)Ax ψ(x)

+

 =

33

d 3 x ψ + (x)A+ x ψ(x)

d 3 x ψ + (x)Ax ψ(x) = Aψ .

(2.43)

Observable quantities like energy or momentum or location of a particle are therefore implemented through self-adjoint operators, e.g. momentum p is implemented through the self-adjoint differential operator − ih∇. We have seen one method to ¯ figure this out in equation (1.73). We will see another method in Eqs. (4.64) and (4.66). Self-adjoint operators have the further important property that their eigenfunctions yield complete sets of functions. Schematically this means the following: Suppose we can enumerate all constants an and functions ψn (x) which satisfy the equation Ax ψn (x) = an ψn (x)

(2.44)

with the set of discrete indices n. The constants an are eigenvalues and the functions ψn (x) are eigenfunctions of the operator Ax . Hermiticity of the operator Ax implies orthogonality of eigenfunctions for different eigenvalues,  an

d

3

+ x ψm (x)ψn (x)

 =

d

3

 =

x ψn+ (x)Ax ψm (x)

+ d 3 x ψm (x)Ax ψn (x)

+

 = am

+ (x)ψn (x) d 3 x ψm

(2.45)

and therefore 

+ d 3 x ψm (x)ψn (x) = 0 if an = am .

(2.46)

However, even if an = am for different indices n = m (i.e. if the eigenvalue an is degenerate because there exist at least two eigenfunctions with the same eigenvalue), one can always chose orthonormal sets of eigenfunctions for a degenerate eigenvalue. We therefore require 

+ (x)ψn (x) = δm,n . d 3 x ψm

(2.47)

Completeness of the set of functions ψn (x) means that an “arbitrary” function f (x) can be expanded in terms of the eigenfunctions of the self-adjoint operator Ax in the form

sometimes defined as equivalent to symmetry or as equivalent to the more restrictive notion of self-adjointness of operators. We define Hermiticity as self-adjointness.

34

2 Self-Adjoint Operators and Eigenfunction Expansions

f (x) =



cn ψn (x)

(2.48)

n

with expansion coefficients  cn =

d 3 x ψn+ (x)f (x).

(2.49)

If we substitute Eq. (2.49) into (2.48) and (in)formally exchange integration and summation, we can express the completeness property of the set of functions ψn (x) in the completeness relation

ψn (x)ψn+ (x  ) = δ(x − x  ).

(2.50)

n

Both the existence and the meaning of the series expansions (2.48) and (2.49) depends on what large a class of “arbitrary” functions f (x) one considers. Minimal constraints require boundedness of f (x), and continuity if the series (2.48) is supposed to converge pointwise. The default constraints in non-relativistic quantum mechanics are continuity of wave functions ψ(x) to ensure validity of the Schrödinger equation with at most finite discontinuities in potentials V (x), and normalizability. Under these circumstances the expansion (2.48) and (2.49) for a wave function f (x) ≡ ψ(x) will converge pointwise to ψ(x). However, it is convenient for many applications of quantum mechanics to use limiting forms of wave functions which are not normalizable in the sense of Eq. (2.33) any more, e.g. plane wave states ψk (x) ∝ exp(ik · x), and we will frequently also have to expand non-continuous functions, e.g. functions of the form f (x) = V (x)ψ(x) with a discontinuous potential V (x). However, finally we only have to use expansions of the form (2.48) and (2.49) in the evaluation of integrals of the form d 3 xg + (x)f (x), and here the concept of convergence in the mean comes to our rescue in the sense that substitution of the series expansion (2.48) and (2.49) in the integral will converge to the same value of the integral, even if the expansion (2.48) and (2.49) does not converge pointwise to the function f (x). A more thorough discussion of completeness of sets of eigenfunctions of selfadjoint operators in the relatively simple setting of wave functions confined to a finite one-dimensional interval is presented in Appendix C. However, for a first reading I would recommend to accept the series expansions (2.48) and (2.49) with the assurance that substitutions of these series expansions is permissible in the calculation of observables in quantum mechanics.

2.3 Problems

35

2.3 Problems 2.1 Suppose the function f (x) has only first order zeros, i.e. we have non-vanishing slope at all nodes xi of the function,  df (x)   0. = f (xi ) = 0 ⇒ f (xi ) ≡ dx x=xi 

(2.51)

Prove the following property of the δ function: δ(f (x)) =

i

1 δ(x − xi ). |f  (xi )|

(2.52)

2.2 Calculate the Fourier transforms of the following functions, where in all cases −∞ < x < ∞. Do not use any electronic integration program. 2.2a ψ1 (x) = exp(− ax 2 ), a > 0, 2.2b ψ2 (x) = 1/(a 2 + x 2 ), a > 0 ∈ R, 2.2c ψ3 (x) = x n exp(− a|x|), a > 0 ∈ R, where n is a natural number. 2.3 The functions f1 (x) = exp(− x 2 ) and f2 (x) = exp(− |x|) are normalizable to functions d(x) in the sense of Eq. (2.15). Use this to find other derivations of the Fourier representation of the δ function similar to Eq. (2.23). 2.4 We consider a finite interval [a, b] together with the set C (1,α) [a, b] of complex valued functions which are continuous in [a, b] and differentiable in (a, b), and satisfy the pseudo-periodicity condition ψ(b) = exp(iα)ψ(a),

α ∈ R.

(2.53)

Show that the differential operator − id/dx is self-adjoint on C (1,α) [a, b]. Give a complete set of eigenstates of − id/dx in C (1,α) [a, b]. 2.5 We consider the finite interval [a, b] together with the set C (2),0 [a, b] of complex valued functions which are continuous in [a, b] and second order differentiable in (a, b), and satisfy the boundary conditions ψ(a) = ψ(b) = 0.

(2.54)

Show that that the differential operator d 2 /dx 2 is self-adjoint on C (2),0 [a, b]. Give a complete set of eigenstates of d 2 /dx 2 in C (2),0 [a, b]. 2.6 We consider the finite interval [a, b] together with the set C (1),0 [a, b] of complex valued functions which are continuous in [a, b] and differentiable in (a, b), and satisfy the boundary conditions

36

2 Self-Adjoint Operators and Eigenfunction Expansions

ψ(a) = ψ(b) = 0.

(2.55)

Show that the symmetric differential operator h1 = − id/dx with domain C (1),0 [a, b] is not self-adjoint in the sense that h+ 1 can be defined on the larger set L2 [a, b] of square integrable functions over [a, b]. Show also that h1 has no eigenstates, while h+ 1 has complex eigenvalues and an overcomplete set of eigenstates.

Chapter 3

Simple Model Systems

One-dimensional models and models with piecewise constant potentials have been used as simple model systems for quantum behavior ever since the inception of Schrödinger’s equation. These models vary in their levels of sophistication, but their generic strength is the clear demonstration of important general quantum effects and effects of dimensionality of a quantum system at very little expense in terms of effort or computation. Simple model systems are therefore more than just pedagogical tools for teaching quantum mechanics. They also serve as work horses for the modeling of important quantum effects in nanoscience and technology, see e.g. [13, 43, 92].

3.1 Barriers in Quantum Mechanics Widely used models for quantum behavior in solid state electronics are described by piecewise constant potentials V (x). This means that V (x) attains constant values in different regions of space, and the transition between those regions of constant V (x) appears through discontinuous jumps in the potential. Figure 3.1 shows an example of a piecewise constant potential. The Schrödinger equation with a piecewise constant potential is easy to solve, and the solutions provide instructive examples for the impact of quantum effects on the motion of charge carriers through semiconductors and insulating barriers. We will first discuss the case of a rectangular barrier. Figure 3.1 shows a cross section of a non-symmetric rectangular square barrier.

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 R. Dick, Advanced Quantum Mechanics, Graduate Texts in Physics, https://doi.org/10.1007/978-3-030-57870-1_3

37

38

3 Simple Model Systems

V(x)

incident wave A exp(ikx) Φ1

B exp(−ikx) reflected wave

F exp(ikx) transmitted wave Φ2 x

L Fig. 3.1 A non-symmetric square barrier

The piecewise constant potential has values ⎧ ⎨ 0, x < 0, V (x) = 1 , 0 ≤ x ≤ L, ⎩ 2 , x > L.

(3.1)

with 1 > 2 > 0. This barrier impedes motion in the x direction. It can be used e.g. as a simple quantum mechanical model for a metal coated with an insulating layer. The region x < 0 would be inside the metal and the potential 2 would be the energy which is required to liberate an electron from the metal if there would not be the insulating layer of thickness L. The energy 1 is the energy which would classically be required for an electron to penetrate the layer. Quantum problems with time-independent potentials are conveniently analyzed by using a Fourier transformation1 from time t to energy E,   i ψ(x, t) = √ dE exp − Et ψ(x, E), h¯ 2π h¯ −∞    ∞ 1 i ψ(x, E) = √ dt exp Et ψ(x, t). h¯ 2π −∞ 1



∞

(3.2) (3.3)

1 The normalization condition (1.72) implies that the function ψ(x, E) does not exist in the sense of classical Fourier theory. We will therefore see in Sect. 5.2 that ψ(x, E) is rather a series of δ-functions of the energy. This difficulty is usually avoided by using an exponential ansatz ψ(x, t) = ψ(x, E) exp(− iEt/h¯ ) instead of a full Fourier transformation. However, if one accepts the δ-function and corresponding extensions of classical Fourier theory, the transition to the timeindependent Schrödinger equation through a formal Fourier transformation to the energy variable is logically more satisfactory.

3.1 Barriers in Quantum Mechanics

39

Substitution into the time-dependent Schrödinger equation (1.2) yields the timeindependent Schrödinger equation2 Eψ(x, E) = −

h¯ 2 ψ(x, E) + V (x)ψ(x, E). 2m

(3.4)

The potential on Fig. 3.1 depends only on x. In this case we can also eliminate the derivatives with respect to y and z through further Fourier transformations, 1 ψ(x, E) = 2π





∞

−∞

dk2

∞ −∞

dk3 exp[i(k2 y + k3 z)] ψ(x, k2 , k3 , E)

(3.5)

to find the time-independent Schrödinger equation for motion in the x direction, E1 ψ(x, E1 ) = −

h¯ 2 ∂ 2 ψ(x, E1 ) + V (x)ψ(x, E1 ). 2m ∂x 2

(3.6)

Here E1 ≡ E − h¯ 2

k22 + k32 , 2m

ψ(x, E1 ) ≡ ψ(x, k2 , k3 , E).

(3.7)

E1 is the kinetic energy for motion in the x direction in the region x < 0. Within each of the three separate regions x < 0, 0 < x < L, and x > L the potential attains a constant value, and Eq. (3.6) can be solved with a final Fourier transformation from x to k1 , ⎧ ⎪ A exp(ik1 x) + B exp(− ik1 x), ⎪ √ ⎪ ⎪ ⎪ k1 = 2mE1 /h, ¯ x < 0, ⎪ ⎪ ⎪ ⎪ ⎨  C exp(ik1 x) + D exp(− ik1 x), √ (3.8) ψ(x, E1 ) = ⎪ k1 = 2m(E1 − 1 )/h, ¯ 0 < x < L, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ F exp(ik1 x) + G exp(− ik1 x), ⎪ ⎪ √ ⎩ k1 = 2m(E1 − 2 )/h, ¯ x > L. We must have E1 > 0 because the absolute minimum of the potential determines a lower bound for the energy of a particle moving in the potential. However, the wave numbers k1 and k1 can be real or imaginary depending on the magnitude of E1 . We define k1 = − iκ,

k1 = iκ  ,

(3.9)

with the conventions κ > 0, κ  > 0, if k1 or k1 are imaginary.

2 Schrödinger

[149], Schrödinger found the time-indepedent equation first and published the timedependent Eq. (1.2) 5 months later.

40

3 Simple Model Systems

The wave function (3.8) is not yet the complete solution to our problem, because we have to impose junction conditions on the coefficients at the transition points x = 0 and x = L to ensure that the Schrödinger equation is also satisfied in those points. This will be done below. However, we can already discuss the meaning of the six different exponential terms appearing in (3.8). The wave function ψ(x, E1 ) is multiplied by the time-dependent exponential exp(− iE1 t/h) ¯ in the transition from ψ(x, E1 ) to the time-dependent wave function ψ(x, t) for motion in x direction, ψ(x, t) = √

1 2π h¯

 0

∞

  i dE1 exp − E1 t ψ(x, E1 ). h¯

(3.10)

A single monochromatic component therefore corresponds to a time-dependent wave funtion proportional to ψ(x, E1 ) exp(− iE1 t/h). ¯ The term A exp[i(k1 x − E1 t/h)] ¯ corresponds to a right moving wave in the region x < 0, while the term B exp[− i(k1 x + E1 t/h)] ¯ is a left moving wave. Similar identifications apply to the C and D components if k1 is real, and to the F and G components if k1 is real. Otherwise, these components will correspond to exponentially damped or growing wave functions, which requires G = 0 if κ  = −ik1 > 0 is real, to avoid divergence of the wave function for x → ∞. There is a subtle point here that needs to be emphasized because it is also relevant for potential scattering theory in three dimensions. We have just realized that the monochromatic wave function ψ(x, E1 ) describes a particle of energy E1 (for the motion in x direction) simultaneously as left and right moving particles in the regions where the wave number is real. The energy dependent wave function always simultaneously describes all states of the particle with energy E1 , but does not yield a time resolved picture of what happens to a particle in the presence of the potential V (x). Let us e.g. assume that we shoot a particle of energy E1 at the potential V (x) from the left. The component A exp[i(k1 x − E1 t/h)] ¯ describes the initially incident particle, while the component B exp[− i(k1 x + E1 t/h)] ¯ describes a particle that is reflected by the barrier. The component F exp[i(k1 x − E1 t/h)], ¯ on the other hand, describes a particle which went across the barrier (if E1 > 1 ), or a particle that penetrated the barrier (without damaging the barrier!) if 1 > E1 > 2 . The calculation of expectation values sheds light on the property of the monochromatic wave function ψ(x, E1 ) exp(− iE1 t/h) ¯ to describe all states of a particle of energy E1 simultaneously. The expectation values both for location x and momentum p of a particle described by a monochromatic wave function are timeindependent, i.e. a single monochromatic wave function can never describe the time evolution of motion of a particle in the sense of first corresponding to an incident wave from the left, and later either to a reflected wave or a transmitted wave. A time resolved picture describing sequential events really requires superposition of several monochromatic components (3.10) with contributions from many different energies. Stated differently, the wave function of a moving particle can never correspond to only one exact value for the energy of the particle. Building wave functions for moving particles will always require superposition of different energy values, which

3.1 Barriers in Quantum Mechanics

41

corresponds to an uncertainty in the energy of the particle. Stated in yet another way: The energy resolved picture described by the Schrödinger equation in the energy domain (3.4) describes all processes happening with energy E, whereas the time-dependent Schrödinger equation describes processes happening at time t. If the time-dependent wave function of the system is indeed monochromatic, ψ(x, t) = ψ(x, E1 ) exp(− iE1 t/h), ¯ then we imply that all these processes at energy E1 happen simultaneously, e.g. because we have a continuous particle beam of energy E1 incident on the barrier. The monochromatic wave function can still tell us a lot about the behavior of particles in the presence of the potential barrier V (x). We choose as an initial condition a particle moving against the barrier from the left. Then we have to set G = 0 in the solution above irrespective of whether k1 is real or imaginary, because in the real case this component would correspond to a particle hitting the barrier from the right, and in the imaginary case G = 0 was imposed anyway from the requirement that the wave function cannot diverge. Before we can proceed, we have to discuss junction conditions for wave functions at points where the potential is discontinuous. A finite jump in V (x) translates through the time-independent Schrödinger equation into a finite jump in d 2 ψ(x)/dx 2 , which means a jump in the slope of dψ(x)/dx, but not a discontinuity in dψ(x)/dx. Therefore both ψ(x) and dψ(x)/dx have to remain continuous across a finite jump in the potential.3 This means that the wave function ψ(x) remains smooth across a finite jump in V (x). On the other hand, an infinite jump in V (x) only requires continuity, but not smoothness of ψ(x). The requirement of smoothness of the wave function yields the junction conditions A+B = C+D k1 (A − B) =

k1 (C

− D)

C exp(ik1 L) + D exp(− ik1 L) = F exp(ik1 L) k1 [C exp(ik1 L) − D exp(− ik1 L)]

=

k1 F

exp(ik1 L)

(3.11) (3.12) (3.13) (3.14)

Elimination of C and D yields   2k1 k1 A = k1 (k1 + k1 ) cos(k1 L) − i(k1 k1 + k12 ) sin(k1 L) F exp(ik1 L),   2k1 k1 B = k1 (k1 − k1 ) cos(k1 L) − i(k1 k1 − k12 ) sin(k1 L) F exp(ik1 L).

3 The

time-dependent Schrödinger equation permits discontinuous wave functions ψ(x, t) even for smooth potentials, because there can be a trade-off between the derivative terms, see e.g. Problem 3.15.

42

3 Simple Model Systems

Note that cos(k1 L) = cosh(κL),

sin(k1 L) = − i sinh(κL).

(3.15)

If we decompose the wave function to the left and the right of the barrier into incoming, reflected, and transmitted components ψin (x) = A exp(ik1 x),

ψre (x) = B exp(− ik1 x),

ψtr (x) = F exp(ik1 x),

the probability current density (1.69) yields jin =

h¯ k1 2 |A| , m

jre = −

hk ¯ 1 2 |B| , m

jtr =

h¯ |F |2 k1 . m

(3.16)

In the last equation we used that k1 is either real or imaginary. The reflection and transmission coefficients from the barrier are then R=

|jre | |B|2 = , |jin | |A|2

T =

|F |2 k1 |jtr | = . |jin | |A|2 k1

(3.17)

This yields in all cases 0 ≤ T = 1 − R ≤ 1. The transmission coefficient is T = 0 for 0 < E1 ≤ 2 ,  T = 4 E1 (E1 − 2 )(1 − E1 ) 

  × (1 − E1 ) 2E1 − 2 + 2 E1 (E1 − 2 ) +1 (1 − 2 ) sinh2



−1 2m(1 − E1 )L/h¯

(3.18)

for 2 ≤ E1 ≤ 1 , and  T = 4 E1 (E1 − 2 )(E1 − 1 ) 

  × (E1 − 1 ) 2E1 − 2 + 2 E1 (E1 − 2 ) +1 (1 − 2 ) sin2



−1 2m(E1 − 1 )L/h¯

(3.19)

for E1 ≥ 1 . Classical mechanics, on the other hand predicts T = 0 for E1 < 1 and T = 1 for E1 > 1 , in stark contrast to the quantum mechanical transmission coefficient shown in Fig. 3.2. The phenomenon that particles can tunnel through regions even when they do not have the required energy is denoted as tunnel effect. It has been observed in many instances in nature and technology, e.g. in the α decay of radioactive nuclei (Gamow, 1928) or electron tunneling in heavily doped pn junctions (Esaki, 1958). Esaki diodes actually provide a beautiful illustration of the interplay of two quantum

3.1 Barriers in Quantum Mechanics

43

Fig. 3.2 The transmission coefficient for a non-symmetric square barrier. The curve calculated here corresponds to m = 511 keV/c2 , 1 = 10 eV, 2 = 3 eV, L = 2 Å

effects, viz. energy bands in solids and tunneling. Charge carriers can tunnel from one energy band into a different energy band in heavily doped pn junctions. We will discuss energy bands in Chap. 10. Quantum mechanical tunneling is also used e.g. in scanning tunneling microscopes (Binnig and Rohrer, 1982), and in flash memory and magnetic tunnel junction devices.4 It is easy to understand from our results for the transmission probability why quantum mechanical tunneling plays such an important role in modern memory devices. If we want to have a memory device which is electrically controlled, then apparently the information bits 0 and 1 can be encoded through the two states of a device being electrically charged or neutral. If we also want to maintain storage of the information even when the power supply is switched off (a nonvolatile memory), then the device should not discharge spontaneously, i.e. it should be electrically insulated. The device should therefore be a conductor which is surrounded by insulating material. Such a device is called a floating gate in flash memory devices, see Fig. 3.3. However, we do want to be able to charge or discharge the floating gate, i.e. eventually we want to run a current through the surrounding insulator without destroying it. Using a tunneling current through the insulator is an elegant way

4 Magnetic

tunnel junctions provide yet another beautiful example of the interplay of two quantum effects—tunneling and exchange interactions. Exchange interactions will be discussed in Chap. 17.

44

3 Simple Model Systems

Fig. 3.3 A simplified schematic of a flash memory cell. The tunneling barrier is the thin section of the insulator between the floating gate and the semiconductor

Control Gate Dielectric Insulator Floating Gate

Semiconductor

to achieve this. Our results for the tunneling probability tell us how to switch a tunneling current. If we substitute m = 511 keV/c2 , 1 − E1  1 eV, and L  10 nm, we find  2m(1 − E1 )L/h¯  51 (3.20) and therefore sinh2



  1  2m(1 − E1 )L/h¯  exp 2 2m(1 − E1 )L/h¯ , 4

(3.21)

i.e. in excellent approximation √ T  16

  E1 (E1 − 2 )(1 − E1 ) exp −2 2m(1 − E1 )L/h¯ . 1 (1 − 2 )

(3.22)

√ The exponential dependence on 1 − E1 implies that decreasing 1 − E1 by increasing E1 will have a huge impact on the tunneling current through the insulator. We can control the energy E1 of the electrons in the floating gate through the electron concentration in a nearby control gate. Presence of a negative charge on the nearby control gate will increase the energy of any electrons stored in the floating gate and allow them to tunnel into a conducting sink (usually a semiconductor) opposite to the control gate. This process will discharge the floating gate. On the other hand, a positive charge on the control gate will attract electrons from an electron current through the semiconductor towards the insulating barrier and help them to tunnel into the floating gate.

3.2 Box Approximations for Quantum Wells, Quantum Wires and Quantum Dots A particle in three dimensions which can move freely in two directions, but is confined in one direction, is said to be confined in a quantum well. A particle which can move freely only in one direction but is confined in two directions is confined in a quantum wire. Finally, a particle which is confined to a small region of space

3.2 Box Approximations for Quantum Wells, Quantum Wires and Quantum Dots

45

is confined to a quantum dot. We will discuss energy levels and wave functions of particles in all three situations in the approximation of confinement to rectangular (box-like) regions. For the quantum well this means that our particle will be confined to the region 0 < x < L1 , but it can move freely in y and z direction. The particle in the quantum wire is confined in x and y direction to 0 < x < L1 , 0 < y < L2 , but it can move freely in the z direction. Finally, box approximation for a quantum dot means that the particle is confined to the box 0 < x < L1 , 0 < y < L2 , 0 < z < L3 . We will assume strict confinement in this section, i.e. the wave function of the particle vanishes outside of the allowed region while the wave function inside the region must continuously go to zero at the boundaries of the allowed region. We gauge the energy axis such that in the allowed region the potential energy of the particle vanishes, V (x) = 0, i.e. the time-independent three-dimensional Schrödinger equation in the allowed region takes the form Eψ(x) = −

h¯ 2 ψ(x). 2m

(3.23)

Substitution of the Fourier decomposition ψ(x) = √ yields k =



1 2π

3

d 3 k ψ(k) exp(ik · x)

(3.24)

√ 2mE/h¯ and the general solution for given energy E takes the form    √ ˆ exp i 2mE kˆ · x , kˆ 2 = 1. (3.25) ψ(x) = d 2 kˆ A(k) h¯

On the other hand, Eq. (3.23) tells us that the energy of a plane wave ψ(x) = exp(ik· √ 3 x)/ 2π of momentum p = hk ¯ is E=

h¯ 2 k 2 . 2m

(3.26)

If we have no confinement condition at all, our particle is a free particle and Eq. (3.26) is the kinetic energy of a free non-relativistic particle of momentum p = hk. ¯

Energy Levels in a Quantum Well If we have a confinement condition in x-direction, e.g. ψ(0, y, z) = 0 and ψ(L1 , y, z) = 0, then we have to superimpose plane wave solutions in x direction to form a standing wave with nodes at the boundary points, and we find solutions

46

3 Simple Model Systems

  n1 π x 1 ψn1 ,k2 ,k3 (x) = √ exp[i(k2 y + k3 z)] sin , L1 π 2L1

(3.27)

with integer n1 ∈ N and energy En1 ,k2 ,k3

  n21 π 2 h¯ 2 2 2 = . k + k3 + 2m 2 L21

(3.28)

The energy of the particle is therefore determined by the discrete quantum number n1 and the continuous wave numbers k2 and k3 .

Energy Levels in a Quantum Wire If the particle is confined both in the x-direction to the region 0 < x < L1 and in the y-direction to the region 0 < y < L2 , the boundary conditions ψ(0, y, z) = 0, ψ(L1 , y, z) = 0, ψ(x, 0, z) = 0 and ψ(x, L2 , z) = 0 yield ψn1 ,n2 ,k3 (x) =

    n2 π x 2 n1 π x sin , exp(ik3 z) sin π L1 L2 L1 L2

(3.29)

and the energy of the particle is determined by the discrete quantum numbers n1 and n2 and the continuous wave number k3 for motion in z direction, En1 ,n2 ,k3

π 2 h¯ 2 = 2m



n21 L21

+

n22



L22

+

h¯ 2 k32 . 2m

(3.30)

Energy Levels in a Quantum Dot If the particle is confined to the region 0 < x < L1 , 0 < y < L2 , 0 < z < L3 , the conditions of vanishing wave function on the boundaries yields normalized states ψn1 ,n2 ,n3 (x) =

      n2 πy n3 π z 8 n1 π x sin sin , sin L1 L2 L3 L1 L2 L3

(3.31)

and the energy levels are determined in terms of three discrete quantum numbers, En1 ,n2 ,n3

π 2 h¯ 2 = 2m



n21 L21

+

n22 L22

+

n23 L23

 .

(3.32)

3.2 Box Approximations for Quantum Wells, Quantum Wires and Quantum Dots

47

Degeneracy of Quantum States If two or more different quantum states have the same energy, the quantum states are said to be degenerate, and the corresponding energy level is also denoted as degenerate. This happens e.g. for the quantum wire and the quantum dot if at least two of the length scales Li have the same value. We will discuss the quantum dot (3.32) and (3.31) with L1 = L2 = L3 ≡ L as an example. This cubic quantum dot has energy levels  π 2 h2

¯ En1 ,n2 ,n3 = n21 + n22 + n23 . 2mL2

(3.33)

The lowest energy level E1,1,1 = 3

π 2 h¯ 2 2mL2

(3.34)

corresponds to a unique quantum state ψ1,1,1 (x) and is therefore not degenerate. However, the next allowed energy value E1,1,2 = E1,2,1 = E2,1,1 = 6

π 2 h¯ 2 2mL2

(3.35)

is realized for three different wave functions ψ1,1,2 (x), ψ1,2,1 (x) and ψ2,1,1 (x), and is therefore three-fold degerate. Three-fold degeneracy is also realized for the next two energy levels E1,2,2 = E2,1,2 = E2,2,1 = 9

π 2 h¯ 2 2mL2

(3.36)

π 2 h¯ 2 . 2mL2

(3.37)

and E1,1,3 = E1,3,1 = E3,1,1 = 11 The next energy level is again non-degenerate, E2,2,2 = 12

π 2 h¯ 2 . 2mL2

(3.38)

Then follows a six-fold degenerate energy level, E1,2,3 = E2,3,1 = E3,1,2 = E1,3,2 = E3,2,1 = E2,1,3 = 14

π 2 h¯ 2 . 2mL2

(3.39)

48

3 Simple Model Systems

3.3 The Attractive δ Function Potential The attractive δ function potential V (x) = − Wδ(x),

W > 0,

(3.40)

provides a simple model system for co-existence of free states and bound states of particles in a potential. Positive energy solutions of the stationary Schrödinger equation for the δ function potential must have the form ψk (x) =



! " (±x) A± exp(ikx) + B± exp(− ikx) ,

±

hk ¯ =

√ 2mE,

(3.41)

and nomalizability limits the negative energy solutions to the from ψκ (x) =



h¯ κ =

(±x)C± exp(∓κx),

±

√ −2mE.

(3.42)

These solutions must be continuous in order not to generate δ  (x) terms which would violate the Schrödinger equation, A+ + B+ = A− + B− ,

C + = C− .

(3.43)

On the other hand, integrating the Schrödinger equation from x = − to x =  and taking the limit  → 0+ yields the junction conditions  lim

→0+

   dψ(x)  2m dψ(x)  = − 2 Wψ(0). −   dx x= dx x=− h¯

(3.44)

This implies ik(A+ − B+ − A− + B− ) = −

m h¯ 2

W(A+ + B+ + A− + B− )

(3.45)

for the free states and κ = mW/h¯ 2

(3.46)

for the bound states. Equation (3.46) tells us that there exists one bound state for W > 0 with energy Eκ = −

m 2h¯ 2

W 2.

(3.47)

3.3 The Attractive δ Function Potential

49

The normalized bound state is ψκ (x) =

√

κ exp(− κ|x|).

(3.48)

For the free states, √ we first look at solutions which are right or left moving plane waves exp(± ikx)/ 2π on the semiaxis √ x > 0, i.e. we solve Eqs. (3.43) and (3.45) = 1/ 2π , B+ = 0, and then under the conditions first under the conditions A + √ A+ = 0, B+ = 1/ 2π . This yields solutions 1 ψ+k (x) = √ exp(ikx) + 2π



1 ψ−k (x) = √ exp(− ikx) + 2π

κ 2 (−x) sin(kx), π k



κ 2 (−x) sin(kx). π k

(3.49)

(3.50)

The free solutions can be unified if we also allow √ for negative values of k (recall that until now k was defined positive from h¯ k = 2mE), 1 ψk (x) = √ exp(ikx) + 2π



κ 2 (−x) sin(kx). π k

(3.51)

These states are free right or left moving plane waves for x > 0, but they do not provide orthonormal bases for the scattering states in the attractive δ potential. We will construct two orthonormal bases below in (3.56), (3.57), (3.60), and (3.61), respectively. Another useful representation for the free states is motivated by considering the outgoing waves with amplitudes A+ and B− as consequences of the incident waves with amplitudes A− and B+ . The junction conditions (3.43) and (3.45) yield 

A+ B−



1 = k − iκ



k iκ iκ k

   A− . · B+

(3.52)

The unitary matrix 1 S= k − iκ



k iκ iκ k



√  √ E i √B √ =√ √ E E−i B i B 1

(3.53)

50

3 Simple Model Systems

is also known as a scattering matrix because it describes scattering of incoming waves off the potential. Here B ≡ −Eκ is the binding energy of the bound state. The scattering matrix can be used to read off the reflection and transmission coefficients for the δ function potential,     2  ∂B− 2  ∂A+ 2 B   = κ   R= = = ,    2 2 ∂A− ∂B+ E+B k +κ

(3.54)

     ∂A+ 2  ∂B− 2 k2 E   =   . T = = =   ∂A− ∂B+ E+B k2 + κ 2

(3.55)

In many situations it is also convenient to use even and odd solutions of the Schrödinger equation. Odd (or negative parity) solutions ψ(x) = −ψ(−x) must satisfy A+ = −B− , B+ = −A− . Solving equations (3.43) and (3.45) with these conditions yields the negative parity solutions 1 ψk,− (x) = √ sin(kx). π

(3.56)

The positive energy solutions of positive parity follow from A+ = B− , B+ = A− and Eqs. (3.43) and (3.45) in the form 1 k cos(kx) − κ sin(k|x|) ψk,+ (x) = √ . √ π κ 2 + k2

(3.57)

The √ wave number k in (3.56) and (3.57) is constrained to the positive semiaxis k = 2mE/h¯ > 0. The solutions (3.48), (3.56) and (3.57) satisfy the usual orthonormalization conditions for bound or free states, respectively (see Problem 3.9), and their completeness relation is ψκ (x)ψκ (x  ) +



∞

! " dk ψk,− (x)ψk,− (x  ) + ψk,+ (x)ψk,+ (x  ) = δ(x − x  ).

0

(3.58) The state (3.49) describes a situation in which a particle is incident on the δ potential from the left, and therefore on the right side of the potential (x > 0) we only have the right moving transmitted component, whereas the wave function for x < 0 contains both incoming and reflected components. We can find a corresponding normalized solution and construct a basis of scattering states which describe scattering of particles incident from the left or from the right by applying a unitary transformation on the basis of even and odd scattering states (3.57) and (3.56): 

ψk,l (x) ψk,r (x)



√   2 2 κ + ik ψk,+ (x) √κ + k = . ψk,− (x) κ 2 + k 2 − κ − ik 2(κ 2 + k 2 ) 1

(3.59)

3.4 Evolution of Free Schrödinger Wave Packets

51

This yields a basis with states describing incidence of particles from the left, ψk,l (x) =

k exp(ikx) + 2κ(−x) sin(kx)  , 2π(κ 2 + k 2 )

(3.60)

and incidence of particles from the right, k exp(− ikx) − 2κ(x) sin(kx)  . 2π(κ 2 + k 2 )

(3.61)

! " dk ψk,l (x)ψk,l (x  ) + ψk,r (x)ψk,r (x  ) = δ(x −x  ).

(3.62)

ψk,r (x) = The completeness relation is ψκ (x)ψκ (x  )+



∞ 0

There is no bound state solution for a repulsive δ potential V (x) = Wδ(x) =

h¯ 2 κ δ(x) m

(3.63)

and the even parity energy eigenstates become 1 k cos(kx) + κ sin(k|x|) φk,+ (x) = √ . √ π κ 2 + k2

(3.64)

The completeness relation for the eigenfunctions of the repulsive δ potential is therefore  ∞ ! " dk ψk,− (x)ψk,− (x  ) + φk,+ (x)φk,+ (x  ) = δ(x − x  ). (3.65) 0

3.4 Evolution of Free Schrödinger Wave Packets Another important model system for quantum behavior is provided by free wave packets. We will discuss in particular free Gaussian wave packets because they provide a simple analytic model for dispersion of free wave packets. This example will also demonstrate that the spatial and temporal range of free particle models is constrained in quantum physics. We will see that free wave packets of subatomic particles disperse on relatively short time scales, which are however too long to interfere with lab experiments involving free electrons or nucleons. Nevertheless, the discussion of the dispersion of free wave packets makes it also clear that simple interpretations of particles in quantum mechanics as highly localized free wave packets which every now and then get disturbed through interactions with other

52

3 Simple Model Systems

wave packets are not feasible. Particles can exist in the form of not too small free wave packets for a little while, but atomic or nuclear size wave packets must be stabilized by interactions to avoid rapid dispersion. We will see examples of stable wave packets in Chaps. 6 and 7.

The Free Schrödinger Propagator Substitution of a Fourier ansatz  ∞  ∞ 1 dk dω ψ(k, ω) exp[i(kx − ωt)] ψ(x, t) = 2π −∞ −∞

(3.66)

into the free Schrödinger equation shows that the general solution of that equation in one dimension is given in terms of a wave packet ψ(k, ω) = 1 ψ(x, t) = √ 2π

√

  h¯ k 2 , 2π ψ(k)δ ω − 2m

  h¯ k 2 t . dk ψ(k) exp i kx − 2m −∞



(3.67)

∞

(3.68)

The amplitude ψ(k) is related to the initial condition ψ(x, 0) through inverse Fourier transformation  ∞ 1 ψ(k) = √ dx ψ(x, 0) exp(− ikx), (3.69) 2π −∞ and substitution of ψ(k) into (3.68) leads to the expression  ψ(x, t) =

∞

dx  U (x − x  , t)ψ(x  , 0)

(3.70)

  hk ¯ 2 t . dk exp i kx − 2m −∞

(3.71)

−∞

with the free propagator U (x, t) =

1 2π



∞

This is sometimes formally integrated as5

5 The propagator is commonly denoted as K(x, t). However, we prefer the notation U (x, t) because

we will see in Chap. 13 that the propagator is nothing but the x representation of the time evolution operator U (t).

3.4 Evolution of Free Schrödinger Wave Packets

 U (x, t) =

53

  mx 2 m exp i . 2π iht 2ht ¯ ¯

(3.72)

The propagator is the particular solution of the free Schrödinger equation ih¯

∂ h¯ 2 ∂ 2 U (x, t) U (x, t) = − ∂t 2m ∂x 2

(3.73)

with initial condition U (x, 0) = δ(x). It yields the corresponding retarded Green’s function ih¯

h¯ 2 ∂ 2 ∂ G(x, t) + G(x, t) = δ(t)δ(x), ∂t 2m ∂x 2   = 0, G(x, t) t − 1 − 2, i.e. the lowest integer ν that can appear in (7.163) is ν = 2 and the lowest exponent that can be reached through the third term is νmin = − 2−2. However, Eq. (7.163) cannot be used to calculate the expectation value for ν = − 2, r −2 n, =

2 n3 (2 + 1)a 2

.

(7.165)

This can be calculated e.g. by a method which Waller introduced to calculate the expectation values (7.162), see Problem 7.11. The uncertainty in distance between the proton and the electron (r)n, =



r 2 n, − r2n, =

a 2 2 n (n + 2) − 2 ( + 1)2 2

(7.166)

is relatively large for most states in the sense that (r/r)n, is not small, except for large  n states with large angular momentum. For√example, we have (r/r)n,0 = 1 + (2/n2 )/3 > 1/3 but (r/r)n,n−1 = 1/ 2n + 1. However, even for large n and , the particle could still have magnetic quantum number m = 0, whence its probability density would be uniformly spread over directions (ϑ, ϕ). This means that a hydrogen atom with sharp energy generically cannot be considered as consisting of a well localized electron near a well localized proton. This is just another illustration of the fact that simple particle pictures make no sense at the quantum level. We also note from (7.152) or (7.153) that the bound eigenstates ψn,,m (r) = ψn, (r)Y,m (ϑ, ϕ) have a typical linear scale na = n

9 See

μ−1 4π 0 h¯ 2 ∝ n . μZe2 Ze2

(7.167)

e.g. Drake and Swainson [44], and references there. See Problem 7.10 for the derivation of the Kramers-Pasternak relation.

7.8 Bound Energy Eigenstates of the Hydrogen Atom

161

Here we have generalized the definition of the Bohr radius a to the case of an electron in the field of a nucleus of charge Ze. Equation (7.167) is another example of the competition between the kinetic term p2 /2μ driving wave packets apart, and an attractive potential, here V (r) = −Ze2 /4π 0 r, trying to collapse the wave function into a point. Metaphorically speaking, pressure from kinetic terms stabilizes the wave function. For given ratio of force constant Ze2 and kinetic parameter μ−1 the attractive potential cannot compress the wave packet to sizes smaller than a, and therefore there is no way for the system to release any more energy. Superficially, there seems to exist a classical analog to the quantum mechanical competition between kinetic energy and attractive potentials in the Schrödinger equation. In classical mechanics, competition between centrifugal terms and attractive potentials can yield stable bound systems. However, the classical analogy is incomplete in a crucial point. The centrifugal term for  = 0 is also there in Eq. (7.140) exactly as in the classical Coulomb or Kepler problems. However, what stabilizes the wave function against core collapse in the crucial lowest energy case with  = 0 is the radial kinetic term, whereas in the classical case bound Coulomb or Kepler systems with vanishing angular momentum always collapse. To understand the quantum mechanical stabilization of atoms against collapse a little better, let us repeat Eq. (7.140) for  = 0 and nuclear charge Ze, and for low values of r, where we can assume ψ(r) = 0: h¯ 2 1 d 2 Ze2 rψ(r) = − Er − . 2μ ψ(r) dr 2 4π 0

(7.168)

The radial probability amplitude rψ(r) must satisfy ψ −1 (r)d 2 (rψ(r))/dr 2 < 0 near the origin, to bend the function around to eventually yield limr→∞ rψ(r) = 0, which is necessary for normalizability of r 2 ψ 2 (r) on the semiaxis r > 0. However, near r = 0, the only term that bends the wave function in the right direction for normalizability is essentially the ratio Ze2 /μ−1 , 1 d2 Ze2 μ rψ(r)  − . ψ(r) dr 2 2π 0 h¯ 2

(7.169)

If we want to concentrate as much as possible of the radial probability density r 2 ψ 2 (r) near the origin r  0, we have to start out with a large slope and then bend around the wave function rψ(r) already close to r = 0 to reach (and maintain) small values ar 2 ψ 2 (r) % 1 very early. However, the only parameter that bends the wave function near the origin r  0 is the ratio between attractive force constant and kinetic parameter, Ze2 /μ−1 . This limits the minimal spatial extension of the wave function and therefore prevents the classically inevitable core collapse in the bound Coulomb system with vanishing angular momentum M = 0. In a nutshell, there is only so much squeezing of the wave function that Ze2 /μ−1 can do. See also Problem 7.18 for squeezing or stretching of a hydrogen atom near its ground state.

162

7 Central Forces in Quantum Mechanics

Fig. 7.7 The functions

√

arψ1,0 (r) (blue),

√

arψ2,0 (r) (orange), and

√

arψ2,1 (r) (green)

The radial probability amplitudes rψ1,0 (r), rψ2,0 (r), and rψ2,1 (r) are plotted in Fig. 7.7.

7.9 Spherical Coulomb Waves Now we assume E > 0. Recall that the asymptotic solutions for rψ(r) for r → 0 were of the form Ar +1 +Br − . Let us initially focus on the solutions which remain regular in the origin. The symptotic behavior for large r seems to correspond to outgoing and incoming spherical waves rψ± (r) → A± exp(±ikr),

k=



2μE/h. ¯

(7.170)

Therefore we try an ansatz rψ± (r) = w± (r)r +1 exp(±ikr),

w± (r) =



c±,ν r ν .

(7.171)

ν≥0

Instead of the recursion relation (7.146) we now find c±,ν+1 = − c±,ν

2 ν+1

1 a

± ik(ν +  + 1) ν + 2 + 2

(7.172)

7.9 Spherical Coulomb Waves

163

and therefore

w± (r) ∝ 1 ∓ ∓

+1∓ 2 + 2

i ka

2ikr +

+1∓

i ka



+2∓

i ka

 (2ikr)2 2!

(2 + 2)(2 + 3) 

 i  + 2 ∓ kai  + 3 ∓ kai (2ikr)3 ka + ... (2 + 2)(2 + 3)(2 + 4) 3!

+1∓



= 1 F1 ( + 1 ∓ (i/ka); 2 + 2; ∓2ikr).

(7.173)

However, the confluent hypergeometric function satisfies Kummer’s identity 1 F1 (α; β; z)

= exp(z)1 F1 (β − α; β; − z),

(7.174)

and this implies in particular exp(ikr)1 F1 ( + 1 − (i/ka); 2 + 2; − 2ikr) = exp(− ikr)1 F1 ( + 1 + (i/ka); 2 + 2; 2ikr),

(7.175)

i.e. there is only one regular solution for given quantum numbers (k, ), and it corresponds neither to an outgoing nor to an incoming spherical wave, but is apparently rather a superposition of incoming and outgoing waves. For applications in scattering theory one often also has to look at solutions which are irregular in the origin, rψ(r) = r −  v± (r) exp(± ikr),

v± (r) =



d±,ν r ν .

(7.176)

ν≥0

In this case, the radial Schrödinger equation yields d±,ν+1 = ∓ d±,ν

2ik ν −  ∓ kai . ν + 1 ν − 2

(7.177)

This recursion relation follows also directly from (7.172) with the substitution  → −  − 1. The solution is v± (r) ∝ 1 F1 (−  ∓ (i/ka); − 2; ∓ 2ikr),

(7.178)

and we have again from Kummer’s identity exp(ikr)1 F1 (−  − (i/ka); − 2; − 2ikr) = exp(− ikr)1 F1 (−  + (i/ka); − 2; 2ikr),

(7.179)

164

7 Central Forces in Quantum Mechanics

i.e. there is also only one irregular solution for given quantum numbers (k, ), as expected. Regular and irregular solutions can be combined to form outgoing or incoming spherical waves, see e.g. [1] or [114]. This is relevant when the long range Coulomb potential is combined with a short range scattering potential, because the short range part will modify the short distance properties of the states and both the regular and irregular spherical Coulomb waves are then needed to model the asymptotic behavior of incoming and scattered waves far from the short range scattering potential. This is relevant for scattering in nuclear physics, when short range scattering is effected by nuclear forces. However, for us the regular solutions are more interesting because together with the bound states ψn,,m (r) = ψn, (r)Y,m (ϑ, ϕ), the regular unbound states ψk,,m (r) = ψk, (r)Y,m (ϑ, ϕ) form a complete set in Hilbert space. We use the normalization10 

 i

   + 1 − ka π 2 exp (2ikr) exp(− ikr) ψk, (r) = π 2ak (2 + 1)! ×1 F1 ( + 1 + (i/ka); 2 + 2; 2ikr).

(7.180)

With this phase choice the regular spherical Coulomb waves become the free waves with sharp angular momentum (7.120) in the limit of vanishing Coulomb potential e2 → 0 ⇒ a → ∞, or if the energy E = h¯ 2 k 2 /2μ of the spherical Coulomb waves is much larger than the binding energy EB = − E1 = h¯ 2 /2μa 2 of hydrogen, E  EB ⇒ ka  1. For the overall normalization, see also the note after Eq. (7.186) and Problem 7.22. Apart from the normalization, the spherical Coulomb waves ψk, (r) become the radial bound state wave functions ψn, (r) through the substitution ik → (na)−1 . This is expected since this substitution takes the positive energy Schrödinger equation into the negative energy Schrödinger equation. The regular spherical Coulomb waves satisfy the orthogonality properties (cf. (7.158)) 

∞

0



0

∞

+ dr r 2 ψk, (r)ψk  , (r) =

1 δ(k − k  ), k2

+ dr r 2 ψk, (r)ψn, (r) = 0,

(7.181) (7.182)

and together with the radial bound state wave functions they satisfy the completeness relation11

10 Gordon

[65], Stobbe [165], see also [12]. Gordon and Stobbe normalized in the k scale, i.e. to δ(k − k  ) instead of δ(k − k  )/k 2 . 11 See e.g. Mukunda [122]. Note that we derived the Coulomb waves (7.180) for the attractive Coulomb potential of the hydrogen atom. For a general potential with charges q1 and q2 , we would

7.9 Spherical Coulomb Waves ∞

165

ψn, (r)ψn, (r  ) +



∞ 0

n=+1

+ dk k 2 ψk, (r)ψk, (r  ) =

1 δ(r − r  ). r2

(7.183)

Together with the completeness relation (7.94) for the spherical harmonics, this implies completeness of the regular hydrogen states,  ∞ =0 m=−



∞

+ ψn,,m (r)ψn,,m (r  ) +

n=+1





∞

dk k

2

0

+ ψk,,m (r)ψk,,m (r  )

= δ(r − r  ).

(7.184)

The spherical Coulomb wave functions in the representation (7.180) diverge like k −1/2 for k → 0, but this is only an artefact of the normalization (7.181) to the radial δ function in k space and is compensated by the measure k 2 in k-space integrals. For calculations of transitions between free and bound states, e.g. for capture of an electron by an ion, one needs free eigenstates which are not radially symmetric but approximate plane waves at large separations. To construct such a state from the spherical Coulomb waves, we can use that Eq. (7.126) tells us the decomposition of the plane wave exp(ikz) in terms of the free angular momentum eigenstates (7.120),  exp(ikz) =

∞ π (e2 =0) (2 + 1)P (cos ϑ)ψk, (r). 2

(7.185)

=0

The superposition of spherical Coulomb waves (7.180),  r|kMG =

∞ π (2 + 1)P (cos ϑ)ψk, (r), 2

(7.186)

=0

corresponds to a free energy eigenstate of hydrogen with energy h¯ 2 k 2 /2μ which up to logarithmic corrections approximates a superposition of a plane wave exp(ikz) with outgoing spherical waves. The normalization and the phase in (7.180) were (e2 =0) determined such that Eq. (7.186) holds without extra factors. Both ψk, (r) (7.120) and ψk, (r) have asymptotic expansions for kr → ∞ in terms of incoming and outgoing spherical waves, see e.g. [1] for the asymptotic expansions of spherical Bessel functions and hypergeometric functions. The terms with incoming spherical (e2 =0) waves in the expansions of ψk, (r) and ψk, (r) must agree, because otherwise (7.186) could not approach a superposition of exp(ikz) with outgoing spherical

need to replace a −1 = μe2 /4π 0 h¯ 2 with − μq1 q2 /4π 0 h¯ 2 . For repulsive potential q1 q2 > 0 we would only have the continuum contribution to the completeness relation (7.184).

166

7 Central Forces in Quantum Mechanics

waves for kr → ∞. The states (7.186) were discussed for the first time by Mott and Gordon [64, 120, 163], and we denote them as Mott-Gordon states.

7.10 Problems 7.1 Show that the transformations (7.3) and (7.4) imply K · R + k · r = k1 · x 1 + k2 · x 2.

(7.187)

What is then the proper boundary condition for limr→∞ ψ(r) for an unbound twoparticle state of the form (7.11) if we assume that the two particles have asymptotic momenta hk ¯ 1 and hk ¯ 2 for large separation? 7.2 How large is the minimal value of the product Mx My of uncertainties of angular momentum components in a state |, m ? 7.3 Why does Eq. (7.113) imply that there is no other choice but (7.114) for the regular solution fk, (r) of (7.110)? 7.4 Prove Eqs. (7.117) and (7.133). Hint: For the proof of (7.117), use  lim x(−x)

x→∞

1 d x dx



  sin x d  sin x. = − x dx

(7.188)

Then prove  −

d dx



  π sin x = sin x − 2

(7.189)

by induction. 7.5 A simple spherical model for a color center or a quantum dot consists of an electron confined to a sphere of radius R. Inside the sphere the electron can move freely because the potential energy vanishes there, V (r) = 0 for r < R. The wave function in the sphere for given angular momentum quantum numbers will therefore have the form ψ(r) ∝ j (kr)Y,m (ϑ, ϕ). Which energy quantization conditions will we get from the condition that the wave function vanishes for r ≥ R? How large is the radius R if the electron absorbs photons of energy 2.3 eV? Zeros xn, of spherical Bessel functions, j (xn, ) = 0, n = 1, 2, . . . can be found e.g. in Chapter 10 of [1]. Which relation between R and lattice constant d follows from Mollwo’s relation νd 2 = 5.02 × 10−5 m2 Hz?

7.10 Problems

167

In hindsight, color centers could be considered as the first realization of atomic scale quantum dots. 2 (r) for bound states of hydrogen has 7.6 Show that the radial density profile ψn, maxima at the extrema of the radial wave function ψn, (r).

Remark ψn, (r) and dψn, (r)/dr have no common zeros, because this would contradict the radial Schrödinger equation. (n,n−1)

7.7 Calculate the radius rmax where the radial wave function ψn,n−1 (r) has a maximum. Compare your result to rn,n−1 ± (r)n,n−1 . (n,n−2)

7.8 For n ≥ 2 calculate the radius rmax ψn,n−2 (r) has a maximum.

where the radial wave function

7.9 As a rule of thumb, quantum systems tend to approach classical behavior for large quantum numbers. We have seen that for large quantum number n the radial wave function ψn,n−1 (r) is localized in a spherical shell rn,n−1 ±(r)n,n−1 which √ is “thin” in the sense of (r/r)n,n−1 = 1/ 2n + 1 → 0. For sharp energy En,,m , could we ever hope to find an approximately localized electron in a hydrogen atom? 7.10 Derive the Kramers-Pasternak relation (7.163) using the equation d2 rψn, (r) = dr 2



( + 1) 2 1 − + 2 2 ar r2 n a

 rψn, (r).

(7.190)

This is the radial Schrödinger equation (7.140) if we substitute the energy levels 2 a and express all the coefficients in terms of the En = − αS2 μc2 /2n2 = − αS hc/2n ¯ Bohr radius. Hint: Show through integration by parts that the radial integral of r ν [rψn, (r)]2 can be written in the following two forms, 

∞ 0

 ∞ ν 2 − ν ν−2 r n, − dr r ν+1 ψn, (r)[rψn, (r)] 2 0  ∞ 2 =− dr r ν+1 [rψn, (r)] [rψn, (r)] , (7.191) ν+1 0

dr r ν [rψn, (r)]2 =

if ν > − 1 − 2. Substitute (7.190) into the two representations and use another integration by parts to express the integrals resulting from the last line in (7.191) as radial expectation values. Remark The condition ν > − 1 − 2 arises from the requirement of vanishing boundary terms at r = 0 in the partial integrations in the proof of (7.191) and also in the partial integrations used to convert the integral in the last line of (7.191) into radial expectation values after substitution of (7.190). However, is this condition simply an artifact of the proof? Could the Kramers-Pasternak relation also hold for

168

7 Central Forces in Quantum Mechanics

ν ≤ − 1 − 2? You can easily check that the Kramers-Pasternak relation is incorrect for ν = − 1 − 2 by substituting that value into the equation, setting  = 0, and using the result for r −2 n,0 from Problem 7.11. 7.11 Waller [173] introduced a method to calculate the radial expectation values (7.162) for the hydrogen atom using that for |u| < 1 ∞

LN (x)uN =

N =0

exp[−ux/(1 − u)] . 1−u

(7.192)

This implies ∞

N m Lm N (x)u = (−)

N =0

∞ ∞ m ∂m N m ∂ −m L (x)u = (−) u LN (x)uN N +m ∂x m ∂x m N =0

N =0

exp[−ux/(1 − u)] u−m ∂x m 1−u exp[−ux/(1 − u)] = , (1 − u)m+1

= (−)m

∂m

(7.193)

and therefore also ∞



N,N  =0

N m N Lm = N (x)u LN  (x)v



u exp −x 1−u +



v 1−v . (1 − u)m+1 (1 − v)m+1

(7.194)

To see how this could help us in the calculation of the expectation values (7.162), we note that in terms of the variable x = 2r/na the expectation values are r n, ν

nν−1 a ν (n −  − 1)! = ν+1 (n + )! 2



∞ 0

2+1 dx x 2+ν+2 exp(−x)[Ln−−1 (x)]2 .

(7.195)

However, Eq. (7.194) implies ∞  N,N  =0 0

=

∞

2+1 2+1 dx x 2+ν+2 exp(−x)LN (x)uN LN (x)v N 

1 2+2 (1 − u) (1 − v)2+2

= (2 + ν + 2)!



∞

  dx x 2+ν+2 exp −x

0

(1 − u)ν+1 (1 − v)ν+1 . (1 − uv)2+ν+3



1 − uv (1 − u)(1 − v)



(7.196)

We can therefore calculate the integral in (7.195) by identifying the expansion coefficient multiplying the term (uv)n−−1 in the Taylor expansion of the right-hand

7.10 Problems

169

side of (7.196). Use this method to calculate r −2 n, . Hint: The geometric series ∞ 1 = uN 1−u

(7.197)

N =0

and the fact that you only need the terms with equal powers of u and v effectively reduces the term for ν = − 2 on the right-hand side of (7.196) to ∞ (2)! (2)! (2)! ⇒ (uv)N = 2+1 2+1 (1 − u)(1 − v)(1 − uv) (1 − uv) (1 − uv)2+2 N =0

∞

=

N =0

(2 + 1 + N)! (uv)N . (2 + 1) · N!

(7.198)

7.12 Calculate the radial expectation value r −3 n, both from the KramersPasternak relation (7.163) and using Waller’s method. What happens for  = 0? Hint: For the calculation with Waller’s method it is helpful to know the relation ∞

(N + 1)2 (uv)N =

N =0

1 + uv . (1 − uv)3

(7.199)

7.13 We have seen how the expectation value r for the separation between the electron and the proton depends on the quantum numbers n and . How large are the corresponding expectation values for the distances of the two particles from the center of mass of the hydrogen atom? 7.14 Discuss the uncertainties [r −1 ]n, and [r −1 ]n, /r −1 n, similar to the discussion of [r]n, and [r]n, /rn, in Sect. 7.8. 7.15 We cannot construct energy eigenstates of the hydrogen atom which separate in the coordinates x e and x p of the electron and the proton. If we want to have a representation which factorizes in electron and proton wave functions, the best we can do is to expand the energy eigenstates K,n,,m (R, r) in terms of complete sets of functions fe (x e )gp (x p ) which arise from complete sets of functions f (x), g(x) for single-particle states. Expand the ground state of a hydrogen atom with center of mass momentum hK, ¯ K,1,0,0 (R, r) = √

r exp iK · R − 3 a 2a π 2 1

(7.200)

in terms of the complete basis of factorized plane electron and proton waves, x e , x p |k e , k p  =

& % 1 exp ik e · x e + ik p · x p . (2π )3

(7.201)

170

7 Central Forces in Quantum Mechanics

Solution Fourier transformation of   m e x e + mp x p |x e − x p | x e , x p |1, 0, 0; K = √ 3 exp iK · − me + mp a 2a π 2 1

yields √

k e , k p |1, 0, 0; K =

3

2a δ(K − k e − k p ) π    2 −2 a2   mp k e − me k p × 1+ , (7.202) (me + mp )2

i.e. the decomposition of the 1s hydrogen state with center of mass momentum h¯ K in terms of electron and proton plane wave states is √ 3 2a |1, 0, 0; K = d 3 k e |k e  ⊗ |K − k e  π   2 −2   m e × 1 + a 2 k e − K  . me + mp

(7.203)

Hint for the Fourier transformation: It is advantageous to use center of mass and relative coordinates for the calculation of the Fourier integrals, d 3 x e ∧ d 3 x p = d 3 r ∧ d 3 R. Remark We can use d 3 k e ∧ d 3 k p = d 3 k ∧ d 3 K to express the decomposition also in the form √ 3 x e |(me /M)K + kx p |(mp /M)K − k 2a x e , x p |1, 0, 0; K = , d 3k π (1 + a 2 k 2 )2 i.e. the hydrogen atom in the ground state and in the center of mass frame corresponds to a superposition of electron and proton waves with small opposite momenta k  a −1 . In this picture the bound state arises as an exponentially narrow wave packet exp(−|x e − x p |/a) from the isotropic superposition of plane waves exp[ik · (x e − x p )] with the weight factor (1 + a 2 k 2 )−2 . The dominant contributions come from plane waves where the amplitude to find the electron and proton at separation x e − x p is sizable and virtually constant on length scales which are large compared to a, but interference between all those waves localizes the electronproton pair within the distance a. 7.16 Show that in general the Fourier transform of a hydrogen atom state x e , x p |n, , m ; K ≡ R|Kr|n, , m , R = (mp x p + me x e )/(mp + me ), r = x e − x p , in terms of electron and proton momenta is

7.10 Problems

171

k e , k p |n, , m ; K = δ(K − k e − k p )k|n, , m ,

(7.204)

where k = (mp k e − me k p )/(mp + me ) and k|n, , m  = √



1 2π

d 3 r r|n, , m  exp(− ik · r).

3

(7.205)

7.17 Show that the k representation of the 2s state is k|2, 0, 0 =

16 √ 3 (2ka)2 − 1 a . π [1 + (2ka)2 ]3

(7.206)

7.18 Suppose we force a hydrogen atom into a 1s type state ψ(r, t) = √

1 π b3

exp(− r/b) exp(− iE1 t/h), ¯

(7.207)

where E1 = −

μe4 32π 2 02 h¯ 2

=−

e2 8π 0 a

(7.208)

is the ground state energy of the hydrogen atom, but the length parameter b is not the Bohr radius a. 7.18a How do the expectation values for kinetic, potential and total energy in the state (7.207) compare to the corresponding values in the ground state of the unperturbed hydrogen atom? 7.18b How do we have to change the potential energy of the system to force the hydrogen atom into the state (7.207)? Show that the change in potential energy can be written as    h¯ 2 1 1 1 1 1 V = − − − . (7.209) μ a b r 2a 2b 7.19 Solve the differential equation (6.8) for the harmonic oscillator not by the operator method, but by the same methods which we have used to solve the radial equation (7.140) for the hydrogen atom. 7.20 Solve the differential equation (6.5) for the isotropic harmonic oscillator in three dimensions in polar coordinates. Solution The radial equation (7.104) for the isotropic oscillator is   h¯ 2 d 2 h¯ 2 ( + 1) μ 2 2 − rψ(r) + + ω r rψ(r) = Erψ(r), 2μ dr 2 2 2μr 2

(7.210)

172

7 Central Forces in Quantum Mechanics

where μ is either the mass of a single oscillator in the external oscillator potential or the reduced mass of two particles interacting through the potential V (x 1 , x 2 ) = K(x 1 − x 2 )2 /2. Investigating the asymptotic form of (7.210) for small and large r yields the ansatz rψ(r) = r +1

  μω 2 r . cν r ν exp − 2h¯ ν≥0



(7.211)

Substitution into (7.210) then yields a recursion relation h¯ 2 cν ν(ν + 2 + 1) = (ν ≥ 2)μcν−2 [hω(2ν + 2 − 1) − 2E]. ¯

(7.212)

This is identically fulfilled for ν = 0, i.e. c0 is a free normalization constant, while ν = 1 implies c1 = 0 and therefore cν = 0 for all odd indices ν. The recursion relation for the higher even indices can also be written as cν+2 = cν

μ hω(2ν + 2 + 3) − 2E ¯ . 2 h¯ (ν + 2)(ν + 2 + 3)

(7.213)

Termination of the series cN +2 = 0 for maximal even index N = 2n implies energy quantization, En,

  3 , = hω ¯ 2n +  + 2

(7.214)

and the recursion relation takes the form c2ν+2 = c2ν

2ν − 2n ν+α μω μω = c2ν . h¯ (ν + 1)(2ν + 2 + 3) h¯ (ν + 1)(ν + β)

(7.215)

This is the recursion relation for the confluent hypergeometric function 1 F1 (α; β; x)

1 α(α + 1) 2 1 α(α + 1)(α + 2) 3 α x + x +. . . = 1+ x + β 2! β(β + 1) 3! β(β + 1)(β + 2)

(7.216)

with α = −n, β =  + 3/2 and x = μωr 2 /h. ¯ The eigenfunctions are therefore up to normalization  

 μω 2  2 (7.217) r , ψn, (r) = Cn, r 1 F1 −n;  + 3/2; μωr /h¯ exp − 2h¯ ψn,,m (r) = ψn, (r)Y,m (ϑ, ϕ).

(7.218)

7.21 The proposal of Bohmian mechanics [16] asserts that quantum mechanics with the Born probability interpretation should be replaced by a pilot wave theory.

7.10 Problems

173

The wave function would still satisfy the Schrödinger equation. However, instead of serving as a probability amplitude for the outcome of single measurements, the wave function provides a pilot wave for particles in the sense that an N particle wave function determines the velocity field for the particles through the equation dx I (t) h¯ = dt 2imI |ψ(x 1 (t), . . . x N (t); t)|2   ↔ + × ψ (x 1 (t), . . . x N (t); t) ∇ I ψ(x 1 (t), . . . x N (t); t) , (7.219) ↔

where ψ + ∇ ψ ≡ ψ + ∇ψ − (∇ψ + )ψ. It has been claimed that this leads to predictions which are indistinguishable from quantum mechanics, at least as long as we are only concerned with motion of non-relativistic particles. We consider a hydrogen atom with center of mass velocity V = hK/(m ¯ e + mp ). Which velocities would equation (7.219) predict for the velocities of the proton and the electron in the ground state of the atom? How would the proton and the electron then be arranged in the ground state of a hydrogen atom? Solution The ground state wave function in terms of electron and proton coordinates is     |x e − x p | me x e + m p x p x e , x p |1, 0, 0; K(t) = exp iK · exp − me + mp a   ih¯ i 1 K 2 t − E1 t . × √ 3 exp − 2(me + mp ) h¯ π 2 2a Equation (7.219) then yields ve = vp = V .

(7.220)

This result agrees with the corresponding expectation values for particle velocities in quantum mechanics. However, here we assume that both the electron and the proton have well defined (although not individually observable) trajectories, and their velocities are sharply defined. Therefore the electron and the proton would both move with the constant center of mass velocity V along straight lines. Motion with a fixed distance between the two particles seems hardly compatible with their electromagnetic attraction, but Bohmian mechanics explains this in terms of an additional quantum potential generated by the wave function, Vψ (x) = −

h¯ 2 |ψ(x)|, 2m|ψ(x)|

(7.221)

174

7 Central Forces in Quantum Mechanics

i.e. the wave function would also induce an additional force field in Bohmian mechanics. However, motion with fixed separation between the electron and the proton should imply observation of an electric dipole moment for individual hydrogen atoms, contrary to ordinary quantum mechanics. On the other hand, motion of the electron and the proton right on top of each other is an appealing classical picture, but is incompatible with the positive nucleus plus negative electron hull structure of atoms that follows e.g. from the van der Waals equation of state for gases (which gives atomic radii between 1 and 2 Å) and Rutherford scattering (which tells us two things: only the positive charge is concentrated in the nucleus, and the nucleus has only a radius of a few femtometers). To avoid this negative verdict, we might argue that we should rather consider a cold gas of Bohmian hydrogen atoms to understand the implications of the Bohmian interpretation for the ground state wave function. In a cold gas of Bohmian hydrogen atoms the static distance between the electron and the proton would be distributed according to |x e − x p |1, 0, 0|2 . There would be many hydrogen atoms with the electron sitting right on top of the proton, but there would also be a lot of hydrogen atoms with a large separation and a corresponding static electric dipole moment d = e(x p − x e ). Standard quantum mechanics in Born’s interpretation does not predict an electric dipole moment in any of the hydrogen atoms, because an electron would only appear to have a particular location if we specifically perform a measurement asking for the location. However, in Bohmian mechanics, the electron and proton would exist as particles at all times with fixed relative location, and therefore there should be an average dipole moment per atom in the ground state with magnitude |d| = er = 3ea/2. These dipole moments might be randomly distributed and therefore we might not observe a macroscopic dipole moment. However, we could align these dipole moments with a weak static external electric field. The field strength would be much weaker than the internal field strength in hydrogen, to ensure that the ground state wave function is not perturbed. In addition to any induced electric dipole moment in the Bohmian hydrogen atoms (which would also exist in the same way for the standard quantum mechanical hydrogen atoms) there would be a macroscopic dipole moment from orientation polarization. This would be a real difference from the standard quantum mechanical cold hydrogen gas. Therefore I disagree with claims that Bohmian mechanics is just a different ontological interpretation of nonrelativistic quantum mechanics. Trying to make pilot wave theories work is certainly tempting, but I cannot consider Bohmian mechanics as a serious competitor to standard quantum mechanics with the Born interpretation of quantum states. 7.22 Use the asymptotic expansions of spherical Bessel functions and confluent hypergeometric functions from Ref. [1] for kr  1 to write down the corresponding (e2 =0) expansions of the free angular momentum eigenstates ψk, (r) (7.120) and the Coulomb waves ψk, (r) (7.180). Compare the coefficients of the incoming and outgoing spherical waves in the expansions.

Chapter 8

Spin and Addition of Angular Momentum Type Operators

We have seen in Sect. 7.4 that representations of the angular momentum Lie algebra (7.51) are labeled by a quantum number  which can take half-integer or integer values. However, we have also seen in Sect. 7.5 that  is limited to integer values when the operators M actually refer to angular momentum, because the wave functions1 x|n, , m  or x|k, , m  for angular momentum eigenstates must be single valued. It was therefore very surprising when Stern, Gerlach, Goudsmit, Uhlenbeck and Pauli in the 1920s discovered that half-integer values of  are also realized in nature, although in that case  cannot be related to an angular momentum any more. Half-integer values of  arise in nature because leptons and quarks carry a representation of the “covering group” SU(2) of the proper rotation group SO(3), where SU(2) stands for the group which can be represented by special unitary 2 × 2 matrices.2 The designation “special” refers to the fact that the matrices are also required to have determinant 1. The generators of the groups SU(2) and SO(3) satisfy the same Lie algebra (7.51), but for every rotation matrix ˆ there are two unitary 2 × 2 matrices U (ϕ) = − U (ϕ + 2π ϕ). ˆ R(ϕ) = R(ϕ + 2π ϕ) In that sense SU(2) provides a double cover of SO(3). We will use the notations l and M for angular momenta, and s or S for spins.

1 We

denote the magnetic quantum number with m in this chapter because m will denote the mass of a particle. 2 Ultimately, all particles carry representations of the covering group SL(2,C) of the group SO(1,3) of proper orthochronous Lorentz transformations, see Appendices B and H. © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 R. Dick, Advanced Quantum Mechanics, Graduate Texts in Physics, https://doi.org/10.1007/978-3-030-57870-1_8

175

176

8 Spin and Addition of Angular Momentum Type Operators

8.1 Spin and Magnetic Dipole Interactions A particle of charge q and mass m which moves with angular momentum l through a constant magnetic field B has its energy levels shifted through a Zeeman term in the Hamiltonian, HZ = −

q l · B. 2m

(8.1)

We will explore the origin of this term in Chap. 15, see Problem 15.2, but for now we can think of it as a magnetic dipole term with a dipole moment μl =

q l. 2m

(8.2)

The relation between μl and l can be motivated from electrodynamics, but is actually a consequence of the coupling to magnetic vector potentials in the Schrödinger equation. The quantization , m |lz |, m  = hm ¯  for angular momentum components in a fixed direction yields a Zeeman shift E = −

q h¯ Bm , 2m

− ≤ m ≤ ,

(8.3)

of the energy levels of a charged particle in a magnetic field. For orbital momentum the resulting number 2 + 1 of energy levels is odd. However, the observation of motion of Ag atoms through an inhomogeneous field by Stern and Gerlach in 1921 revealed a split of energy levels of these atoms into two levels in a magnetic field. This complies with a split into 2s + 1 levels only if the angular momentum like quantum number s is 1/2. This additional angular momentum type quantum number is denoted as spin. Spin behaves in many respects similar to angular momentum, but it cannot be an orbital angular momentum because that would exclude half-integer values for s. Another major difference to angular momentum concerns the fact that the spectroscopically observed splitting of energy levels due to spin complies with a magnetic dipole type interaction only if the corresponding Zeeman type term is increased by a factor gs , H = − μs · B,

μs = g s

q s. 2m

(8.4)

This “anomalous g factor” is in very good approximation gs  2. The relation between μs and s is a consequence of relativistic quantum mechanics and will be explained in Sect. 22.5. The important observation for now is that there exist operators which satisfy the angular momentum Lie algebra (7.51),

8.1 Spin and Magnetic Dipole Interactions

177

[Si , Sj ] = ih ¯ ij k Sk ,

(8.5)

and therefore have representations of the form (7.69)–(7.74), Sz |s, ms  = hm ¯ s |s, ms ,

(8.6)

 S± |s, ms  = h¯ s(s + 1) − ms (ms ± 1)|s, ms ± 1,

(8.7)

S 2 |s, ms  = h¯ 2 s(s + 1)|s, ms .

(8.8)

However, these operators are not related to orbital angular momentum and therefore can have half-integer values of the quantum number s in their representations. Our previous calculations of matrix representations of the rotation group in Sect. 7.4 imply that spin is related to transformation properties of particle wave functions under rotations. However, before we can elaborate on this, we have to take a closer look at the representations with s = 1/2. In the following mapping between matrices we use an index mapping for the magnetic quantum numbers ms = ±1/2 to indices a(ms ) = (3/2) − ms ,

(8.9)

i.e. ms = 1/2 → a(ms ) = 1, ms = −1/2 → a(ms ) = 2. Substitution of s = 1/2 in Eqs. (8.6) and (8.7) yields 1/2, ms |S3 |1/2, ms  = hm ¯ s δms ,ms = 1/2, ms |S1 |1/2, ms  =

h¯ (σ3 )a(ms ),a(ms ) , 2

(8.10)

& h¯ h¯ % δms ,ms +1 + δms ,ms −1 = (σ1 )a(ms ),a(ms ) , 2 2

(8.11)

& h¯ h¯ % δms ,ms +1 − δms ,ms −1 = (σ2 )a(ms ),a(ms ) , 2i 2

(8.12)

and 1/2, ms |S2 |1/2, ms  = with the Pauli matrices   01 σ1 = , 10

 σ2 =

 0 −i , i 0

 σ3 =

 1 0 . 0 −1

(8.13)

The Pauli matrices provide a basis for hermitian traceless 2 × 2 matrices and satisfy the relation σ i · σ j = δij 1 + iij k σ k .

(8.14)

178

8 Spin and Addition of Angular Momentum Type Operators

The index mapping ms → a(ms ) is employed in the notation of spin states as |1/2, ms  → |a(ms ) such that a general s = 1/2 state is |ψ =

−1/2

|1/2, ms 1/2, ms |ψ =

ms =1/2

2

|aa|ψ

(8.15)

a=1

Knowledge of a spin 1/2 state |ψ is equivalent to the knowledge of its two components 1/2, 1/2|ψ ≡ 1|ψ ≡ ψ1 , 1/2, −1/2|ψ ≡ 2|ψ ≡ ψ2 . In column notation this corresponds to the 2-spinor ψ  ψ=

 ψ1 , ψ2

(8.16)

such that application of a spin operator Si −1/2

1/2, ms |ψ   = 1/2, ms |Si |ψ =

1/2, ms |Si |1/2, ms 1/2, ms |ψ

ms =1/2

corresponds to the matrix multiplication ψ =

h¯ σ · ψ. 2 i

(8.17)

For example, a general electron state |ψ corresponds to a superposition of spin orientations ±1/2 and a superposition of x eigenstates,  |ψ =

d 3x



−1/2

|x; ms x; ms |ψ ≡

d 3x

ms =1/2

−1/2

|x; ms ψa(ms ) (x),

ms =1/2

and is given in 2-spinor notation (listing all common index conventions) as  ψ(x) =

ψ1 (x) ψ2 (x)



 ≡

ψ1/2 (x) ψ−1/2 (x)



 ≡

ψ+ (x) ψ− (x)



 ≡

 ψ↑ (x) . ψ↓ (x)

(8.18)

The normalization is 



d 3 x |ψ1 (x)|2 + |ψ2 (x)|2 = 1.

(8.19)

The probability densities for finding the electron with spin up or down in the location x are |ψ1 (x)|2 and |ψ2 (x)|2 , respectively, while the probability density to find the electron in the location x in any spin orientation is |ψ1 (x)|2 + |ψ2 (x)|2 . Note that

8.2 Transformation of Scalar, Spinor, and Vector Wave Functions Under. . .

179

these three probability densities can have maxima in three different locations, which reminds us how questionable the concept of a particle is in quantum mechanics.

8.2 Transformation of Scalar, Spinor, and Vector Wave Functions Under Rotations We will now examine transformations of wave functions under coordinate rotations x  = R(ϕ) · x similar to the analysis of translations x  = x −  in Problem 6.6. The commutation relations between angular momentum M = x × p and x, [Mi , xj ] = ih ¯ ij k xk

(8.20)

imply with the rotation generators (Li )j k = ij k and the rotation matrices from Sect. 7.4     i i exp ϕ · M x exp − ϕ · M = exp(− ϕ · L) · x = R(− ϕ) · x, (8.21) h¯ h¯ and therefore  x| exp

i ϕ·M h¯

 = R(− ϕ) · x|.

(8.22)

Rotation of a state 

 i ϕ · M |ψ(t) h¯

(8.23)

x  |ψ  (t) = R(ϕ) · x|ψ  (t) = x|ψ(t),

(8.24)

|ψ(t) → |ψ  (t) = exp

therefore implies for the rotated wave function

where x  = R(ϕ) · x is the rotated set of coordinates. A transformation behavior like (8.24) tells us that the transformed wave function at the transformed set of coordinates is the same as the original wave function at the original set of coordinates. Such a transformation behavior is denoted as a scalar transformation law, and the corresponding wave functions are scalar functions. On the other hand, spinor wave functions have two components which denote probability amplitudes for spin orientation along a given spatial axis, conventionally chosen as the z axis. The z axis of the rotated frame will generically have a direction which is different from the z axis, and the probability amplitudes for spin along the z direction will be different from the probability amplitudes along the z direction.

180

8 Spin and Addition of Angular Momentum Type Operators

The rotated 2-spinor state |ψ(t) → |ψ  (t) = exp



 i ϕ · (M + S) |ψ(t) h¯

(8.25)

has components  i x , a|ψ (t) ≡ = R(ϕ) · x, a| exp ϕ · (M + S) |ψ(t) h¯      i i = x, a| exp ϕ · S |ψ(t) = exp ϕ · σ x, b|ψ(t) 2 h¯ ab 





ψa (x  , t)

or in terms of the column 2-spinor (8.18), ψ  (x  , t) = exp



i ϕ·σ 2

 · ψ(x, t).

(8.26)

For comparison, we also give the result if we use the representation (8.6) and (8.7) with s = 1 for the spin operators S on wave functions. In that case the matrix correspondence s = 1/2, ms |S|s = 1/2, ms  = hσ ¯ a(ms ),a(ms ) /2

(8.27)

is replaced in a first step by s = 1, ms |S|s = 1, ms  = h ¯ j (ms ),j (ms )

(8.28)

with j (ms ) = 2 − ms , ⎛ ⎞ 0 1 0 1 ⎝ 1 = √ 1 0 1 ⎠, 2 0 1 0 ⎛ ⎞ 1 0 0  3 = ⎝ 0 0 0 ⎠. 0 0 −1

⎛ ⎞ 0 −1 0 i ⎝ 2 = √ 1 0 −1 ⎠, 2 0 1 0 (8.29)

However, this is still not the standard matrix representation for spin s = 1. The connection with the conventional representation (7.44) of vector rotation operators is achieved through the similarity transformation L= with the unitary matrix

i M = iA ·  · A−1 h¯

(8.30)

8.3 Addition of Angular Momentum Like Quantities

⎛ ⎞ −1 0 1 1 ⎝ A= √ −i √0 −i ⎠, 2 0 2 0

A−1

⎛ ⎞ −1 i √0 1 ⎝ =√ 0 0 2 ⎠. 2 1 i 0

181

(8.31)

The transformation law for vector wave functions x, i|A(t) ≡ Ai (x, t)% under& rotations is then given in terms of the same rotation matrices R(ϕ) = exp ϕ · L which effect rotations of the vector x, & % x  = exp ϕ · L · x,

& % A (x  , t) = exp ϕ · L · A(x, t).

(8.32)

We will see in Chap. 18 that photons are described by vector wave functions.

8.3 Addition of Angular Momentum Like Quantities In classical mechanics, angular momentum is an additive vector quantity which is conserved in rotationally symmetric systems. Furthermore, the transformation equation (8.25) for spinor states involved addition of two different operators which both satisfy the angular momentum Lie algebra (7.51). However, before immersing ourselves into the technicalities of how angular momentum type operators are combined in quantum mechanics, it is worthwhile to point out that interactions in atoms and materials provide another direct physical motivation for addition of angular momentum like quantities. We have seen in Sect. 7.1 that relative motion of two interacting particles with an interaction potential V (x 1 − x 2 ) can be described in terms of effective singleparticle motion of a (quasi)particle with location r(t) = x 1 (t) − x 2 (t), mass m = m1 m2 /(m1 + m2 ), momentum p = (m2 p 1 − m1 p2 )/(m1 + m2 ) and angular momentum l = r × p. Furthermore, if m2  m1 , but the charge q2 is not much larger than q1 and the spin |s 2 | is not much larger than |s 1 |, then we can assign a charge3 q = q1 and a spin s = s 1 to the quasiparticle with mass m  m1 . A particle of charge −e and mass m with angular momentum operators l and spin s experiences a contribution to its energy levels from an interaction term Hl·s =

μ0 e2 l·s 8π m2 r 3

(8.33)

in its Hamiltonian, if it is moving in the electric field E = rˆ e/(4π 0 r 2 ) of a much heavier particle of charge e. One can think of Hl·s as a magnetic dipoledipole interaction (μ0 /4π r 3 )μl · μs , but finally it arises as a consequence of a 3 We

will return to the question of assignment of charge and spin to the quasiparticle for relative motion in Sect. 18.4.

182

8 Spin and Addition of Angular Momentum Type Operators

relativistic generalization of the Schrödinger equation. We will see this in Chap. 22, in particular Eq. (22.211). However, for the moment we simply accept the existence of terms like (8.33) as an experimental fact. These terms contribute to the fine structure of spectral lines. The term (8.33) is known as a spin-orbit coupling term or ls coupling term, and applies in this particular form to the energy levels of the quasiparticle which describes relative motion in a two-particle system. However, if there are many charged particles like in a many-electron atom, then there will also be interaction terms between angular momenta and spins of different particles in the system, i.e. we will have terms of the form Hj 1 ·j 2 = f (r12 )j 1 · j 2 ,

(8.34)

where j i are angular momentum like operators. We will superficially denote all these operators (including spin) simply as angular momentum operators in the following. Diagonalization of Hamiltonians like (8.33) or (8.34) requires us to combine two operators to a new operator according to j = l + s or j = j 1 + j 1 , respectively. From the perspective of spectroscopy, terms like (8.33) or (8.34) are the very reason why we have to know how to combine two angular momentum type operators in quantum mechanics. Diagonalization of (8.33) and (8.34) is important for understanding the spectra of atoms and molecules, and spin-orbit coupling also affects energy levels in materials. Furthermore, Hamiltonians of the form −2J s 1 ·s 2 provide an effective description of interactions in magnetic materials, see Sect. 17.7, and they are important for spin entanglement and spintronics. The advantage of introducing the combined angular momentum operator j = l + s is that it also satisfies angular momentum commutation rules (7.51) [ja , jb ] = ih ¯ abc jc and therefore should have eigenstates |j, mj , j 2 |j, mj  = h¯ 2 j (j + 1)|j, mj ,

jz |j, mj  = h¯ mj |j, mj .

(8.35)

However, j commutes with l 2 and s 2 , [ja , l 2 ] = [ja , s 2 ] = 0, and therefore we can try to construct the states in (8.35) such that they also satisfy the properties l 2 |j, mj , , s = h¯ 2 ( + 1)|j, mj , , s,

(8.36)

s 2 |j, mj , , s = h¯ 2 s(s + 1)|j, mj , , s.

(8.37)

The advantage of these states is that they are eigenstates of the coupling operator (8.33), j 2 − l2 − s2 |j, mj , , s 2 j (j + 1) − ( + 1) − s(s + 1) = h¯ 2 |j, mj , , s, (8.38) 2

l · s|j, mj , , s =

8.3 Addition of Angular Momentum Like Quantities

183

and therefore the energy shifts from spin-orbit coupling in these states are E =

μ0 e2 h¯ 2 −3 r  [j (j + 1) − ( + 1) − s(s + 1)]. 16π m2

(8.39)

The states that we know for the operators l and s are the eigenstates |, m  for l 2 and lz , and |s, ms  for s 2 and sz , respectively. We can combine these states into states |, m  ⊗ |s, ms  ≡ |, m ; s, ms 

(8.40)

which will be denoted as a tensor product basis of angular momentum states. The understanding in the tensor product notation is that l only acts on the first factor and s only on the second factor. Strictly speaking the combined angular momentum operator should be written as j = l ⊗ 1 + 1 ⊗ s,

(8.41)

which automatically ensures the correct rule j (|, m  ⊗ |s, ms ) = l|, m  ⊗ |s, ms  + |, m  ⊗ s|s, ms ,

(8.42)

but we will continue with the standard physics notation j = l + s. The main problem for combination of angular momenta is how to construct the eigenstates |j, mj , , s for total angular momentum from the tensor products (8.40) of eigenstates of the initial angular momenta, |j, mj , , s =



|, m ; s, ms , m ; s, ms |j, mj , , s.

(8.43)

m ,ms

We will denote the states |j, mj , , s as the combined angular momentum states. There is no summation over indices  =  or s  = s on the right-hand side because all states involved are eigenstates of l 2 and s 2 with the same eigenvalues h¯ 2 ( + 1) or h¯ 2 s(s + 1), respectively. The components , m ; s, ms |j, mj , , s of the transformation matrix from the initial angular momenta states to the combined angular momentum states are known as Clebsch–Gordan coefficients or vector addition coefficients. The notation , m ; s, ms |j, mj , , s is logically satisfactory by explicitly showing that the Clebsch–Gordan coefficients can also be thought of as the representation of the combined angular momentum states |j, mj , , s in the basis of tensor product states |, m ; s, ms . However, the notation is also redundant in terms of the quantum numbers  and s, and a little clumsy. It is therefore convenient to abbreviate the notation by setting , m ; s, ms |j, mj , , s ≡ , m ; s, ms |j, mj .

(8.44)

184

8 Spin and Addition of Angular Momentum Type Operators

The new angular momentum eigenstates must also be normalizable and orthogonal for different eigenvalues, i.e. the transformation matrix must be unitary,

j, mj |, m ; s, ms , m ; s, ms |j  , mj  = δj,j  δmj ,mj ,

(8.45)

, m ; s, ms |j, mj j, mj |, m ; s, ms  = δm ,m δms ,ms .

(8.46)

m ,ms

j,mj

The hermiticity properties jz = (lz + sz )+ ,

j± = (l∓ + s∓ )+

(8.47)

imply with the definition (4.80) of adjoint operators the relations mj , m ; s, ms |j, mj  = (m + ms ) , m ; s, ms |j, mj 

(8.48)

and 

j (j + 1) − mj (mj ± 1), m ; s, ms |j, mj ± 1  = ( + 1) − m (m ∓ 1), m ∓ 1; s, ms |j, mj   + s(s + 1) − ms (ms ∓ 1), m ; s, ms ∓ 1|j, mj .

(8.49)

Equation (8.48) yields , m ; s, ms |j, mj  = δm +ms ,mj , m ; s, ms |j, m + ms .

(8.50)

The highest occuring value of mj which is also the highest occuring value for j is therefore  + s, and there is only one such state. This determines the state | + s,  + s, , s up to a phase factor as | + s,  + s, , s = |, ; s, s,

(8.51)

i.e. we choose the phase factor as , ; s, s| + s,  + s = 1.

(8.52)

Repeated application of j− = l− +s− on the state (8.51) then yields all the remaining states of the form | + s, mj , , s or equivalently the remaining Clebsch–Gordan coefficients of the form , m ; s, ms | + s, mj = m + ms  with − − s ≤ mj <  + s. For example, the next two lower states with j =  + s are given by

8.3 Addition of Angular Momentum Like Quantities

185



2( + s)| + s,  + s − 1, , s √ √ = 2|,  − 1; s, s + 2s|, ; s, s − 1

j− | + s,  + s, , s =

(8.53)

and  √ j−2 | + s,  + s, , s = 2  + s 2( + s) − 1| + s,  + s − 2, , s  √ = 2 (2 − 1)|,  − 2; s, s + 4 s|,  − 1; s, s − 1  + 2 s(2s − 1)|, ; s, s − 2. (8.54) However, we have two states in the |, m ; s, ms  basis with total magnetic quantum number  + s − 1, but so far discovered only one state in the |j, mj , , s basis with this magnetic quantum number. We can therefore construct a second state with mj =  + s − 1, which is orthogonal to the state | + s,  + s − 1, , s,  |+s−1, +s−1, , s =

 s  |, −1; s, s− |, ; s, s−1. +s +s

(8.55)

Application of j 2 would show that this state has j =  + s − 1, which was already anticipated in the notation. Repeated application of the lowering operator j− on this state would then yield all remaining states of the form | + s − 1, mj , , s with 1 −  − s ≤ mj <  + s − 1, e.g. √  + s − 1| + s − 1,  + s − 2, , s =  −



 s

2 − 1 |,  − 2; s, s +s

s− 2s − 1 |, ; s, s − 2 + √ |,  − 1; s − 1, s. +s +s

(8.56)

We have three states with mj =  + s − 2 in the direct product basis, viz. |,  − 2; s, s, |, ; s, s − 2 and |,  − 1; s − 1, s, but so far we have only constructed two states in the combined angular momentum basis with mj =  + s − 2, viz. | + s,  + s − 2, , s and | + s − 1,  + s − 2, , s. We can therefore construct a third state in the combined angular momentum basis which is orthogonal to the other two states, | + s − 2,  + s − 2, , s ∝ |,  − 1; s − 1, s − | + s,  + s − 2, , s + s,  + s − 2, , s|,  − 1; s − 1, s − | + s − 1,  + s − 2, , s + s − 1,  + s − 2, , s|,  − 1; s − 1, s. Substitution of the states and Clebsch–Gordan coefficients from (8.54) and (8.56) and normalization yields

186

8 Spin and Addition of Angular Momentum Type Operators

(2 − 1)(2s − 1) |,  − 1; s − 1, s (2 + 2s − 1)( + s − 1) √ √ (2 − 1)|, ; s, s − 2 − s(2s − 1)|,  − 2; s, s + . (8.57) √ (2 + 2s − 1)( + s − 1)

| + s − 2,  + s − 2, , s =

Application of j− then yields the remaining states of the form | + s − 2, mj , , s. This process of repeated applications of j− and forming new states with lower j through orthogonalization to the higher j states terminates when j reaches a minimal value j = | − s|, when all (2 + 1)(2s + 1) states |, m ; s, ms  have been converted into the same number of states of the form |j, mj , , s. In particular, we observe that there are 2 × min(, s) + 1 allowed values for j , j ∈ {| − s|, | − s| + 1, . . . ,  + s − 1,  + s}.

(8.58)

The procedure to reduce the state space in terms of total angular momentum eigenstates |j, mj , , s through repeated applications of j− and orthogonalizations is lengthy when the number of states (2 + 1)(2s + 1) is large, and the reader will certainly appreciate that Wigner [179] and Racah [140] have derived expressions for general Clebsch–Gordan coefficients. Racah derived in particular the following expression (see also [47, 145]) , m ; s, ms |j, mj  = δm +ms ,mj √ ν2 (2j + 1) · ( + s − j )! · (j +  − s)! · (j + s − )! ν × (−) √ (j +  + s + 1)! · ν! · ( − m − ν)! · (s + ms − ν)! ν=ν 1

  ( + m )! · ( − m )! · (s + ms )! · (s − ms )! · (j + mj )! · (j − mj )! × . (j − s + m + ν)! · (j −  − ms + ν)! · ( + s − j − ν)! (8.59)

The boundaries of the summation are determined by the requirements max[0, s − m − j,  + ms − j ] ≤ ν ≤ min[ + s − j,  − m , s + ms ].

(8.60)

Even if we decide to follow the standard convention of using real Clebsch–Gordan coefficients, there are still sign ambiguities for every particular value of j in |−s| ≤ j ≤  + s. This arises from the ambiguity of constructing the next orthogonal state when going from completed sets of states |j  , mj  , , s, j < j  ≤  + s to the next lower level j , because a sign ambiguity arises in the construction of the next orthogonal state |j, j, , s. For example, Racah’s formula (8.59) would give us the state | + s − 2,  + s − 2, , s constructed before in Eq. (8.57), but with an overall minus sign.

8.4 Problems

187

Tables of Clebsch–Gordan coefficients had been compiled in the olden days, but nowadays these coefficients are implemented in commercial mathematical software programs for numerical and symbolic calculation, and there are also free online applets for the calculation of Clebsch–Gordan coefficients.

8.4 Problems 8.1 Calculate the spinor rotation matrix  U (ϕ) = exp

 i ϕ·σ . 2

(8.61)

Hint: Use the expansion of the exponential function and consider odd and even powers of the exponent separately. Verify the property ˆ U (ϕ) = − U (ϕ + 2π ϕ).

(8.62)

8.2 We perform a rotation of the reference frame by an angle ϕ around the x-axis. How does this change the coordinates of the vector x? Suppose we have a spinor which has only a spin up component in the old reference frame. How large are the spin up and spin down components of the spinor with respect to the rotated z axis? 8.3 The Cartesian coordinates {x, y, z} transform under rotations according to & % x → x  = exp ϕ · L · x.

(8.63)

Construct coordinates {X, Y, Z} which transform with the matrices (8.29) under rotations, & % X → X = exp iϕ ·  · X.

(8.64)

8.4 Construct the matrices s, ms |S|s, ms  = h ¯ j (ms ),j (ms ) for s = 3/2. Choose the index mapping ms → j (ms ) such that ⎛

3 1⎜ 0 3 = ⎜ ⎝ 2 0 0

⎞ 0 0 0 1 0 0⎟ ⎟. 0 −1 0 ⎠ 0 0 −3

(8.65)

Suppose we have an excited Lithium atom in a spin s = 3/2 state, which is described by the 4-component wave function j (x 1 , x 2 , x 3 ), 1 ≤ j ≤ 4. How does this wave function transform under a rotation around the x axis by an angle ϕ = π/2?

188

8 Spin and Addition of Angular Momentum Type Operators

8.5 Construct all the states |j, mj ,  = 1, s = 1/2 as linear combinations of the tensor product states | = 1, m ; s = 1/2, ms , using either the recursive construction from the state |j = 3/2, mj = 3/2,  = 1, s = 1/2 = | = 1, m = 1; s = 1/2, ms = 1/2 or Racah’s formula (8.59). Compare with the results from a symbolic computation program or an online applet for the calculation of Clebsch– Gordan coefficients.

Chapter 9

Stationary Perturbations in Quantum Mechanics

We denote a quantum system with a time-independent Hamiltonian H0 as solvable (or sometimes also as exactly solvable) if we can calculate the energy eigenvalues and eigenstates of H0 analytically. The harmonic oscillator and the hydrogen atom provide two examples of solvable quantum systems. Exactly solvable systems provide very useful models for quantum behavior in physical systems. The harmonic oscillator describes systems near a stable equilibrium, while the Hamiltonian with a Coulomb potential is an important model system for atomic physics and for every quantum system which is dominated by Coulomb interactions. However, in many cases the Schrödinger equation will not be solvable, and we have to go beyond solvable model systems to calculate quantitative properties. In these cases we have to resort to the calculation of approximate solutions. The methods developed in the present chapter are applicable to perturbations of discrete energy levels by timeindependent perturbations V of the Hamiltonian, H0 → H = H0 + V .

9.1 Time-Independent Perturbation Theory Without Degeneracies We consider a perturbation of a solvable time-independent Hamiltonian H0 by a time-independent term V , and for bookkeeping purposes we extract a coupling constant λ from the perturbation, H = H0 + V → H = H0 + λV .

(9.1)

After the relevant expressions for shifts of states and energy levels have been calculated to the desired order in λ, we usually subsume λ again in V , such that e.g. λφ (0) |V |ψ (0)  → φ (0) |V |ψ (0) . © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 R. Dick, Advanced Quantum Mechanics, Graduate Texts in Physics, https://doi.org/10.1007/978-3-030-57870-1_9

189

190

9 Stationary Perturbations in Quantum Mechanics

We know the unperturbed energy levels and eigenstates of the solvable Hamiltonian H0 , H0 |ψj(0)  = Ej(0) |ψj(0) .

(9.2) (0)

In the present section we assume that the energy levels Ej are not degenerate, and we want to calculate in particular approximations for the energy level Ei which arises from the unperturbed energy level Ei(0) due to the presence of the perturbation V . We will see below that consistency of the formalism requires that the differences (0) (0) |Ei − Ej | for j = i must have a positive minimal value, i.e. the unperturbed (0)

energy level Ei for which we want to calculate corrections has to be discrete.1 Orthogonality of eigenstates for different energy eigenvalues implies (0)

(0)

ψi |ψj  = δij .

(9.3)

In the most common form of time-independent perturbation theory we try to find an approximate solution to the equation H |ψi  = Ei |ψi 

(9.4)

in terms of power series expansions in the coupling constant λ, |ψi  =



λn |ψi(n) , ψi(0) |ψi(n≥1)  = 0, Ei =

n≥0



λn Ei(n) .

(9.5)

n≥0

Depending on the properties of V , these series may converge for small values of |λ|, or they may only hold as asymptotic expansions for |λ| → 0. The book by Kato [94] provides results and resources on convergence and applicability properties of the perturbation series. Here we will focus on the commonly used first and second order expressions for wave functions and energy levels. We can require (0)

(n)

ψi |ψi  = δn,0

(9.6)

because the recursion equation (9.12) below, which is derived without the assumption (9.6), does not determine these particular coefficients. One way to understand (n≥1)  into terms parallel and orthogonal this is to observe that we can decompose |ψi (0) to |ψi , (n≥1)

|ψi

1 This

(0)

(0)

(n≥1)

 = |ψi ψi |ψi

(n≥1)

 + |ψi

(0)

(0)

(n≥1)

 − |ψi ψi |ψi

(0)

.

(9.7)

condition is not affected by a possible degeneracy of Ei , as will be shown in Sect. 9.2.

9.1 Time-Independent Perturbation Theory Without Degeneracies (0)

(0)

(n≥1)

Inclusion of the parallel part |ψi ψi |ψi by a rescaling by

191

 in the zeroth order term, followed

−1

= 1 − ψi(0) |ψi(n≥1)  + O(λ2n ) 1 + ψi(0) |ψi(n≥1) 

(9.8)

to restore a coefficient 1 in the zeroth order term, affects only terms of order λn+1 or higher in the perturbation series. This implies that if we have solved the Schrödinger equation to order λn−1 with the constraint (0)

(m)

ψi |ψi

 = δm,0 ,

0 ≤ m ≤ n − 1,

(9.9)

then ensuring that constraint also to order λn preserves the constraint for the lower order terms. Therefore we can fulfill the constraint (9.6) to any desired order in which we wish to calculate the perturbation series. Substitution of the perturbative expansions into the Schrödinger equation H |ψi  = Ei |ψi  yields

(n)

λn H0 |ψi  +

n≥0





(n)

λn+1 V |ψi  =

n≥0

(m)

λm+n Ei

(n)

|ψi 

m,n≥0

=

n

λn Ei(m) |ψi(n−m) .

(9.10)

n≥0 m=0

This equation is automatically fulfilled at zeroth order. Isolation of terms of order λn+1 for n ≥ 0 yields H0 |ψi(n+1)  + V |ψi(n)  =

n+1

Ei(m) |ψi(n−m+1) ,

(9.11)

m=0

and projection of this equation onto |ψj(0)  yields (0)

(0)

(n+1)

Ej ψj |ψi

(0)

(n)

 + ψj |V |ψi  =

n

(m)

(0)

(n−m+1)

Ei

ψj |ψi

(n+1)

δij .



m=0

+Ei

(9.12)

We can first calculate the first order corrections for energy levels and wave functions from this equation, and then solve it recursively to any desired order.

192

9 Stationary Perturbations in Quantum Mechanics

First Order Corrections to the Energy Levels and Eigenstates The first order corrections are found from Eq. (9.12) for n = 0. Substitution of j = i implies for the first order shifts of the energy levels the result (1)

Ei (0)

and j = i yields with Ei

(0)

(0)

= ψi |V |ψi ,

(9.13)

(0)

= Ej the first order shifts of the energy eigenstates (0)

(0)

(1)

ψj |ψi  =

(0)

ψj |V |ψi  (0)

Ei

(0)

− Ej

(9.14)

.

Recursive Solution of Eq. (9.12) for n ≥ 1 We first observe that j = i in Eq. (9.12) implies with the condition (9.6) (n+1) Ei

=

(0) (n) ψi |V |ψi  −

n

(m)

Ei

(0)

(n−m+1)

 = ψi |V |ψi ,

(0)

(n)

ψi |ψi

(0)

(n)

(9.15)

(n−m+1)

(9.16)

m=1

and i = j yields

(0)

Ei

(0)

− Ej



(0)

(n+1)

ψj |ψi

 = ψj |V |ψi  −

n

(m)

Ei

(0)

ψj |ψi

.

m=1

The right-hand side of both equations depends only on lower order shifts of energy levels and eigenstates. Therefore these equations can be used for the recursive solution of Eq. (9.12) to arbitrary order.

Second Order Corrections to the Energy Levels and Eigenstates Substitution of n = 1 into Eq. (9.15) yields with (9.14) and  |ψk(0) ψk(0) | = 1 k

(9.17)

9.1 Time-Independent Perturbation Theory Without Degeneracies

193

the second order shift (2)

Ei

=

 (0) (0) (0) (0) ψi |V |ψk ψk |V |ψi  (0)

k=i

Ei

(0)

− Ek

=

 (0) (0) |ψi |V |ψk |2 k=i

(0)

Ei

(0)

− Ek

.

(9.18)

States in the continuous part of the spectrum of H0 will also contribute to the shifts (0) in energy levels and eigenstates. It is only required that the energy level Ei , for which we want to calculate the corrections, is discrete and does not overlap with any continuous energy levels. Note that Eq. (9.18) implies that the second order correction to the ground state energy is always negative. For the eigenstates, Eq. (9.16) yields with the first order results (9.13,9.14) the equation (recall i = j in (9.16)) (0)

(2)

ψj |ψi  =

 ψj(0) |V |ψk(0) ψk(0) |V |ψi(0)  % (0) & (0) &% (0) k=i Ei − Ej Ei − Ek(0) (0)

−

(0)

(0)

(0)

ψi |V |ψi ψj |V |ψi  . % (0) (0) &2 Ei − Ej

(9.19)

Now we can explain why it is important that our original unperturbed energy level Ei(0) is discrete. To ensure that the n-th order corrections to the energy levels and eigenstates in Eqs. (9.5) are really of order λn (or smaller than all previous terms), (0) (0) the matrix elements |ψj |V |ψk | of the perturbation operator should be at most (0)

of the same order of magnitude as the energy differences |Ei

(0)

− Ej | between the

unperturbed level Ei(0) and the other unperturbed energy levels in the system. This implies in particular that the minimal absolute energy difference between Ei(0) and the other unperturbed energy levels must not vanish, i.e. Ei(0) must be a discrete energy level. Equations (9.13) and (9.18) (and their counterparts (9.32) and (9.44) in degenerate perturbation theory below) used to be the most frequently employed equations of time-independent perturbation theory, because historically many experiments were concerned with spectroscopic determinations of energy levels. However, measurements e.g. of local electron densities or observations of wave functions (e.g. in scanning tunneling microscopes or through X-ray scattering using synchrotrons) are very common nowadays, and therefore the corrections to the states are also directly relevant for the interpretation of experimental data.

194

9 Stationary Perturbations in Quantum Mechanics

Summary of Non-degenerate Perturbation Theory in Second Order If we include λ with V , the states and energy levels in second order are (0)

(1)

(2)

(0)

|ψi  = |ψi  + |ψi  + |ψi  = |ψi  +

−

 j =i

 j =i

(0)

|ψj(0) 

(0)

Ei

(0)

− Ej

(0)

(0)

|ψj 



(0)

ψj |V |ψi 

(0)

(0)

(0)

(0)

(0)

ψj |V |ψk ψk |V |ψi  |ψj(0)  % (0) (0) &% (0) (0) & j,k=i Ei − Ej Ei − Ek

+

(0)

(0)

ψj |V |ψi ψi |V |ψi  % (0) &2 Ei − Ej(0)

(9.20)

and Ei =

Ei(0)

+ ψi(0) |V |ψi(0)  +

 |ψi(0) |V |ψj(0) |2 (0)

j =i

Ei

(0)

− Ej

.

(9.21)

The second order states |ψi  are not normalized any more, ψi |ψj  = δij + O(λ2 )δij .

(9.22)

Normalization is preserved in first order due to ψi(0) |ψj(1)  + ψi(1) |ψj(0)  = 0,

(9.23)

but in second order we have (0)

(2)

(1)

(1)

(2)

(0)

ψi |ψj  + ψi |ψj  + ψi |ψj  =

 (0) (0) |ψi |V |ψk |2 δ . % (0) (0) &2 ij k=i E − E i k

(9.24)

However, we can add to the leading term |ψi(0)  in |ψi  a term of the form |ψi(0) O(λ2 ) and still preserve the master equation (9.12) to second order. We can therefore rescale (9.20) by a factor [1 + O(λ2 )]−1/2 to a normalized second order state   |ψi(0) |V |ψj(0) |2 ψj(0) |V |ψi(0)  1 (0) |ψi  = |ψi(0)  − |ψi(0)  + |ψ  % (0) j (0) &2 2 j =i E j =i Ei(0) − Ej(0) i − Ej +

 j,k=i

(0) |ψj 

ψj(0) |V |ψk(0) ψk(0) |V |ψi(0)  % (0) &% & Ei − Ej(0) Ei(0) − Ek(0)

9.2 Time-Independent Perturbation Theory with Degenerate Energy Levels

−

 j =i

(0)

(0) |ψj 

(0)

(0)

195

(0)

ψj |V |ψi ψi |V |ψi  . % (0) (0) &2 Ei − Ej

(9.25)

Now the second order shift is not orthogonal to |ψi(0)  any more, but we still have a solution of Eq. (9.12) to second order.

9.2 Time-Independent Perturbation Theory With Degenerate Energy Levels Now we admit degeneracy of energy levels of our unperturbed Hamiltonian H0 . Time-independent perturbation theory in the previous section repeatedly involved division by energy differences [Ei(0) − Ej(0) ]i=j . This will not be possible any more for pairs of degenerate energy levels, and we have to carefully reconsider each step in the previous derivation if degeneracies are involved. The full Hamiltonian and the 0-th order results are now (0)

(0)

(0)

H = H0 + λV , H0 |ψj α  = Ej |ψj α ,

(9.26)

where Greek indices denote sets of degeneracy indices. For example, if H0 would correspond to a hydrogen atom, the quantum number j would correspond to the principal quantum number n of a bound state or the wave number k of a spherical Coulomb wave, and the degeneracy index α would correspond to the set of angular momentum quantum number, magnetic quantum number, and spin projection, α = {, m , ms }. For the same reasons as in Eq. (9.19), the energy level for which we wish to calculate an approximation must be discrete, i.e. the techniques developed in this chapter can be used to study perturbations of the bound states of hydrogen atoms, but not perturbations of Coulomb waves. (0) We denote the degeneracy subspace to the energy level Ej as Ej and the projector on Ej is (0)

Pj

=



(0)

(0)

|ψj α ψj α |.

(9.27)

α

As in the previous section, we wish to calculate approximations for the energy level Eiα and corresponding eigenstates |ψiα , H |ψiα  = Eiα |ψiα , which arise from the (0) energy level Ei(0) and the eigenstates |ψiα  due to the perturbation V . The energy (0) level Ei may split into several energy levels Eiα because the perturbation might lift the degeneracy of Ei(0) . We will actually assume that the perturbation V lifts the (1) (1) degeneracy of the energy level Ei(0) already at first order, Eiα = Eiβ if α = β.

196

9 Stationary Perturbations in Quantum Mechanics

The Rayleigh-Ritz-Schrödinger ansatz is |ψiα  =





(n) (0) (n≥1) λn |ψiα , ψiα |ψiα  = 0, Eiα =

n≥0

(n) λn Eiα .

(9.28)

n≥0

Substitution into the full time-independent Schrödinger equation yields

(n)

λn H0 |ψiα  +

n≥0





(n)

λn+1 V |ψiα  =

n≥0

(m)

(n)

λm+n Eiα |ψiα 

m,n≥0 n

=

(m)

(n−m)

λn Eiα |ψiα

.

(9.29)

n≥0 m=0

This yields in (n + 1)-st order for n ≥ 0 the equation (n+1)

H0 |ψiα

(n)

 + V |ψiα  =

n+1

(m)

(n−m+1)

Eiα |ψiα

.

(9.30)

m=0 (n≥1)

We determine the corrections |ψiα  to the wave functions through their projec(0) (n≥1) tions ψjβ |ψiα  onto the basis of unperturbed states. Projection of Eq. (9.30) yields (0)

(0)

(n+1)

Ej ψjβ |ψiα

(0)

n

(n)

 + ψjβ |V |ψiα  =

(m)

(0)

(n−m+1)

Eiα ψjβ |ψiα



m=0 (n+1) + Eiα δij δαβ .

(9.31)

First Order Corrections to the Energy Levels The first order equations (n = 0 in Eq. (9.31)) yield for j = i and β = α the equation (1)

(0)

(0)

Eiα = ψiα |V |ψiα ,

(9.32)

while j = i, α = β imposes a consistency condition on the choice of basis of unperturbed states,  (0) (0)  ψiβ |V |ψiα 

β=α

= 0,

(9.33)

9.2 Time-Independent Perturbation Theory with Degenerate Energy Levels

197

This condition means that we have to diagonalize V first within each degeneracy subspace Ei in the sense (0)

(1)

(0)

V |ψiα  = Eiα |ψiα  +

 j =i β

(0)

(0)

(0)

|ψjβ ψjβ |V |ψiα ,

(9.34)

before we can use the perturbation ansatz (9.28), and according to (9.32) the (1) first order energy corrections Eiα are the corresponding eigenvalues in the i-th (1) are all we care about, degeneracy subspace. If the first order energy corrections Eiα this means that we can calculate them from the eigenvalue conditions   (0) (0) (1) |V |ψiα  − Eiα δαβ = 0, det ψiβ

(9.35)

using any initial choice of unperturbed orthogonal energy eigenstates. But that would achieve only a very limited objective. As also indicated in Eq. (9.34), diagonalization within the subspaces means (0) (0) only diagonalization of the operators Pi V Pi , which does not amount to total diagonalization of V ,

(0)

(0)

Pi V Pi

= V =

i



(0)

(0)

Pi V Pj .

(9.36)

i,j (0)

(0)

We still will have non-vanishing transition matrix elements ψjβ |V |ψiα  = 0 between different degeneracy subspaces i = j .

First Order Corrections to the Energy Eigenstates Setting i = j in Eq. (9.31) yields a part of the first order corrections to the wave functions, (0) (1) ψjβ |ψiα  =

(0) (0) |V |ψiα  ψjβ

Ei(0) − Ej(0) (0)

.

(9.37)

(1)

However, this yields only the projections ψjβ |ψiα  of the first order corrections (1)  onto the unperturbed states for j = i. We need to use j = i in the second |ψiα (0) (1) |ψiα , (β = α), for the first order equations to calculate the missing terms ψiβ order corrections.

198

9 Stationary Perturbations in Quantum Mechanics

Equation (9.31) yields for n = 1, j = i and β = α the equation  (0) (1)  ψiβ |V |ψiα 

β=α

 (1) (0) (1)  = Eiα ψiβ |ψiα 

(9.38)

β=α

and after substitution of Eqs. (9.32), (9.33), and (9.37)

(1) Eiα

(1) − Eiβ





(0) (1)  ψiβ |ψiα  β=α

 (0) (0) (0) (1) = ψiβ |V |ψj γ ψj γ |ψiα  j =i γ

(0) (0) (0) (0)  ψiβ |V |ψj γ ψj γ |V |ψiα 

=

Ei(0) − Ej(0)

j =i γ

, (9.39)

i.e. we find the missing pieces of the first order corrections to the states  (0) (1)  ψiβ |ψiα 

β=α

=

1 (0) (0) (0) (0) ψiα |V |ψiα  − ψiβ |V |ψiβ 

×

 ψ (0) |V |ψ (0) ψ (0) |V |ψ (0)  iβ jγ jγ iα (0)

j =i γ

Ei

(9.40)

(0)

− Ej

(1)

(1)

if V has removed the degeneracy between |ψiα  and |ψiβ  in first order, Eiα = Eiβ .

Recursive Solution of Eq. (9.31) for n ≥ 1 We first rewrite Eq. (9.31) by inserting 1=

 k,γ

(0)

(0)

|ψkγ ψkγ |

(9.41)

in the matrix element of V , and using Eqs. (9.32) and (9.33): (0) (0) (n+1) (1) (0) (n) Ej ψjβ |ψiα  + Ejβ ψjβ |ψiα  +

(0)

(0)

(n+1)

= Ei ψjβ |ψiα

(1)

(0)

(n)

 (0) (0) (0) (n) ψjβ |V |ψkγ ψkγ |ψiα  k=j γ

 + Eiα ψjβ |ψiα  + (n ≥ 2)

n

(m)

(0)

(n−m+1)

Eiα ψjβ |ψiα



m=2 (n+1)

+ Eiα

δij δαβ .

(9.42)

9.2 Time-Independent Perturbation Theory with Degenerate Energy Levels

199

Substitution of j = i and β = α yields (n+1)

Eiα

 (0) (0) (0) (n) ψiα |V |ψkγ ψkγ |ψiα ,

=

(9.43)

k=i γ

where Eqs. (9.28) and (9.33) have been used. The second order correction is in particular with Eq. (9.37): (2)

Eiα =

(0) (0)  |ψjβ |V |ψiα |2 (0)

j =i β

Ei

(0)

− Ej

(9.44)

.

We find again that the second order correction to the ground state energy is always negative. For the higher order shifts of the states we find for j = i in Eq. (9.42) n

 (0) (n+1) (0) (n) (m) (0) (n−m+1) Ei(0) − Ej(0) ψjβ |ψiα  = ψjβ |V |ψiα − Eiα ψjβ |ψiα  m=1

=

(0) (n) (0) (0) (0) (n) ψjβ |V |ψiα  − ψiα |V |ψiα ψjβ |ψiα 

 n−1 (0) (0) (0) (m) (0) (n−m) ψiα |V |ψkγ ψkγ |ψiα ψjβ |ψiα ,

− (n ≥ 2)

m=1

(0)

(n+1)

which gives us the contributions ψjβ |ψiα

  

function corrections. Substitution of i = j , α = β yields finally

(1) Eiα

(1) − Eiβ

− (n ≥ 2)





(0) (n≥1)  ψiβ |ψiα  β=α

n

(m)

(0)

 n−1 m=1

k=i γ

(0)

=



(0)

(1)

(1)

k=i γ (0)

(m)

(0)

(0)

β=α

(0)

(n)

ψiβ |V |ψkγ ψkγ |ψiα  (0)

(n−m)

ψiα |V |ψkγ ψkγ |ψiα ψiβ |ψiα

 (0) (n)  This gives us the missing pieces ψiβ |ψiα  correction for Eiα = Eiβ .

to the (n + 1)-st order wave

k=i γ

(n−m+1)

Eiα ψiβ |ψiα

j =i

 (0) (0) (0) (n) = ψiβ |V |ψkγ ψkγ |ψiα 

m=2

− (n ≥ 2)

(9.45)

k=i γ

.

(9.46)

of the n-th order wave function

200

9 Stationary Perturbations in Quantum Mechanics

Summary of First Order Shifts of the Level Ei(0) if the Perturbation Lifts the Degeneracy of the Level We must diagonalize the perturbation operator V within the degeneracy subspace Ei (0) in the sense of (9.34), i.e. we must choose the unperturbed eigenstates |ψiα  such that the equation (0)

(0)

(1)

ψiα |V |ψiβ  = Eiα δαβ

(9.47)

also holds for α = β. The projections of the first order shifts of the energy eigenstates onto states in other degeneracy sectors are 

(0) (1)  ψjβ |ψiα  j =i

(0)

=

(0)

ψjβ |V |ψiα  (0)

Ei

(0)

− Ej

(9.48)

,

and the projections within the degeneracy sector are  (0) (1)  ψiβ |ψiα 

β=α

=

1 (0) (0) (0) (0) ψiα |V |ψiα  − ψiβ |V |ψiβ 

×

 ψ (0) |V |ψ (0) ψ (0) |V |ψ (0)  iβ jγ jγ iα j =i γ

(0)

Ei

(0)

− Ej

.

(9.49)

This requires that the first order shifts have completely removed the degeneracies in (1) (1) the i-th energy level, Eiβ = Eiα for β = α.

9.3 Problems 9.1 A one-dimensional harmonic oscillator is perturbed by a term V = λ[(a + )2 + a 2 ]2 .

(9.50)

Calculate the first and second order corrections to the ground state energy and wave function. 9.2 An atom on a surface is prevented from moving along the surface through a two-dimensional potential V (x, y) =

1 mω2 (x 2 + y 2 ) + Ax 4 + By 4 , 2

A ≥ 0,

B ≥ 0.

(9.51)

9.3 Problems

201

Find an approximation H0 for the Hamiltonian of the atom where you can write down exact energy levels and eigenstates for the atom. Use the remaining terms in H − H0 to calculate first order corrections to the energy levels and eigenstates of the atom. 9.3 Which results do you get for the perturbed system from 9.2 in second order perturbation theory? 9.4 Suppose that the perturbation V has removed all degeneracies in all energy (0) (1) levels of an unperturbed system. Show that all the first order states |ψiα  + |ψiα  are orthonormal in first order. 9.5 A hydrogen atom is perturbed by a static electric field E = Eez in z direction. This field induces an extra potential V = − e = eEz

(9.52)

in the Hamiltonian for relative motion. 9.5a Calculate the shift of the ground state energy up to second order in E. 9.5b Calculate the shift of the ground state wave function up to second order in E. 9.5c Which constraints on E do you find from the requirement of applicability of perturbation theory? 9.6 Calculate the first order shifts of the n = 2 level of hydrogen under the perturbation (9.52). 9.7 A two-level system has two energy eigenstates |E±  with energies   E |E± , H0 |E±  = E0 ± 2

E = 0.

(9.53)

We can use 2-spinor notation such that a general state in the two-level system is |ψ =



 |E± E± |ψ → ψ =

±

 ψ1 , ψ2

ψ1 = E+ |ψ,

ψ2 = E− |ψ.

The Hamiltonian in 2-spinor notation is H0 = E0 1 +

E σ . 2 3

(9.54)

We now perturb the Hamiltonian H0 → H = H0 + V through a term V =

V1 V2 σ1 + σ . 2 2 2

(9.55)

202

9 Stationary Perturbations in Quantum Mechanics

9.7a Calculate the first order corrections to the energy levels and eigenstates due to the perturbation V . 9.7b Calculate the second order corrections to the energy levels and eigenstates due to the perturbation V . 9.7c The Hamiltonian H is a hermitian 2 × 2 matrix which can be diagonalized exactly. Calculate the exact energy levels and eigenstates of H . Compare with the perturbative results from 9.7a and 9.7b. 9.8 Which consistency conditions in the degeneracy subspace Ei would you find if the perturbation V does not remove the degeneracy in that subspace in first order? (1)

(1)

Solution The derivation of (9.40) shows that if we still have Eiα = Eiβ for all degeneracy indices in Ei , consistency of the second order equation requires that not (1) (0) (0) just the operator Vi = Pi V Pi is diagonal, but also that the operator (0)

Vi(2) = Pi(0) V

1 − Pi (0)

Ei

− H0

V Pi(0)

(9.56) (1)

is diagonal. However, consistency of the simultaneous diagonalization of Vi (2) Vi then also implies the condition   (1) (2) (0) Vi , Vi = Pi V

 (0) Pi V

1 − Pi(0) (0)

Ei

− H0

−

1 − Pi(0) (0)

Ei

− H0

and

 (0) V Pi

(0)

V Pi

= 0.

If V preserves the degeneracy in Ei , but these consistency conditions cannot be met, then H = H0 + V apparently does not have a complete set of eigenstates which scale analytically under scaling V → λV of the perturbation.

Chapter 10

Quantum Aspects of Materials I

Quantum mechanics is indispensable for the understanding of materials. In return, solid state physics provides beautiful illustrations for the impact of quantum dynamics on allowed energy levels in a system, for wave-particle duality, and for applications of perturbation theory. In the present chapter we will focus on Bloch’s theorem, the duality between Bloch and Wannier states, the emergence of energy bands in crystals, and the emergence of effective mass in kp perturbation theory. We will do this for onedimensional lattices, since this captures the essential ideas. Students who would like to follow up on our introductory exposition and understand the profound impact of quantum mechanics on every physical property of materials at a deeper level should consult the monographs of Callaway [23], Ibach and Lüth [86], Kittel [95] or Madelung [109], or any of the other excellent texts on condensed matter physics— and they should include courses on condensed matter physics in their curriculum!

10.1 Bloch’s Theorem Electrons in solid materials provide a particularly beautiful realization of waveparticle duality. Bloch’s theorem covers the wave aspects of this duality. From a practical perspective, Bloch’s theorem implies that we can discuss electrons in terms of states which sample the whole lattice of ion cores in a solid material. This has important implications for the energy levels of electrons in materials, and therefore for all physical properties of materials. It is useful to recall the theory of discrete Fourier transforms as a preparation for the proof of Bloch’s theorem. We write the discrete Fourier expansion for functions f (x) with periodicity a as

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 R. Dick, Advanced Quantum Mechanics, Graduate Texts in Physics, https://doi.org/10.1007/978-3-030-57870-1_10

203

204

10 Quantum Aspects of Materials I

f (x) =

nx  . fn exp 2π i a n=−∞ ∞

(10.1)

The orthogonality relation 1 a



a 0



nx  mx  exp −2π i = δmn dx exp 2π i a a

(10.2)

yields the inversion fn =

1 a



a 0

nx  , dx f (x) exp −2π i a

(10.3)

and substituting this back into Eq. (10.1) yields a representation of the δ-function in a finite interval of length a,   ∞ 1 x − x = δ(x − x  ), exp 2π in a n=−∞ a

(10.4)

or equivalently ∞

exp(inξ ) = 2π δ(ξ ).

(10.5)

n=−∞

Equation (10.4) is the completeness relation for the Fourier monomials on an interval of length a. The Hamiltonian for electrons in a lattice with periodicity a is H =

p2 + V (x), 2m

(10.6)

where the potential operator has the periodicity of the lattice,  V (x) = V (x + a) = exp

   i i ap V (x) exp − ap . h¯ h¯

(10.7)

This implies    i i exp ap H = H exp ap , h¯ h¯ 

(10.8)

and therefore the eigenspace of H with eigenvalue En can be decomposed into eigenspaces of the lattice translation operator

10.1 Bloch’s Theorem

205



 i T (a) = exp ap . h¯

(10.9)

The eigenvalues of this unitary operator must be pure phase factors,1 

 i exp ap |En , k = exp(ika) |En , k. h¯

(10.10)

Let us repeat this result in the x-representation:    i d x|En , k = x + a|En , k x| exp ap |En , k = exp a dx h¯ 

= exp(ika) x|En , k.

(10.11)

This means that the energy eigenstate x|En , k ≡ ψn (k, x) has exactly the same periodicity properties under lattice translations as the plane wave x|k = √ exp(ikx)/ 2π . The ratio ψn (k, x)/x|k must therefore be a periodic function! This is Bloch’s theorem in solid state physics:2 Energy eigenstates in a periodic lattice can always be written as the product of a periodic function un (k, x + a) = un (k, x) with a plane wave,  ψn (k, x) =

a exp(ikx)un (k, x). 2π

(10.12)

The quasiperiodicity parameter k (multiplied with h) ¯ has momentum-like properties, but it is not the momentum En , k|p|En , k in the state |En , k. Therefore it is often denoted as a quasimomentum or a pseudomomentum. Periodicity of the modulation factor un (k, x) implies the expansions un (k, x) =

∈Z

1 un, (k) = a



a 0

  x , un, (k) exp 2π i a

 x . dx un (k, x) exp −2π i a

(10.13)



(10.14)

We denote the eigenfunctions ψn (k, x) ≡ x|En , k of the lattice Hamiltonian as Bloch functions or equivalently as the x representation of the Bloch states |En , k. The corresponding periodic functions un (k, x) ≡ x|un (k) will be denoted as

1 This

is a consequence of Schur’s Lemma in group theory: Abelian symmetry groups have onedimensional irreducible representations. 2 Bloch [14]. As a mathematical theorem in the theory of differential equations it is known as Floquet’s theorem due to Floquet [55].

206

10 Quantum Aspects of Materials I

Bloch factors. Equation (10.12) for the Bloch state |En , k reads in basis free notation  a |En , k = (10.15) exp(ikx)|un (k). 2π For arbitrary  ∈ Z the eigenvalues of the lattice translations satisfy    2π  a , exp(ika) = exp i k + a

(10.16)

and therefore the quasimomentum k can be restricted to the region −

π π t0 in time. Of course, the same results apply for backward evolution. However, the discretization into time steps (t − t0 )/N imply that consecutive steps are either always later or always earlier depending on t > t0 or t < t0 , respectively. Therefore path integrals with factors like x(t1 )x(t2 ) in the integrand correpond to time ordered matrix elements in canonical quantization, but whether time ordering refers to later times or earlier times depends on whether we are studying forward or backward evolution in time. Usually we are interested in forward evolution, i.e. we assume t > t0 in the following. A virtue of the path integral is that it explains the principle of stationary action of classical paths as a consequence of dominant contributions from those trajectories where small fluctuations of the path do not yield cancellation of the integral from phase fluctuations. As a relatively simple exercise, let us see how this reproduces the x representation (4.115) of the free propagator. The integrations in (14.34) for V (x) = 0 include a set of N − 1 Gaussian integrals. The first integral over d 3 x 1 yields 3   1 mN mN 2 exp − (x 2 − x 0 ) 2π ih(t 2ih(t ¯ − t0 − i) ¯ − t0 − i) 2      mN x2 + x0 2 3 2 x1 − × d x 1 exp − 2ih(t 2 ¯ − t0 − i)   1 mN 1 = √ 3 exp − (x 2 − x 0 )2 . 2i h(t − t − i) 2 ¯ 0 2

(14.37)

320

14 Path Integrals in Quantum Mechanics

Next we evaluate the x 2 integral and then work consecutively through all the integrals. This reproduces always a similar result with minor variations. One can show by induction with respect to I that the x I integral yields 3   1 mN mN 2 exp − (x I +1 − x 0 ) 2π ih(t 2ih(t ¯ − t0 − i)I ¯ − t0 − i) I + 1      I +1 mN I

x0  2 3 x I +1 + × d x I exp − xI − 2ih(t I +1 I ¯ − t0 − i) I   1 mN 1 (14.38) exp − =√ (x I +1 − x 0 )2 . 3 2i h(t − t − i) I + 1 ¯ 0 I +1

After the final integrations over x N −1 and x N (which is trivial due to the δ function in (14.34)), we are left with x, t|x 0  = x|U (t, t0 )|x 0     3 m m (x − x 0 )2 exp − = , 2π ih(t 2ih(t ¯ − t0 − i) ¯ − t0 − i)

(14.39)

which is indeed the x representation (4.115) of the free propagator. Note that the classical trajectory of the particle from the location x 0 at time t0 to the location x at time t is given by x cl (t  ) = x 0 +

t − t x − x0  t  − t0 , (t − t0 ) = x + x0 t − t0 t − t0 t0 − t

(14.40)

and therefore the factor in the exponent of the free propagator is just the action functional evaluated on the classical trajectory, m (x − x 0 )2 = S[x cl (t  )]. 2 t − t0

(14.41)

This holds in general for propagators where the Lagrange function contains at most second order terms in particle velocities and locations, and the path integral formulation is particularly well suited to prove this. If the Lagrange function contains at most second order terms in x˙ and x, then due to fixed initial and final points x(t0 ) ≡ x 0 and x(t) ≡ x, the action functional for all admissible paths x(t  ) is exactly 1 S[x(t )] = S[x cl (t )] + 2 



×

δ2S δx(t  )δx(t  )



t

dt t0





t

% & dt  x(t  ) − x cl (t  )

t0

% & · x(t  ) − x cl (t  ) ,

(14.42)

14.3 Path Integrals in Scattering Theory

321

see Problem 14.1. Functional integration over exp(iS[x(t  )]/h) ¯ then yields a constant from the Gaussian integral over the fluctuations x(t  ) − x cl (t  ), and a remnant exponential factor,  x|U (t, t0 )|x 0  ∼ exp

 i S[x cl (t  )] . h¯

(14.43)

However, note that this requires vanishing fluctuations at the boundaries, x(t(0) ) − x cl (t(0) ) = 0. Otherwise boundary terms involving x(t(0) ) − x cl (t(0) ) will appear in the exponent. This is important e.g. in scattering theory in the following section, when we are really concerned with fixed initial and final momenta rather than locations.

14.3 Path Integrals in Scattering Theory We have seen in Chap. 13 that the calculation of transition probabilities or scattering cross section from an initial state |ψi (t  ) to a final state |ψf (t) requires the calculation of the scattering matrix element Sf i (t, t  ) = ψf (t)|U (t, t  )|ψi (t  ) = ψf |UD (t, t  )|ψi ,

(14.44)

      i i t i  dτ H (τ ) exp − H0 t UD (t, t ) = exp H0 t T exp − h¯ h¯ t  h¯

(14.45)

where 



is the time evolution operator on the states in the interaction picture. We also recall that the usual default definition of the scattering matrix involves t → ∞, t  → −∞, Sf i ≡ Sf i (∞, −∞). For the following discussion it is convenient to relabel initial and final times as t  → ti , t → tf . Equation (14.35) then implies a connection between scattering matrix elements and path integrals,  i H0 tf |x f  ti →−∞,tf →∞ h¯      x(tf )=x f i i × D 3 x(t) exp S[x(t)] x i | exp − H0 ti |ψi . (14.46) h¯ h¯ x(ti )=x i 

Sf i =

lim





d 3xf

d 3 x i ψf | exp

This is still a mixed formula involving both canonical operators and a path integral. We now assume that our initial and final states are momentum eigenstates |ψi  = |pi  and |ψf  = |pf , and we also assume that the scattering potential V (x, t) is analytic with finite range. The free Hamiltonian for the free-free scattering problem is H0 = p2 /2m. The resulting scattering matrix element is then

322

14 Path Integrals in Quantum Mechanics

 Sf i =

lim

ti →−∞,tf →∞

 d 3xf

 d 3xi

x(tf )=x f

 D 3 x(t) exp

x(ti )=x i

  2 2 i p f tf − p i ti 1 exp × + pi · x i − pf · x f h¯ 2m (2π h) ¯ 3

i S[x(t)] h¯  .



(14.47)

For the perturbative evaluation of (14.47) we introduce an auxiliary external force F (t), such that the Lagrange function including the scattering potential V (x, t) takes the form L=

m 2 x˙ (t) − V (x(t), t) + F (t) · x(t). 2

(14.48)

The path integral in (14.47) then takes the form 

   tf  tf ∞ 1 i 3 D x(t) exp S[x(t)] = D x(t) dt1 . . . dtn (ih) h¯ ¯ n n! ti ti n=0   tf

 i m 2 x˙ (t) + F (t) · x(t) × V (x(t1 ), t1 ) . . . V (x(tn ), tn ) exp dt 2 h¯ ti    tf  tf  ∞ 1 h¯ δ ... , t dt . . . dt V = D 3 x(t) 1 n 1 (ih) i δF (t1 ) ¯ n n! ti ti n=0    tf

  i δ h¯ m 2 , tn exp x˙ (t) + F (t) · x(t) ×V dt i δF (tn ) 2 h¯ ti      i tf  h¯ δ 3  ,t dt V = D x(t) exp − i δF (t  ) h¯ ti   tf

 m 2 i x˙ (t) + F (t) · x(t) . dt (14.49) × exp 2 h¯ ti



3

Evaluation of the Gaussian integrals as in Eq. (14.34) for V (x) = 0 reproduces the canonical perturbation series (13.42). However, a different representation is gotten if we pull the variational derivative operators V (− ihδ/δF (t), t) out of the path ¯ integral, 

     i tf  i h¯ δ  ,t Z[F ], dt V D x(t) exp S[x(t)] = exp − h¯ h¯ ti i δF (t  ) 

3

with  Z[F ] =

D 3 x(t) exp

  tf

 i m 2 x˙ (t) + F (t) · x(t) . dt h¯ ti 2

(14.50)

14.3 Path Integrals in Scattering Theory

323

It is useful to have a convolution notation for the following calculations. We define  (G ◦ F )(t) ≡

∞

−∞

dt  G(t, t  )F (t  )

(14.51)

and ˙ ◦ F )(t) ≡ (G



∞ −∞

∂ G(t, t  )F (t  ). ∂t

dt 

(14.52)

Partial integration yields the following representation of the action of a particle under the influence of a force F (t) for every Green’s function (14.8),  S[x, F ] =

tf

dt

m



x˙ 2 (t) + F (t) · x(t)

2   tf ˙ ◦ F )(t) 2 (G 1 m tf ˙ − dt x(t) + dt F (t) · (G ◦ F )(t) = 2 ti m 2m ti   (G ◦ F )(tf ) ˙ ◦ F )(tf ) + x(tf ) − · (G 2m   (G ◦ F )(ti ) ˙ ◦ F )(ti ). · (G (14.53) − x(ti ) − 2m ti



The trajectory x(t) between x i and x f appears only in the free particle action for the trajectory X(t) = x(t) −

1 (G ◦ F )(t), m

(14.54)

¨ which classically satisfies X(t) = 0. Therefore the path integral (14.50) can be evaluated in terms of the result for the free particle,   3 & m i % ˙ ◦ F )(tf ) − Xi · (G ˙ ◦ F )(ti ) exp X f · (G 2π ih(t h¯ ¯ f − ti )   " i ! ˙ ◦ F )(tf ) − (G ◦ F )(ti ) · (G ˙ ◦ F )(ti ) (G ◦ F )(tf ) · (G × exp 2mh¯  %  &2  tf Xf − Xi i + × exp im dt F (t) · (G ◦ F )(t) 2h(t 2mh¯ ti ¯ f − ti )    tf i = Xf |U0 (tf , ti )|Xi  exp dt F (t) · (G ◦ F )(t) 2mh¯ ti 

Z[F ] =

324

14 Path Integrals in Quantum Mechanics



 " i ! ˙ ˙ (G ◦ F )(tf ) · (G ◦ F )(tf ) − (G ◦ F )(ti ) · (G ◦ F )(ti ) × exp 2mh¯   & i % ˙ ◦ F )(tf ) − Xi · (G ˙ ◦ F )(ti ) . X f · (G (14.55) × exp h¯ We can summarize our results in the equations Sf i =

      i tf h¯ δ , t Sf i [F ] exp − dt V , ti →−∞,tf →∞ i δF (t) h¯ ti F =0 lim

1 Sf i [F ] = (2π h) ¯ 3 



(14.56)

 3

d Xf

d 3 Xi Z[F ](Xf , tf ; X i , ti ) 

  1 i × exp i exp p i · Xi + (G ◦ F )(ti ) 2mh¯ m h¯    i 1 × exp − pf · Xf + (G ◦ F )(tf ) . (14.57) m h¯ p2f tf − p 2i ti



The integrals over Xf and Xi amount to a Gaussian integral involving Xf − Xi and an integral over a Fourier monomial involving Xi . Evaluation of the integrals yields  Sf i [F ] = exp

i 2mh¯



tf

 dt F (t) · (G ◦ F )(t)

ti

 " ! " i ! ˙ ◦ F )(tf ) · tf (G ˙ ◦ F )(tf ) − (G ◦ F )(tf ) 2p f − (G 2mh¯   " ! " i ! ˙ ˙ 2pi − (G ◦ F )(ti ) · ti (G ◦ F )(ti ) − (G ◦ F )(ti ) × exp − 2mh¯ & % ˙ ◦ F )(tf ) − p i + (G ˙ ◦ F )(ti ) . (14.58) × δ p f − (G 

× exp

& % For consistency we note that this reproduces the correct result Sf i = δ pf − p i for the free particle. The δ function implies conservation of the free momentum ˙ ◦ F )(t), or equivalently matching of the external momenta under P = p(t) − (G evolution with the force F (t),  pf = pi +

tf

dt F (t).

(14.59)

ti

Please note that it is not possible to impose simultaneous boundary conditions   ∂   tf = G(tf , t  ) G(t, t ) ∂t t=tf

(14.60)

14.3 Path Integrals in Scattering Theory

325

and   ∂   G(t, t ) ti = G(ti , t  ), ∂t t=ti

(14.61)

because such a Green’s function does not exist. As a consequence it is not possible to eliminate the initial and final state dependent exponentials in the scattering matrix through a clever choice of the Green’s function. This is of course as it should be, because the scattering amplitude Mf i = i(Sf i −δf i )/δ(P f −P i ) generically must depend on the initial and final states. The functionals S[x, F ] (14.53), Z[F ] (14.55) and Sf i [F ] (14.58) are all independent of the boundary functions α(t  ) and β(t  ) in the general Green’s function (14.8). The easiest way to show this is by observing that the functionals are invariant under shifts (G ◦ F )(t) → (G ◦ F )(t) + At + B

(14.62)

with constant vectors A and B. For Z[F ] the demonstration has to take into account that X f and Xi contain (G ◦ F )(tf ) or (G ◦ F )(ti ) according to (14.54). We are therefore free to use e.g. the Green’s functions Gi (t, t  ) (14.3) or Gf i (t, t  ) (14.6), or the retarded Green’s function Gret (t, t  ) = (t − t  )(t − t  ) or a Stückelberg-Feynman type Green’s function with equal contribtions from retarded and advanced components, GSF (t, t  ) = |t − t  |/2, or any other Green’s function of the form (14.8). The limit ti → −∞, tf → ∞ in Eq. (14.58) yields the following representation of the S-matrix element for scattering due to the external force F (t), 

   ∞ pf + pi · dt F (t) exp i dt tF (t) Sf i [F ] = δ pf − pi − 2mh¯ −∞ −∞    ∞  ∞ i    dt dt |t − t |F (t) · F (t ) . (14.63) × exp 4mh¯ −∞ −∞ 

∞



In the next steps we will compare the correlation functions between the canonical and the path integral formalism. The calculation of the one-point function from the path integral (14.55) has to take into account that the generic Green’s function (14.8) shifts x f/i to Xf/i according to Eq. (14.54). This implies for the one-point function in the path integral formalism   δ Z[F ] x f , tf |x(t)|x i , ti  = − ih¯ = x f |U0 (tf , ti )|x i  δF (t) F =0   % & & xf − xi xf + xi δ % . (14.64) + α(t) x f − x i + m Xf − Xi · × 2 δF (t) tf − ti

326

14 Path Integrals in Quantum Mechanics

However, we have m

& tf + ti δ % Xf − Xi = α(t)(ti − tf ) + t − , δF (t) 2

(14.65)

and therefore the path integral result for the one-point function is indeed independent on the gauge functions α(t  ) and β(t  ), as was already clear from the cancellation of those terms in Z[F ],   δ = x f |U0 (tf , ti )|x i  x f , tf |x(t)|x i , ti  = − ih¯ Z[F ] δF (t) F =0   tf − t t − ti , (14.66) + xi × xf tf − ti tf − ti i.e. we do find the same result (14.17) as in the canonical formalism. For the calculation of the two-point functions in the functional formalism x f , tf |x(t2 ) ⊗ x(t1 )|x i , ti  = − h¯ 2

  δ 2 Z[F ]  δF (t2 ) ⊗ δF (t1 ) F =0

(14.67)

it is useful to observe that δ 2 Z[F ] δ 2 ln Z[F ] δ ln Z[F ] δ ln Z[F ] 1 = + ⊗ . Z[F ] δF (t2 ) ⊗ δF (t1 ) δF (t2 ) ⊗ δF (t1 ) δF (t2 ) δF (t1 )

(14.68)

The factors in the last term were evaluated at F = 0 in (14.66) and reproduce the tensor product of one-point functions in (14.19). The second order variational derivative of ln Z[F ] yields for ti ≤ t1 ≤ t2 ≤ tf (but only in that case) − h¯ 2

ih¯ (tf − t2 )(t1 − ti ) δ 2 ln Z[F ] = 1. δF (t2 ) ⊗ δF (t1 ) m tf − ti

(14.69)

The general result for ti < tf is − h¯ 2

 ih¯ δ 2 ln Z[F ] = (tf − t2 )(t2 − t1 )(t1 − ti ) δF (t2 ) ⊗ δF (t1 ) m(tf − ti )

 × (tf − t2 )(t1 − ti ) + (tf − t1 )(t1 − t2 )(t2 − ti )(tf − t1 )(t2 − ti ) 1. Therefore we cannot in general simply identify −h¯ 2 δ 2 Z[F ]/(δF (t2 ) ⊗ δF (t1 )) at (2) F = 0 with either g (2) f i (t2 , t1 ) or Gf i (t2 , t1 ), but we have   δ 2 Z[F ] (2) (2)  − h¯ = g f i (t2 , t1 ) = Gf i (t2 , t1 ) δF (t2 ) ⊗ δF (t1 ) F =0 if ti ≤ t1 ≤ t2 ≤ tf . 2

(14.70)

14.4 Problems

327

It seems surprising that substitution of (14.63) into Eq. (14.56) and setting F = 0 after evaluation of the functional derivatives yields scattering from the potential V . However, Eqs. (14.56) and (14.63) compare to the practically useful relation (13.42) (or the equivalent relation (14.49)) for the scattering matrix elements like the representation 

t − t  − i ∂ 2 x|U0 (t − t )|x  = exp ih¯ 2m ∂x 2 





δ(x − x  )

(14.71)

for the x matrix elements of the free time evolution operator compares to the practically more useful representation (4.115). Recasting the perturbation series in terms of the operator V (− ihδ/δF (t), t) ¯ instead of V (x, t) does not yield a more efficient or practical representation for potential scattering theory. However, recasting interactions in terms of functional derivatives is useful when interactions are expressed in terms of higher order products of wave functions instead of potentials. Therefore we used the transcription of potential scattering theory in terms of functional derivatives with respect to auxiliary forces as an illustration for functional methods in perturbation theory.

14.4 Problems 14.1 Verify Eq. (14.42) for the general second order particle action S[x(t  )] =



t



1  1 ˙ ) · M · x(t ˙  ) + x(t  ) · F · x(t ˙ ) x(t 2 2  1 − x(t  ) · 2 · x(t  ) + F · x(t  ) , 2 dt 

t0

FT = −F,

x(t0 ) = x cl (t0 ) = x 0 ,

x(t) = x cl (t) = x.

(14.72) (14.73)

14.2 Which exponential factor in the propagator x|U (t, t0 )|x 0  do you find for a harmonic oscillator? 14.3 Derive the particular Green’s functions (14.3) and (14.6) from the general form (14.8). 14.4 Show that the terms A[F ] =

" 1 ! ˙ ◦ F )(tf ) − (G ◦ F )(ti ) · (G ˙ ◦ F )(ti ) (G ◦ F )(tf ) · (G 2m  tf 1 + dt F (t) · (G ◦ F )(t) (14.74) 2m ti

328

14 Path Integrals in Quantum Mechanics

in the exponent in (14.55) are actually the action S[x, F ] for the classical trajectory x(t) = (G ◦ F )(t)/m. Why is Z[F ] not just given by 0|U0 (tf , ti )|0 exp(iA[F ]/h), ¯ in spite of what you might have expected from (14.42)? 14.5 Calculate the time ordered three-point function x f , tf |x(t3 ) ⊗ x(t2 ) ⊗ x(t1 )|x i , ti ,

ti < t1 < t2 < t3 < tf ,

(14.75)

both in the canonical formalism and in the path integral formalism. 14.6 The functional Sf i [F ] ≡ S(p f , p i )[F ] (14.63) is a scattering matrix element between momentum eigenstates. The functional Z[F ] (14.55) on the other hand is not a scattering matrix element in position space because it does not correspond to a position space matrix element of an interaction picture time evolution operator. Instead it corresponds to the path integral result for the position matrix element of the full time evolution operator of a free particle under the influence of a spatially homogeneous force F (t). However, we can derive a position space scattering matrix element through Fourier transformation of S(p f , p i )[F ]. Show that 

 & S(p f , p i )[F ] i % p · x f − pi · x i S(x f , x i )[F ] = d p f d p i exp h¯ f (2π h) ¯ 3      ∞  ∞ xf + xi 1 · = δ xf − xi + dt tF (t) exp i dt F (t) m −∞ 2h¯ −∞    ∞  ∞ i dt dt  |t − t  |F (t) · F (t  ) . (14.76) × exp 4mh¯ −∞ −∞ 



3

3

Show also that with the conditions  lim t

t→±∞

t

−∞

dt  F (t  ) = 0

(14.77)

the δ function implies xf = xi +

1 m





∞ −∞

dt

t

−∞

dt  F (t  ),

(14.78)

while the conditions 

∞

lim t

t→±∞

yield with the δ function the relation

t

dt  F (t  ) = 0

(14.79)

14.4 Problems

329

1 xi = xf + m





−∞

∞

dt

t ∞

dt  F (t  ).

(14.80)

Equation (14.78) describes the asymptotic solution of a classical trajectory of a non-relativistic particle which started out at rest in x i at t → −∞, while (14.80) reconstructs the initial location for t → −∞ of a particle which comes to rest in x f in the limit t → ∞. The conditions (14.77) or (14.79) do not generate extra restrictions on the physical motion of the particle, but are mathematical conditions for convergence of the time integrals in (14.78) or (14.80), respectively, i.e. they are necessary for existence of solutions of the Newton equation for t → ±∞.

Chapter 15

Coupling to Electromagnetic Fields

Electromagnetism is the most important interaction for the study of atoms, molecules and materials. It determines most of the potentials or perturbation operators V which are studied in practical applications of quantum mechanics, and it also serves as a basic example for the implementation of other, more complicated interactions in quantum mechanics. Therefore the primary objective of the current chapter is to understand how electromagnetic fields are introduced in the Schrödinger equation.

15.1 Electromagnetic Couplings The introduction of electromagnetic fields into the Schrödinger equation for a particle of mass m and electric charge q can be inferred from the description of the particle in classical Lagrangian mechanics. The Lagrange function for the particle in electromagnetic fields E(x, t) = − ∇(x, t) −

∂A(x, t) , ∂t

B(x, t) = ∇ × A(x, t)

(15.1)

is L=

m ˙ 2 + q x(t) ˙ · A(x(t), t) − q(x(t), t). x(t) 2

(15.2)

Let us check (or review) that Eq. (15.2) is indeed the correct Lagrange function for the particle. The electromagnetic potentials in the Lagrange function depend on the time t both explicitly and implicitly through the time dependence x(t) of the trajectory of the particle. The time derivative of the conjugate momentum

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 R. Dick, Advanced Quantum Mechanics, Graduate Texts in Physics, https://doi.org/10.1007/978-3-030-57870-1_15

331

332

15 Coupling to Electromagnetic Fields

p=

∂L = mx˙ + qA ∂ x˙

(15.3)

is therefore ∂A dp ∂A +q = mx¨ + q x˙i . dt ∂xi ∂t

(15.4)

According to the Euler-Lagrange equations (cf. Appendix A), this must equal ∂L = q x˙i ∇Ai − q∇. ∂x

(15.5)

The property (7.34) of the  tensor implies % & ei x˙j ∂i Aj − x˙j ∂j Ai = ei ij k klm x˙j ∂l Am = x˙ × B,

(15.6)

and therefore the Euler-Lagrange equation yields the Lorentz force law mx¨ = q(E + v × B),

(15.7)

as required. The classical Hamiltonian for the particle follows as H = p · x˙ − L =

m 1 (p − qA)2 + q = x˙ 2 + q. 2m 2

(15.8)

The Hamilton operator of the charged particle therefore becomes H =

1 [p − qA(x, t)]2 + q(x, t), 2m

(15.9)

and the Schrödinger equation in x representation is ih¯

1 ∂ 2 (x, t) = − [h∇ ¯ − iqA(x, t)] (x, t) + q(x, t)(x, t). ∂t 2m

(15.10)

This is the Schrödinger equation for a charged particle in electromagnetic fields. If we write this in the form ih¯

∂ 1 2  − q = (ih∇ ¯ + qA)  ∂t 2m

(15.11)

we also recognize that this arises from the free Schrödinger equation through the substitutions ih∇ ¯ → ih∇ ¯ + qA,

ih¯

∂ = ihc∂ ¯ 0 → ihc∂ ¯ 0 − q. ∂t

(15.12)

15.1 Electromagnetic Couplings

333

These equations can be combined in 4-vector notation with p0 = −E/c, A0 = −/c, pμ = − ih∂ ¯ μ → pμ − qAμ = − ih∂ ¯ μ − qAμ .

(15.13)

This observation is useful for recognizing a peculiar symmetry property of Eq. (15.10). Classical electromagnetism is invariant under gauge transformations of the electromagnetic potentials (here we use f (x) ≡ f (x, t)), (x) →  (x) = (x)−c∂0 ϕ(x), A(x) → A (x) = A(x)+∇ϕ(x),

(15.14)

where the arbitrary function ϕ(x) has the dimension of a magnetic flux, i.e. it comes in units of Vs. Eqs. (15.14) read in 4-vector notation, Aμ (x) → Aμ (x) = Aμ (x) + ∂μ ϕ(x).

(15.15)

The Schrödinger equation (15.10) should respect the gauge invariance of classical electromagnetism to comply with classical limits, and indeed it does. If we also transform the wave function according to   q (x) →   (x) = exp i ϕ(x) (x), h¯

(15.16)

then the Schrödinger equation in the transformed fields and wave functions has exactly the same form as the Schrödinger equation in the original fields, because the linear transformation property ∂  1  2   − q   + (h∇ ¯ − iqA )  ∂t 2m    1 ∂ q 2 (h∇ − iqA) = exp i ϕ(x) ih¯  − q +  ¯ ∂t 2m h¯ ih¯

(15.17)

(15.18)

implies that ih¯

∂  1  2   − q   + (h∇ ¯ − iqA )  = 0 ∂t 2m

(15.19)

holds in the transformed fields if and only if the Schrödinger equation also holds in the original fields, ih¯

∂ 1 2  − q + (h∇ ¯ − iqA)  = 0. ∂t 2m

(15.20)

334

15 Coupling to Electromagnetic Fields

The reason for the linear transformation law is q q q ∂μ − i Aμ = ∂μ − i Aμ − i (∂μ ϕ) h¯ h¯ h¯      q q q ∂μ − i Aμ exp − i ϕ , = exp i ϕ h¯ h¯ h¯

(15.21)

which implies that the covariant derivatives  q Dμ  = ∂μ − i Aμ  h¯ 

(15.22)

transform exactly like the fields,   q (15.23) (x) →  (x) = exp i ϕ(x) (x), h¯   q   Dμ (x) → Dμ  (x) = exp i ϕ(x) Dμ (x), (15.24) h¯   q Dμ Dν . . . Dρ (x) → Dμ Dν . . . Dρ   (x) = exp i ϕ(x) Dμ Dν . . . Dρ (x). h¯ 

This implies preservation of every partial differential equation which like the Schrödinger equation uses only covariant derivatives, ihcD ¯ 0 (x) = −

h¯ 2 2 D (x) 2m

⇔

  ihcD ¯ 0  (x) = −

h¯ 2 2  D  (x). 2m

(15.25)

Coupling of matter wave functions to electromagnetic potentials through covariant derivatives is known as minimal coupling. Observables are gauge invariant, too. For example, the mechanical momentum of the charged particle in electromagnetic fields is m

d x(t) = dt =

 

d 3 x  + (x, t) [− ih∇ ¯ − qA(x, t)] (x, t) ! "   d 3 x  + (x, t) − ih∇ ¯ − qA (x, t)  (x, t).

(15.26)

Electromagnetic interactions ensure local phase invariance of nature. We can rotate the wave function with an arbitrary local phase factor without changing the dynamics or observables of a physical system, due to the presence of the electromagnetic potentials. In hindsight, we should consider this as the reason for the peculiar coupling of the electromagnetic potentials in the Schrödinger equation (15.10).

15.1 Electromagnetic Couplings

335

The presence of the electromagnetic potentials in the observables will of course affect conservation laws. E.g. the mechanical momentum (15.26) of the particle will generically not be conserved, because it can exchange momentum with the electromagnetic field which carries momentum p em (t) = 0 d 3 x E(x, t) × B(x, t). The conserved momentum1 of the coupled system of non-relativistic charged particle and electromagnetic fields is P =m

d x(t) + p em (t) = dt

 d 3 x P(x, t),

(15.27)

where the momentum density with symmetrized action of the derivatives on the wave functions is given by P(x, t) =

" h¯ ! +  (x, t) · ∇(x, t) − ∇ + (x, t) · (x, t) 2i − q + (x, t)A(x, t)(x, t) + 0 E(x, t) × B(x, t).

(15.28)

The derivation of momentum conservation for the classical particle-field system in the full relativistic setting can be found in Appendix B, see in particular Eq. (B.106). Systematic derivations of momentum densities in the coupled system of charged particles and electromagnetic fields in the framework of relativistic spinor quantum electrodynamics (QED) can be found in Sects. 22.4 and 22.5, see in particular Eqs. (22.149) and (22.172). Problem 22.6 and Eq. (22.249) provide the corresponding results in relativistic scalar QED. Both the non-relativistic limits for bosons and fermions lead to (15.28) for the conserved momentum density in non-relativistic QED, which is also known as quantum electronics.

Multipole Moments In many applications of quantum mechanics, simplifications of the electromagnetic coupling terms in Eq. (15.10) can be employed if the electromagnetic fields have large wavelengths compared to the wave functions in the Schrödinger equation. The leading order and most common approximation is related to the electric dipole moment of charge distributions, and therefore we will briefly discuss the origin of multipole moments in electromagnetism.

= − ih¯ d 3 x  + (x, t)∇(x, t) = m(dx(t)/dt)+qA(x, t) is also generically not conserved, except if the particle moves in a spatially homogeneous electric field E(t) = − dA(t)/dt, e.g. in a plate capacitor. However, note that this is an artifact of the gauge  = 0.

1 The canonical momentum p(t)

336

15 Coupling to Electromagnetic Fields

Suppose that we probe the electromagnetic potential of a charge q which is located at x. We are interested in the potential at location r, where |r|  |x|. Second order Taylor expansion of the Coulomb term in the variables x yields q q r ·x 3(r · x)2 − r 2 x 2 q r ·d 1 ≈ +q 3 +q = + 3 + 5r ·Q·r |r − x| r r r 2r 5 r 2r

(15.29)

with the dipole and quadrupole terms d = qx,



Q = q 3x ⊗ x − x 2 1 .

(15.30)

For an extended charge distribution (x) this implies at large distance a representation of the potential  4π 0 (r) =

d 3x

(x) 1 q r ·d ≈ + 3 + 5r · Q · r |r − x| r r 2r

(15.31)

in terms of the monopole, dipole, and quadrupole moments  q=

 d 3 x (x),

d=

 d 3 x (x)x,

Q=

d 3 x (x)[3x ⊗ x − x 2 1].

We will find that the leading order coupling of long wavelength electromagnetic fields to charges appears through electric dipole moments of the charges.

Semiclassical Treatment of the Matter-Radiation System in the Dipole Approximation In the semiclassical treatment the electromagnetic fields are considered as external classical fields with which the quantum mechanical matter (atom, nucleus, molecule, solid) interacts. If we consider e.g. an atom with an internal (average or effective) potential Vint (x) experienced by the electrons, the Schrödinger equation for these electrons in the external electromagnetic fields is ih¯

1 ∂ 2 =− (h∇ ¯ − iqA)  + (q + Vint ). ∂t 2m

(15.32)

If the electromagnetic fields vary weakly over the extension a of the wave functions (corresponding to approximately homogeneous field over the extension of the atom or molecule under consideration), we can effectively assume a spatially homogeneous field E = E(t) corresponding to a potential (x, t) = −E(t) · x. If we assume that our material probes range over length scales from 1 Å (corresponding

15.1 Electromagnetic Couplings

337

to the size of atoms) to several Å (corresponding to molecules containing e.g. several Benzene rings), electromagnetic fields with wavelengths larger than 100 nm or photon energies smaller than 12 eV can be considered as approximately spatially homogeneous over the size of the probe. Furthermore the magnetic field in the electromagnetic wave satisfies B(t) =

1 ∇ × (B(t) × x) 2

(15.33)

and 1 |E|, c

ω 2π |E| = |E|. c λ

(15.34)

   ∂A  1   πa   ˙   ∂t  = 2 B × x  λ |E| % |E| ,

(15.35)

|B| =

˙ = |B|

We have

and therefore the description of E only through the electric potential, E(t) = −∇(x, t) = ∇(E(t) · x),

(15.36)

is justified for λ  a. Furthermore, the magnitudes of magnetic contributions to the Schrödinger equation are of order q h¯ q h¯ q h¯ |A · ∇|  |E| |x| |∇|  |E| || , m 2mc 2mc

(15.37)

q2 2 q2 q 2a2 2 2 2 || A ||   E x E || . 2m 8mc2 8mc2

(15.38)

For comparison, the electric contribution has a magnitude of order q |E| |x| ||  qa |E| || .

(15.39)

The ratio of the linear magnetic term (15.37) to the electric term is h/(2mca). If we ¯ use the electron mass for m, we find h¯ 1Å ≤ 2 × 10−3 × , 2mca a

(15.40)

i.e. the linear magnetic term is often negligible compared to the electric term. The ratio of the second magnetic term (15.38) to the electric term is approximately qa |E| /8mc2 . Validity of the non-relativistic approximation requires that the electrostatic energy qa |E| due to the electric field should be small compared

338

15 Coupling to Electromagnetic Fields

to mc2 . Therefore we also find that the second magnetic term should be negligible compared to the electric term. Quantitatively, if we assume mc2 = 511 keV, we have ea |E| /8mc2 % 1 for |E| %

8mc2 V 1Å = 4 × 1016 × . ea m a

(15.41)

For comparison, the internal field strength in hydrogen is of order e/(4π 0 a02 )  5 × 1011 V/m. We conclude that for λ  a the effect of external electromagnetic fields can be approximated by the addition of a term V (x, t) = q(x, t) = − qE(t) · x = − d · E(t)

(15.42)

in the Schrödinger equation. The approximation of spatially homogeneous external field yields a perturbation proportional to the dipole operator and is therefore denoted as dipole approximation. Two cautionary remarks are in order at this point. The term dipole approximation is nowadays more widely used for the long wavelength approximation exp(ik · x)  1 in matrix elements irrespective of whether the perturbation operator has the dipole form (15.42) or is given in terms of the coupling to the vector potential A(x, t) in (15.32). Furthermore, if we describe electromagnetic interactions at the level of photonmatter interactions, the dipole approximation (15.42) is generically limited to first order perturbation theory, and holds in second order perturbation theory only if additional conditions are met, see Sect. 18.9 and Problem 18.17.

Dipole Selection Rules The first √ order scattering matrix elements in dipole approximation are given by Sf i = 2π iqE(ωf i )·f |x|i/h, ¯ i.e. only transitions |i → |f  with non-vanishing dipole matrix elements qf |x|i are allowed in this approximation. This yields straightforward selection rules for states which are eigenstates of M 2 and Mz . The commutator relation [Mz , z] = 0 implies     n ,  , m |[Mz , z]|n, , m = hn ¯ ,  , m |z|n, , m(m − m) = 0,

(15.43)

and therefore an electric field component in z direction can only induce transitions between states with the same magnetic quantum number. In the same way, the commutators [Mz , x ± iy] = ±h(x ¯ ± iy) imply n ,  , m |x ± iy|n, , m(m − m ∓ 1) = 0,

(15.44)

15.2 Stark Effect and Static Polarizability Tensors

339

such that electric field components in the (x, y) plane can only induce transitions which increase or decrease the magnetic quantum number by one unit. Finally, the fairly complicated relation [M 2 , [M 2 , x]] = 2h¯ 2 {M 2 , x} yields n ,  , m |x|n, , m( +  )( +  + 2)[( −  )2 − 1] = 0.

(15.45)

This implies that the matrix element can be non-vanishing only if  =  ± 1.  =  = 0 is not a solution, because in this case the wave functions depend only on r and the angular integrations then show that the matrix element vanishes. Equations (15.43)–(15.45) imply the dipole selection rules  = ±1 and m = 0, ±1.

15.2 Stark Effect and Static Polarizability Tensors Polarizability tensors characterize the response of a quantum system to an external electric field E. The calculation of polarizability tensors is another example of applications of second order perturbation theory in materials science. It also illustrates the role of perturbation theory in derivations of quantum mechanical expressions for measurable physical quantities, which were first introduced in classical electrodynamics and were initially approximated by means of simple mechanical models. The calculation of polarizabilities generically involves many particles and related  dipole operators V (t) = − N i=1 qi E(t)·xi , where it is assumed that all particles are confined to a region which is still small compared to the wavelength of the electric field. We will develop the theory in a single-particle approximation in the sense that we only use the single charged (quasi)particle operator V (t) = −qE(t) · x. In the present section we will do this for time-independent external field, where we can use the techniques of time-independent perturbation theory. The case of dynamical polarizability for time-dependent external fields will be discussed in Sect. 15.3.

Linear Stark Effect Before we jump into the second order calculation of the response to an electric field, we consider the implications of first order perturbation theory for the dipole approximation. An external static electric field shifts the Hamilton operator according to H0 → H = H0 + V = H0 − qE · x.

(15.46)

Time-independent perturbation theory tells us that the first order shifts of atomic or molecular energy levels due to the external field have to be determined as the eigenvalues of the matrix

340

15 Coupling to Electromagnetic Fields (0)

(0)

(0) (0) n,α |V (x)|n,β  = − qn,α |x|n,β  · E,

(15.47)

and when the nth degeneracy subspace has been internally diagonalized with respect to V (x), the first order shifts are (1) (0) (0) En,α = − qn,α |x|n,α  · E = − d n,α · E,

(15.48)

(0)

with the intrinsic dipole moment in the state |n,α  (0) (0) |x|n,α . d n,α = qn,α

(15.49)

The perturbation V has odd parity under x → −x, while atomic states of opposite parity are usually not degenerate. Therefore in systems which are symmetric under the parity transformation x → −x, the states in the nth energy level usually satisfy (0) (0) |x|n,β =0 n,α

(15.50)

because the integrand is odd under the parity transformation. Usually this implies absence of a linear Stark effect in atoms, and the same remark applies to molecules with parity symmetry. An important exception is the hydrogen atom due to degeneracy of its energy levels (if the matrix elements of V are larger than the fine structure of the hydrogen levels). States with angular momentum quantum number  have parity (−1) , so that the nth hydrogen level with n > 1 contains degenerate states of opposite parity. Diagonalization of V in that degeneracy subspace then (0) (0) (0) yields states |n,α  with n,α |x|n,α  = 0.

Quadratic Stark Effect and the Static Polarizability Tensor Second order perturbation theory yields the following corrections to discrete atomic or molecular energy levels, (2) Enα

=



(0) (0) |mβ |V |nα |2

m=n β

=q E· 2



(0)

(0)

En − Em

(0) (0) (0) (0) nα |x|mβ mβ |x|nα 

m=n β

(0)

(0)

En − Em

· E.

(15.51)

The notation takes into account that the intermediate levels can be continuous, but degeneracy indices are always discrete.

15.3 Dynamical Polarizability Tensors

341

We can write the second order shifts in the form (2) Enα =−

1 1 d (nα) · E = − E · α (nα) · E, 2 2

(15.52)

where d (nα) = α (nα) · E

(15.53)

is the induced dipole moment and α (nα) is the static electronic polarizability tensor (0)

in the state |nα , α (nα) = − q 2





m=n β

1 (0) En(0) − Em

(0) (0) (0) (0) nα |x|mβ  ⊗ mβ |x|nα 

 (0) (0) (0) (0) |x|nα  ⊗ nα |x|mβ  . + mβ

(15.54)

Note that in the ground state αii > 0 (no summation convention), i.e. in second order perturbation theory, which usually should capture all linear contributions from a weak external electric field to the induced dipole moment, there is no electronic dia-electricity for the ground state.

15.3 Dynamical Polarizability Tensors We cannot use time-independent perturbation theory if the perturbation operator V (t) = −qx · E(t) varies with time. Application of our results from Chap. 13 for time-dependent perturbations implies that the first order transition probability from a state |m into a state |n under the action of the electric field E(t) between times t  and t is proportional to2 (1) Pm→n (t, t  )

  t 2 q   = dτ exp(iωnm τ )E(τ ) · n|x|m , h¯ t 

(15.55)

where ωnm =

1 (En − Em ). h¯

(15.56)

that |Snm |2 is a true transition probability only if the initial and final state are discrete, while otherwise it enters into decay rates or cross sections.

2 Recall

342

15 Coupling to Electromagnetic Fields

For t  → −∞, t → ∞, this becomes in particular (see our previous results (13.65) and (13.66)) (1) Pm→n

 2 q   = 2π  E(ωnm ) · n|x|m , h¯

(15.57)

i.e. long term action of an external electric field can induce a transition in first order between energy levels Em and En only if the field contains a Fourier component of the corresponding frequency ωnm . However, at this time we are interested in the problem how Eq. (15.54) can be generalized to a dynamical polarizability in the presence of a time-dependent external field E(t). (0) (0) Suppose the system was in the state |n,α (0) ≡ |n,α  at t = 0, when it begins (0) to experience the effect of the electric field. The shift of the wave function |n,α (t) under the influence of the external field is (0) (0) (t) = (t) [U (t) − U0 (t)] |n,α  |n,α (t) − |n,α ! + " (0) = (t)U0 (t) U0 (t)U (t)U0 (0) − 1 |n,α  (0) = (t)U0 (t) [UD (t) − 1] |n,α ,

(15.58)

and the first order shift is therefore (1) (t) |n,α

i = − (t)U0 (t) h¯

 0

t

(0) dτ HD (τ )|n,α 

 t i (0)  = − (t)U0 (t) dτ U0+ (τ )V (τ )U0 (τ )|n,α h¯ 0  t i (0) . = − (t) dτ U0 (t − τ )V (τ )U0 (τ )|n,α h¯ 0

(15.59)

(0)  is then given in leading order by the The induced dipole moment in the state |n,α first order terms (recall that the 0th order term corresponds to the intrinsic dipole moment (15.49)) (0) (1) (1) (0) d (nα) (t) = n,α (t)|qx|n,α (t) + n,α (t)|qx|n,α (t)  t i (0) (0) |U0+ (t)xU0 (t − τ )x · E(τ )U0 (τ )|n,α  = q 2 (t) dτ n,α h¯ 0  t i (0) (0) |U0+ (τ )x · E(τ )U0+ (t − τ )xU0 (t)|n,α . − q 2 (t) dτ n,α h¯ 0

This becomes after insertion of complete sets of unperturbed states in U0 (t − τ ) = U0 (t)U0+ (τ ) and U0+ (t − τ ) = U0 (τ )U0+ (t)

15.3 Dynamical Polarizability Tensors

343

  t i (0) (0) d (nα) (t) = q (t) dτ exp[iωnm (t − τ )]n,α |x|m,β  h¯ m,β 0   t (0) (0) 2 i ×m,β |x · E(τ )|n,α  − q (t) dτ exp[− iωnm (t − τ )] h¯ m,β 0 2

(0) (0) (0) (0) |x · E(τ )|m,β m,β |x|n,α  ×n,α  ∞ dτ α (nα) (t − τ ) · E(τ ), =

(15.60)

0

with a dynamical polarizability tensor   i (0) (0) (0) (0) exp(iωnm t)n,α |x|m,β  ⊗ m,β |x|n,α  α (nα) (t) = q (t) h¯ m,β   (0) (0) (0) (0) exp(− iωnm t)m,β |x|n,α  ⊗ n,α |x|m,β  . (15.61) − 2

m,β

Now we assume harmonic time dependence of an electric field which is switched on at t = 0, E(τ ) ≡ E ω (τ ) = E(t) sin(ωτ ) = E(t)

exp(iωτ ) − exp(− iωτ ) . 2i

(15.62)

The time integrals in the two terms for d (nα) (t) then yield ±



q2 2h¯

=± =±

t 0

 dτ exp[±iωnm (t − τ ) + iωτ ] − exp[±iωnm (t − τ ) − iωτ ] 

q2 2ih¯

exp(iωt) − exp(±iωnm t) exp(− iωt) − exp(±iωnm t) + ω ∓ ωnm ω ± ωnm

q 2 ω cos(ωt) ± iωnm sin(ωt) − ω exp(±iωnm t) . 2 ih¯ ω2 − ωnm



(15.63)

We also assume slowly oscillating field in the sense ω % |ωnm | for all quantum (0) (0) |x|n,α |. This numbers m which correspond to large matrix elements |m,β means that the external field is not likely to induce direct transitions between different energy levels. Under these conditions, the contribution from the integrals in Eq. (15.63) to d (nα) (t) will be dominated by the term which is in phase with the external field, ± →

q2 2h¯ q2



t

 dτ exp[±iωnm (t − τ ) + iωτ ] − exp[±iωnm (t − τ ) − iωτ ]

0

ωnm sin(ωt) , 2 h¯ ω2 − ωnm

(15.64)

344

15 Coupling to Electromagnetic Fields

and the induced dipole moment in this approximation is  ωmn (0) q2 (0) (0) (0) n,α |x|m,β m,β |x · E ω (t)|n,α  d (nα)ω (t) = 2 − ω2 h¯ m,β ωmn  (0) (0) (0) (0) + m,β |x|n,α n,α |x · E ω (t)|m,β  . (15.65) This can also be written as d (nα)ω (t) = α (nα) (ω) · E ω (t)

(15.66) (0)

with the frequency dependent polarizability tensor for the state |n,α  (usually the ground state) 

ωmn q2 (0) (0) (0) (0) n,α |x|m,β  ⊗ m,β |x|n,α  α (nα) (ω) = 2 − ω2 h¯ m,β ωmn  (0) (0) (0) (0) + m,β |x|n,α  ⊗ n,α |x|m,β  . (15.67) The zero frequency polarizability tensor α (nα) (0) is the static tensor (15.54), as expected. The frequency dependent polarizability tensor is not only relevant for slowly oscillating fields, but appears implicitly already in the Eqs. (15.60) and (15.61), which do not include a restriction to slowly oscillating external field. If we agree to shift the denominator in (15.67) by small imaginary numbers according to α (nα) (ω) =

 ωnm q2 (0) (0) (0) (0) n,α |x|m,β  ⊗ m,β |x|n,α  2 − ω2 − i h¯ ω m,β nm   ωnm (0) (0) (0) (0)  |x|  ⊗  |x|  , (15.68) + n,α n,α m,β m,β 2 2 m,β ω − ωnm + i

we find that the dynamical polarizability tensors in Eqs. (15.61) and (15.68) are related via  (t) ∞ α (nα) (t) = dω α (nα) (ω) exp(− iωt). (15.69) π −∞

Oscillator Strength Equation (15.68) yields an averaged polarizability

15.3 Dynamical Polarizability Tensors

345

 ωmn 1 2q 2 (0) (0) 2 tr α (nα) (ω) = |m,β |x|n,α | 2 3 3h¯ m,β ωmn − ω2  q 2 fm,β;n,α = (15.70) 2 − ω2 m m,β ωmn

α(nα) (ω) =

(0)

(0)

with the oscillator strength for the transition |n,α  → |m,β : fm,β;n,α =

2m (0) (0) 2 | = − fn,α;m,β . ωmn |m,β |x|n,α 3h¯

(15.71)

We use m both for the mass of the charged (quasi)particle which has its wave functions shifted due to the external field, and as a label for the intermediate states. Since mass never appears as an index in Eq. (15.71) or the following equations, this should not cause confusion. The polarizability is also often averaged over degenerate initial states. If the degeneracy of the nth energy level is gn , then αn (ω) =

 1 q 2 fm|n α(nα) (ω) = 2 − ω2 gn α m m ωmn

(15.72)

with an effective oscillator strength which is averaged over degenerate initial states and summed over degenerate final states, fm|n =

1 gm fm,β;n,α = − fn|m . gn gn

(15.73)

α,β

With these conventions, positive oscillator strength corresponds to absorption and negative oscillator strength corresponds to emission. Oscillator strengths are sometimes also defined through absolute values, but for the f -sum rules below it plays a role that emission transitions contribute with negative sign. For an explanation of the name oscillator strength for fm,β;n,α , we observe that a classical isotropic harmonic oscillator model for polarizability ¨ + mω02 x(t) = qE sin(ωt) mx(t)

(15.74)

yields an induced dipole moment d ω (t) = qx(t) =

1 q2 E sin(ωt) = α(ω)E ω (t) 2 m ω0 − ω2

(15.75)

346

15 Coupling to Electromagnetic Fields

with the polarizability α(ω) =

1 q2 , m ω02 − ω2

(0)

(15.76)

(0)

i.e. every virtual transition |n,α  → |m,β  contributes effectively like an    (0)  oscillator of frequency |ωmn | = Em − En(0)  /h¯ to the polarizability α(nα) (ω) of (0) the state |n,α , but the contribution of that transition is weighted with the oscillator strength (15.71).

Thomas-Reiche-Kuhn Sum Rule (f -Sum Rule) for the Oscillator Strength Kuhn, Reiche and Thomas found a sum rule for the oscillator strength already in the framework of old quantum theory [103, 142]. The quantum mechanical proof is based on the fact that the Hamiltonian operator H = (p2 /2m) + V (x) yields a commutator [H, x] =

h¯ p . im

(15.77)

(0) This implies for a discrete normalized state |n,α 

 2m (0) (0) (0) (0) fm,β;n,α = ωmn n,α |x|m,β  · m,β |x|n,α  3h¯ m,β m,β   2m (0) (0) (0) (0) (0) Em − En(0) n,α = 2 |x|m,β  · m,β |x|n,α  3h¯ m,β  m (0) (0) (0) (0) = 2 n,α |x|m,β  · m,β |[H0 , x]|n,α  3h¯ m,β  (0) (0) (0) (0) − n,α |[H0 , x]|m,β  · m,β |x|n,α   1 (0) (0) (0) (0) n,α |x|m,β  · m,β |p|n,α  = 3ih¯ m,β  1 (0) (0) (0) (0) (0)  (0) |(x · p − p · x)|n,α |p|m,β  · m,β |x|n,α  =  = 1. − n,α 3ih¯ n,α



15.3 Dynamical Polarizability Tensors

347

This is the3 Thomas-Reiche-Kuhn sum rule,  fm,β;n,α = 1.

(15.78)

m,β

Note that we used the decomposition of the identity in the form 

(0)

m,β

(0)

|m,β m,β | = 1,

(15.79)

 but in applications of the sum rule you need to take into account that m,β for continuous quantum numbers also includes measure factors like e.g. in the completeness relation (7.184) for hydrogen eigenstates,  ∞ =0 m=−



∞

 |n, , mn, , m| +

∞

 dk k |k, , mk, , m| . 2

(15.80)

0

n=+1

Averaging Eq. (15.78) over the initial degeneracy indices yields with the definition (15.73) the sum rule  fm|n = 1. (15.81) m

Equation (15.77) implies a further relation which connects matrix elements of x and p, (0)

(0) ωmn m,β |x|n,α =

1 (0) (0)  |p|n,α . im m,β

(15.82)

This yields an alternative representation of the oscillator strength fm,β;n,α =

2 (0) (0) 2 |m,β |p|n,α | , 3mhω ¯ mn

(15.83)

which is known as the velocity form of the oscillator strength, while Eq. (15.71) is denoted as the length form of the oscillator strength. Yet another common definition in atomic, molecular and optical physics is fm,β;n,α =

3 If

2mωmn Sm,β;n,α , 3hq ¯ 2

fm|n =

2mωmn Sm,n , 3hq ¯ 2

(15.84)

the wave functions are N -particle wave functions and the potential V is the corresponding sum of dipole operators, the number on the right-hand side of the sum rules becomes N .

348

15 Coupling to Electromagnetic Fields (0)

(0)

with the electric dipole line strength of the transition |n,α  → |m,β   2  q  (0) (0) (0) 2 (0)  Sm,β;n,α = |m,β |qx|n,α | =  m,β |p|n,α  , mωmn Sm,n =

1 gm Sm,β;n,α = Sn,m . gn gn

(15.85)

(15.86)

α,β

Tensorial Oscillator Strengths and Sum Rules We can define oscillator strength tensors through the relations

α n (ω) =

 q 2 f m,β;n,α α (nα) (ω) = , 2 − ω2 m m,β ωmn

(15.87)

1 q 2 f m,n α (nα) (ω) = , 2 − ω2 gn α m m ωmn

(15.88)

i.e. we have representations for oscillator strength tensors f m,β;n,α =

=

m (0) (0) (0) (0) |x|m,β  ⊗ m,β |x|n,α  ωmn n,α h¯  (0) (0) (0) (0) + m,β |x|n,α  ⊗ n,α |x|m,β  1 (0) (0) (0) (0) n,α |x|m,β  ⊗ m,β |p|n,α  2ih¯ (0) (0) (0) (0) − n,α |p|m,β  ⊗ m,β |x|n,α  (0)

(0)

(0) (0)  ⊗ n,α |x|m,β  + m,β |p|n,α

 (0) (0) (0) (0)  ⊗ n,α |p|m,β  = − f n,α;m,β , (15.89) − m,β |x|n,α and reduced oscillator strength tensors f m|n =

1 gm f m,β;n,α = − f . gn gn n|m α,β

This yields tensorial f -sum rules,

(15.90)

15.4 Problems

m,β

f m,β;n,α =

349

% & (0) 1 (0) n,α | [xi , pj ] − [pi , xj ] |n,α ei ⊗ ej = 1 = f m|n . 2ih¯ m

For comparison, we note that the polarization tensor of an isotropic classical oscillator is easily shown to be α(ω) =

1 q2 1. 2 m ω0 − ω2

(15.91)

The standard oscillator strength is related to the oscillator strength tensor via fm,β;n,α =

1 tr f m,β;n,α . 3

(15.92)

15.4 Problems 15.1 Show that the probability current density in the presence of electromagnetic potentials is given by   h¯ q + + +  ∇ − ∇ ·  − 2i  A . j= h¯ 2im

(15.93)

Is this expression gauge invariant? 15.2 Suppose that a particle is moving in a spatially homogeneous magnetic field B(t). Show that in order |qB| this yields the Zeeman term x|HZ (t)|(t) = − x|

q h¯ q B(t) · l|(t) = i B(t) · (x × ∇)(x, t) 2m 2m

in the Schrödinger equation (15.10). Here l = x × p is the angular momentum operator. Hint: You can use Eq. (15.33). 15.3 A hydrogen atom is initially in its ground state when it is excited by an external electric field E(t). 15.3a Show through direct evaluation of the matrix elements that the dipole term V (t) = −qx · E(t) in first order only excites higher level p states. 15.3b The external field is E(t) = ez E exp(− t 2 /τ 2 ).

(15.94)

How large are the first order transition probabilities P1→n into excited bound energy levels?

350

15 Coupling to Electromagnetic Fields

15.4 How large is the ionization probability for a hydrogen atom in the electric field (15.94) in leading order perturbation theory? You will need to calculate the matrix elements into the Coulomb waves numerically. 15.5 Calculate the ionization probability (13.92) for a hydrogen atom in its ground state which is perturbed by an oscillating electric field in z direction, V (t) = ezE cos(ωt). You will need to calculate the matrix elements into the Coulomb waves numerically. 15.6 Calculate the linear Stark effect for the first excited level of hydrogen due to a homogeneous static electric field E. 15.7 Calculate the static polarizability tensor in the ground state of hydrogen. You will need to calculate the contributions from the Coulomb waves numerically. 15.8 Calculate the oscillator strengths fn ;n = 2mωn n |n |x|n|2 /h¯ for a onedimensional oscillator. Why does the equation for the one-dimensional oscillator strength differ by a factor 3 from the three-dimensional oscillator strength (15.71)? Does the result for fn ;n satisfy the Thomas-Reiche-Kuhn sum rule? 15.9 Calculate the oscillator strengths fn,,m ;1,0,0 for the hydrogen atom. How large is the sum   ∞ =0 m =− 0

∞

dk k 2 fk,,m ;1,0,0

(15.95)

of the oscillator strengths into Coulomb waves? Hint: You do not need to calculate the dipole matrix elements into Coulomb waves to answer this question. 15.10 We consider the transition |m → |n between energy eigenstates due to a time-reversal invariant electric field E(−t) = E(t). Show that the square of the corresponding first order scattering matrix element is related to the oscillator strength tensor of the transition through |Snm |2 =

π q2 E(ωnm ) · f n;m · E(ωnm ). mhω ¯ nm

(15.96)

15.11 Show that energy eigenstates with finite expectation values n|x|n have vanishing momentum expectation values, n|p|n = 0. Why does this equation not hold for plane wave states?

(15.97)

15.4 Problems

351

15.12 We assume that |m,β  is a complete set of energy eigenstates with eigenvalues Em . We also assume that the particular state |n,α  satisfies the normalization condition n,α |n,α  = 1 with finite expectation value n,α |x|n,α . Prove the Bethe sum rule [11],  2  2m ωmn m,β | exp(ik · x)|n,α  = k 2 . (15.98) h¯ m,β

Chapter 16

Principles of Lagrangian Field Theory

The replacement of Newton’s equation by quantum mechanical wave equations in the 1920s implied that by that time all known fundamental degrees of freedom in physics were described by fields like A(x, t) or (x, t), and their dynamics was encoded in wave equations. However, all the known fundamental wave equations 1 can be derived from a field theory version of Hamilton’s principle, i.e. the concept of the Lagrange function L(q(t), q(t)) ˙ and the related action S = dt L generalizes ˙ to a Lagrange density L(φ(x, t), φ(x, t), ∇φ(x, t)) with related action S = 3 dt d x L, such that all fundamental wave equations can be derived from the variation of an action, ∂L ∂L − ∂μ = 0. ∂φ ∂(∂μ φ)

(16.1)

This formulation of dynamics is particularly useful for exploring the connection between symmetries and conservation laws of physical systems, and it also allows for a systematic approach to the quantization of fields, which allows us to describe creation and annihilation of particles.

16.1 Lagrangian Field Theory Irrespective of whether we work with relativistic or non-relativistic field theories, it is convenient to use four-dimensional notation for coordinates and partial derivatives,

1 Please

review Appendix A if you are not familiar with Lagrangian mechanics, or if you need a reminder. © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 R. Dick, Advanced Quantum Mechanics, Graduate Texts in Physics, https://doi.org/10.1007/978-3-030-57870-1_16

353

354

16 Principles of Lagrangian Field Theory

x μ = {x 0 , x} ≡ {ct, x},

∂μ =

∂ = {∂0 , ∇}. ∂x μ

(16.2)

We proceed by first deriving the general field equations following from a Lagrangian L(∂φI , φI ) which depends on a set of fields φI (x) ≡ φI (x, t) and their first order derivatives ∂μ φI (x). These fields will be the Schrödinger field (x, t) and its complex conjugate field  + (x, t) in Chap. 17, but in Chap. 18 we will also deal with the wave function A(x) of the photon. We know that the equations of motion for the variables x(t) of classical ˙ x) = 0 in the form of mechanics follow from action principles δS = δ dtL(x, the Euler–Lagrange equations ∂L d ∂L − = 0. ∂xi dt ∂ x˙i

(16.3)

The variation of a field dependent action functional  1 S[φ] = d 4 x L(∂φI , φI ) c V

(16.4)

for fields φI (x) proceeds in the same way as in classical mechanics, the only difference being that we apply the Gauss theorem for the partial integrations. To elucidate this, we require that arbitrary first order variation φI (x) → φI (x) + δφI (x)

(16.5)

with fixed fields at initial and final times t0 and t1 , δφI (x, t0 ) = 0,

δφI (x, t1 ) = 0,

(16.6)

leaves the action S[φ] in first order invariant. We also assume that the fields and their variations vanish at spatial infinity. The first order variation of the action between the times t0 and t1 is δS[φ] = S[φ + δφ] − S[φ]  t1  3 dt [L(∂φI + ∂δφI , φI + δφI ) − L(∂φI , φI )] = d x  =

 3

t0 t1

d x t0

  ∂L ∂L ∂μ δφI . dt δφI + ∂φI ∂(∂μ φI )

(16.7)

Partial integration in the last term yields  δS[φ] =

 3



t1

d x

dt δφI t0

 ∂L ∂L , − ∂μ ∂φI ∂(∂μ φI )

(16.8)

16.1 Lagrangian Field Theory

355

where the boundary terms vanish because of the vanishing variations at spatial infinity and at t0 and t1 . Equation (16.8) implies that we can have δS[φ] = 0 for arbitrary variations δφI (x) between t0 and t1 if and only if the equations ∂L ∂L =0 − ∂μ ∂φI ∂(∂μ φI )

(16.9)

hold for all the fields φI (x). These are the Euler–Lagrange equations for Lagrangian field theory. The derivation of Eq. (16.9) does not depend on the number of four spacetime dimensions, μ ∈ {0, 1, 2, 3}. It would just as well go through in any number d of dimensions, where d could be a number of spatial dimensions if we study equilibrium or static phenomena in field theory, or d can be d − 1 spatial and one time dimension. Relevant cases for observations include d = 1 (mechanics or equilibrium in one-dimensional systems), d = 2 (equilibrium phenomena on interfaces or surfaces, time-dependent phenomena in one-dimensional systems), d = 3 (equilibrium phenomena in three dimensions, time-dependent phenomena on interfaces or surfaces), and d = 4 (time-dependent phenomena in observable spacetime). In particular, classical particle mechanics can be considered as a field theory in one spacetime dimension.

The Lagrange Density for the Schrödinger Field An example is provided by the Lagrange density for the Schrödinger field, L=

  ∂ + h¯ 2 ih¯ ∂ + · − · − ∇ + · ∇ −  + · V · . 2 ∂t ∂t 2m

(16.10)

In the notation of the previous paragraph, this corresponds to fields φ1 (x) =  + (x) and φ2 (x) = (x), or we could also denote the real and imaginary parts of  as the two fields. We have the following partial derivatives of the Lagrange density, ∂L h¯ 2 = − ∂i , ∂(∂i  + ) 2m (16.11) and the corresponding adjoint equations. The Euler–Lagrange equation from variation of the action with respect to  + , ∂L ih¯ ∂ − V , = ∂ + 2 ∂t

∂L ih¯ = − , ∂(∂t  + ) 2

∂L ∂L ∂L − ∂i = 0, − ∂t + + ∂ ∂(∂t  ) ∂(∂i  + )

(16.12)

356

16 Principles of Lagrangian Field Theory

is the Schrödinger equation ih¯

h¯ 2 ∂ +  − V  = 0. ∂t 2m

(16.13)

The Euler–Lagrange equation from variation with respect to  in turn yields the complex conjugate Schrödinger equation for  + . This is of course required for consistency, and for the Lagrange density (16.10) this can also immediately be inferred as a consequence of2 L = L+ . The important conclusion from this section is that Schrödinger’s quantum mechanics is a Lagrangian field theory with a Lagrange density (16.10).

16.2 Symmetries and Conservation Laws We consider an action with fields φ (φI , 1 ≤ I ≤ N) in a d-dimensional space or spacetime: S=

1 c

 d d x L(φ, ∂φ).

(16.14)

To reveal the connection between symmetries and conservation laws, we calculate the first order change of the action S (16.14) if we perform transformations of the coordinates, x  (x) = x − (x).

(16.15)

This transforms the integration measure in the action as & % d d x  = d d x 1 − ∂μ  μ ,

(16.16)

and partial derivatives transform according to & % ∂μ = ∂μ + ∂μ  ν ∂ν .

(16.17)

We also include transformations of the fields, φ  (x  ) = φ(x) + δφ(x).

(16.18)

we could add an arbitrary complex term ∂μ F ( + , ) to the Lagrange density (16.10) and still find the same pair of complex conjugate equations, see Problem 16.1.

2 However,

16.2 Symmetries and Conservation Laws

357

Coordinate transformations often also imply transformations of the fields, e.g. if φ is a tensor field of n-th order with components φα...ν (x), the transformation induced by the coordinate transformation x → x  (x) = x − (x) is φα  ...ν  (x  ) = ∂α  x α · ∂β  x β . . . ∂ν  x ν · φαβ...ν (x).

(16.19)

This yields in first order  (x  ) − φα...ν (x) δφαβ...ν (x) = φα...ν

= ∂α  σ · φσβ...ν (x) + ∂β  σ · φασ ...ν (x) + . . . + ∂ν  σ · φαβ...σ (x). Fields can also transform without a coordinate transformation, e.g. through a phase transformation. We denote the transformations (16.15) and (16.18) as a symmetry of the Lagrangian field theory (16.14) if they leave the volume form d d x L invariant, d d x  L(φ  , ∂  φ  ; x  ) = d d x L(φ, ∂φ; x).

(16.20)

Here we also allow for an explicit dependence of the Lagrange density on the coordinates x besides the implicit coordinate dependence through the dependence on the fields φ(x). If we define a transformed Lagrange density from the requirement of invariance of the action S under the transformations (16.15) and (16.18), L (φ  , ∂  φ  ; x  ) = det(∂  x)L(φ, ∂φ; x),

(16.21)

the symmetry condition (16.20) amounts to form invariance of the Lagrange density. Equations (16.17) and (16.18) imply the following first order change of partial derivative terms: & % % & δ ∂μ φ = ∂μ δφ + ∂μ  ν ∂ν φ.

(16.22)

The resulting first order change of the volume form is (with the understanding that we sum over all fields in all multiplicative terms where the field φ appears twice):     % & % & ∂L ∂L μ σ + δ ∂ρ φ −  δσ L − L L + δφ δ(d x L) = d x 1 − ∂μ  ∂φ ∂(∂ρ φ)      ∂L ∂L − ην μ L + ∂μ δφ = d d x (∂μ  ν ) ∂ν φ · ∂(∂μ φ) ∂(∂μ φ)   ∂L ∂L + δφ − ∂μ ∂φ ∂(∂μ φ)   ∂L ∂L − ∂ μ ∂ν φ · −  μ ∂μ L − ∂μ φ · ∂φ ∂(∂ν φ) d

d

358

16 Principles of Lagrangian Field Theory

   = d x ∂μ  ν ∂ν φ ·

  ∂L ∂L μ − ην L + δφ ∂(∂μ φ) ∂(∂μ φ) )  % & ∂L ∂L . + δφ +  ν ∂ν φ − ∂μ ∂φ ∂(∂μ φ) d

(16.23)

Here δμ L = ∂μ L − ∂μ φ ·

∂L ∂L − ∂μ ∂ν φ · ∂φ ∂(∂ν φ)

(16.24)

is the partial derivative of L with respect to any explicit coordinate dependence. If we have off-shell δ(d d x L) = 0 for the proposed transformations , δφ, we find a local on-shell conservation law ∂μ j μ = 0

(16.25)

with the current density  j μ =  ν ην μ L − ∂ν φ ·

∂L ∂(∂μ φ)

 − δφ ·

∂L . ∂(∂μ φ)

(16.26)

The corresponding charge in a d-dimensional spacetime 1 Q= c



 d

d−1

x j (x, t) = 0

d d−1 x (x, t)

(16.27)

is conserved if no charges are escaping or entering at |x| → ∞:  lim

|x|→∞

d d−2  |x|d−3 x · j (x, t) = 0.

(16.28)

Here d d−2  = dθ1 . . . dθd−2 sind−3 θ1 . . . sin θd−3 is the measure on the (d − 2)dimensional sphere in the d − 1 spatial dimensions, see also (J.43) (note that in (J.43) the number of spatial dimensions is denoted as d). If the off-shell variation of d d xL satisfies δ(d d x L) ≡ d d x ∂μ K μ , the on-shell conserved current is J μ = j μ + K μ and the charge is the spatial integral over J 0 /c. Symmetry transformations which only transform the fields, but leave the coordinates invariant (δφ = 0,  = 0), are denoted as internal symmetries. Symmetry transformations involving coordinate transformations are denoted as external symmetries. The connection between symmetries and conservation laws was developed by Emmy Noether3 and is known as Noether’s theorem.

3 Noether

[126], see also arXiv:physics/0503066.

16.2 Symmetries and Conservation Laws

359

Energy-Momentum Tensors We now specialize to inertial (i.e. pseudo-Cartesian) coordinates in Minkowski spacetime. If the coordinate shift in (16.15) is a constant translation, ∂μ  ν = 0, all fields transform like scalars, δφ = 0, and the conserved current becomes  j μ =  ν ην μ L − ∂ν φ ·

∂L ∂(∂μ φ)

 =  ν ν μ .

(16.29)

Omitting the d irrelevant constants  ν leaves us with d conserved currents (0 ≤ ν ≤ d − 1) ∂μ ν μ = 0,

(16.30)

with components ∂L . ∂(∂μ φ)

ν μ = ην μ L − ∂ν φ ·

(16.31)

The corresponding conserved charges 1 pν = c

 d d−1 x ν 0

(16.32)

are the components of the four-dimensional energy-momentum vector of the physical system described by the Lagrange density L, and the tensor with components ν μ is therefore denoted as an energy-momentum tensor. The spatial components ij of the energy-momentum tensor have dual interpretations in terms of momentum current densities and forces. To explain the meaning of ij , we pick an arbitrary (but stationary) spatial volume V . Since we are talking about fields, part of the fields will reside in V . From Eq. (16.32), the fields in V will carry a part of the total momentum p which is p V = ei

1 c

 d d−1 x i0 .

(16.33)

V

The Eqs. (16.30) and (16.32) imply that the change of pV is given by d p = ei dt V



* d d−1 x ∂0 i0 = − ei V

d d−2 Sj ij ,

(16.34)

∂V

where the Gauss theorem in d − 1 spatial dimensions was employed and d d−2 Sj is the outward bound surface element on the boundary ∂V of the volume. This equation tells us that the component ij describes the flow of the momentum component pi through the plane with normal vector ej , i.e. ij is the flow of

360

16 Principles of Lagrangian Field Theory

momentum pi in the direction ej and j i = ij ej is the corresponding current density. In the dual interpretation, we read Eq. (16.34) with the relation F V = dp V /dt between force and momentum change in mind. In this interpretation, F V is the force exerted on the fields in the fixed volume V , because it describes the rate of change of momentum of the fields in V . −F V is the force exerted by the fields in the fixed volume V . The component ij is then the force per area with normal vector ej , which is exerted by the fields in direction ei . The component −ij is the force per area with normal vector ej , which is acting on the fields in direction ei . This represents strain or pressure for i = j and stress for i = j . The energy-momentum tensor is therefore also known as stress-energy tensor. There is another equation for the energy-momentum tensor in general relativity, which agrees with Eq. (16.31) for scalar fields, but not for vector or relativistic spinor fields. Both definitions yield the same conserved energy and momentum of a system, but improvement terms have to be added to the tensor from Eq. (16.31) in relativistic field theories to get the correct expressions for local densities for energy and momentum. We will discuss the necessary modifications of ν μ for the Maxwell field (photons) in Sect. 18.1 and for relativistic fermions in Sect. 22.4.

16.3 Applications to Schrödinger Field Theory The energy-momentum tensor for the Schrödinger field is found by substituting (16.10) into Eq. (16.31). The corresponding energy density is usually written as a Hamiltonian density H, H = cP 0 = − 0 0 =

h¯ 2 ∇ + · ∇ +  + · V · , 2m

(16.35)

and the momentum density is P=

& 1 h¯ % + ei i0 =  · ∇ − ∇ + ·  . c 2i

(16.36)

The energy current density for the Schrödinger field follows as j H = − c0 i ei = −

  h¯ 2 ∂ + ∂ ∇ + · + · ∇ . 2m ∂t ∂t

(16.37)

The energy E = d 3 x H and momentum p = d 3 x P agree with the corresponding expectation values of the Schrödinger wave function in quantum mechanics. The results of the previous section, or direct application of the Schrödinger equation, tell us that E is conserved if the potential is time-independent, V = V (x), and the momentum component e.g. in x-direction is conserved if the momentum does not depend on x, V = V (y, z).

16.3 Applications to Schrödinger Field Theory

361

Probability and Charge Conservation from Invariance Under Phase Rotations The Lagrange density (16.10) is invariant under phase rotations of the Schrödinger field, q δ(x, t) = i ϕ(x, t), h¯

q δ + (x, t) = − i ϕ + (x, t). h¯

(16.38)

We wrote the constant phase in the peculiar form qϕ/h¯ in anticipation of the connection to local gauge transformations (15.14) and (15.16), which will play a recurring role later on. However, for now we note that substitution of the phase transformations into the Eq. (16.26) yields after division by the irrelevant constant qϕ the density 1 j0 =− = c qϕ

 δ

∂L ∂L + δ + ∂(∂t ) ∂(∂t  + )



=  + =

1 q q

(16.39)

and the related current density j =− =

1 qϕ

 δ

∂L ∂L + δ + ∂(∇) ∂(∇ + )

 =

& h¯ % +  · ∇ − ∇ + ·  2im

1 j . q q

(16.40)

Comparison with Eqs. (1.68) and (1.69) shows that probability conservation in Schrödinger theory can be considered as a consequence of invariance under global phase rotations. Had we not divided out the charge q, we would have drawn the same conclusion for conservation of electric charge with q = q +  as the charge density and j q = qj as the electric current density. The coincidence of the conservation laws for probability and electric charge in Schrödinger theory arises because it is a theory for non-relativistic particles. Only charge conservation will survive in the relativistic limit, but probability conservation for particles will not hold any more, because q (x, t)/q will not be positive definite any more and therefore will not yield a quantity that could be considered as a probability density to find a particle in the location x at time t. Comparison with Eq. (16.36) tells us that j is also proportional to the momentum density, j (x, t) =

1 P(x, t), m

(16.41)

which tells us that the probability current density of the Schrödinger field is also a velocity density.

362

16 Principles of Lagrangian Field Theory

16.4 Problems 16.1 Irrelevance of total derivative terms in the Lagrange density 16.1a Show that addition of a derivative term ∂μ F(φI ) to the Lagrange density L(φI , ∂φI ) does not change the Euler–Lagrange equations. 16.1b Show also that addition of the derivative term to the Lagrange density does not change any conserved currents, i.e. μ

KF = δφI ·

∂F(φ1 , . . . φN ) μ = −jF . ∂φI

(16.42)

16.2 We consider classical particle mechanics with a Lagrangian L(qI , q˙I ). 16.2a Suppose the action is invariant under constant shifts δqJ of the coordinate qJ (t). Which conserved quantity do you find from Eq. (16.26)? Which condition must L fulfill to ensure that the action is not affected by the constant shift δqJ ? 16.2b Now we assume that the action is invariant under constant shifts δt = − of the internal coordinate t. Which conserved quantity do you find from Eq. (16.26) in this case? 16.3 Use the Schrödinger equation to confirm that the energy density (16.35) and the energy current density (16.37) indeed satisfy the local conservation law ∂ H = −∇ · j H ∂t

(16.43)

if the potential is time-independent, V = V (x). How does E change if V = V (x, t) is time-dependent? 16.4 We have only evaluated the components 0 0 , i 0 and 0 i of the energymomentum tensor of the Schrödinger field in Eqs. (16.35)–(16.37). Which momentum current densities j iP do you find from the energy-momentum tensor of the Schrödinger field? 16.5 Schrödinger fields can have different transformation properties under coordinate rotations δx = −ϕ ×x, see Sect. 8.2. In this problem we analyze a Schrödinger field which transforms like a scalar under rotations, δ(x, t) =   (x  , t) − (x, t) = 0.

(16.44)

The Lagrange density (16.10) is invariant under rotations if V = V (r, t). Which conserved quantity do you find from this observation? Solution Equation (16.26) yields with  = ϕ × x a conserved charge density

16.4 Problems

363

  ∂L ∂L j0 + = − (ϕ × x) · ∇ + ∇ = c ∂(∂t ) ∂(∂t  + ) =−

&" % ih¯ ! ϕ · x ×  + · ∇ − ∇ + ·  = ϕ · M, 2

(16.45)

with an angular momentum density M=

% & h¯ x ×  + · ∇ − ∇ + ·  = x × P. 2i

(16.46)

Since the constant parameters ϕ are arbitrary, we find three linearly independent conserved quantities, viz. the angular momentum  M=

d 3 x M = x × p

(16.47)

of the scalar Schrödinger field. 16.6 Now we assume that our Schrödinger field is a 2-spinor with the transformation property δ =

i (ϕ · σ ) · , 2

i δ + = −  + · (ϕ · σ ). 2

(16.48)

Show that the corresponding density of “total angular momentum” of the Schrödinger field in this case consists of an orbital and a spin part, J =

% & h¯ h¯ x ×  + · ∇ − ∇ + ·  +  + · σ ·  2i 2

= x × P +  + · S ·  = M + S.

(16.49)

Rotational invariance implies only conservation of the total angular momentum J = 3 d x J . However, on the level of the Lagrange density (16.10), which does not contain spin-orbit interaction terms (8.33), the orbital and spin parts are preserved separately. We will see in Sect. 22.5 that spin-orbit coupling is a consequence of relativity. 16.7 An electron has a normalized wave function   1 1 (x, t) = ψ(x, t) √ . 2 −1

(16.50)

16.7a Calculate the spin expectation value S and the uncertainty S = {Sx , Sy , Sz } of the electron. 16.7b Do your results comply with the uncertainty relation (5.8)?

364

16 Principles of Lagrangian Field Theory

16.8 Suppose the Hamiltonian has the spin-orbit coupling form H = αM ·S, where Mi and Si are angular momentum and spin operators. How do these operators evolve in the Heisenberg picture? 16.8a Show that the Heisenberg evolution equations for the operators yield ˙ = αS × M, M

S˙ = αM × S.

(16.51)

16.8b Show that J ≡ M + S, M 2 , S 2 and M · S are all constant. 16.8c Show that the evolution equations (16.51) are solved by M(t) = exp(− αJ · Lt − ihαt) ·M ¯

(16.52)

· S, S(t) = exp(− αJ · Lt − ihαt) ¯

(16.53)

and

where M ≡ M(0), S ≡ S(0), and L = (L1 , L2 , L3 ) is the vector of matrices with components (Li )j k = ij k , see Eq. (7.44). 16.8d Show that the solutions (16.52) and (16.53) can also be written in the form M(t) = M · exp(αJ · Lt + ihαt), ¯

(16.54)

S(t) = S · exp(αJ · Lt + ihαt). ¯

(16.55)

16.9 We are dealing with particles with spin, and therefore their states will in general be sums over tensor product states | =



|ψa  ⊗ |χa ,

(16.56)

a

where |ψa  are orbital states and |χa  are spin states. On the other hand, the Eqs. (16.52) and (16.53) seem to suggest that M(t) and S(t) are rotating around the direction of the vector J with angular velocity ω = αJ . This suggestive picture of coupled angular momentum type operators rotating around the total angular momentum vector is often denoted as the vector model of spin-orbit type couplings. However, note that the two components of the total angular momentum vector are acting on different factors in tensor products of orbital and spin states. The vector operator J therefore has the following form in fully explicit notation, J = M ⊗ 1 + 1 ⊗ S.

(16.57)

That does not mean that the results (16.51–16.55) or the conservation laws expressed in 16.8b are incorrect, but we must beware of simple interpretations in terms of

16.4 Problems

365

vectors living within one and the same vector space as long as we are dealing with vector operators acting on different factors in tensor products.4 Repeat the previous Problem 16.8 in terms of explicit tensor product notation with the spin-orbit coupling Hamiltonian .

H = αM ⊗ S = αMi ⊗ Si .

(16.58)

The dot over the tensor product reminds us of the sum over the vector components of the operators in the two factor spaces. 16.10 Electromagnetic potentials in the Schrödinger Lagrangian 16.10a Show that the Lagrange density   ih¯ ∂ + + ∂ L=  · − ·  − q + ·  ·  2 ∂t ∂t    h¯ 2 q + q + ∇ + i  A · ∇ − i A − 2m h¯ h¯

(16.59)

yields the equations of motion for the Schrödinger field in external electromagnetic fields E(x, t) = − ∇(x, t) −

∂ A(x, t), ∂t

B(x, t) = ∇ × A(x, t).

(16.60)

16.10b Derive the Lorentz force law from the results of Problem 16.10a. 16.11 Derive the electric charge and current densities for the Schrödinger field in electromagnetic fields from the phase invariance of (16.59). Answers The charge density is q = q + .

(16.61)

The current density is jq =

& q2 q h¯ % +  · ∇ − ∇ + ·  −  + A. 2im m

(16.62)

Are the charge and current densities gauge invariant?

4 As

soon as we map the operators acting either on the orbital factors or on the spin factors of quantum states into expectation values, we can recover the picture of the vector model, because at least one of M(t) or S(t) is now a classical vector which can be interpreted as an operator M(t)1 or S(t)1, respectively, acting in the complementary factor space. Once both vector operators have been mapped into expectation values, we recover the vector model in its standard interpretation as angular momentum vector M(t) and spin vector S(t) rotating around the constant vector J with angular velocity ω = αJ .

Chapter 17

Non-relativistic Quantum Field Theory

Quantum mechanics, as we know it so far, deals with invariant particle numbers, d (t)|(t) = 0. dt

(17.1)

However, at least one of the early indications of wave-particle duality implies disappearance of a particle, viz. absorption of a photon in the photoelectric effect. This reminds us of two deficiencies of Schrödinger’s wave mechanics: it cannot deal with absorption or emission of particles, and it cannot deal with relativistic particles. In the following sections we will deal with the problem of absorption and emission of particles in the non-relativistic setting, i.e. for slow electrons, protons, neutrons, or nuclei, or quasiparticles in condensed matter physics. The strategy will be to follow a quantization procedure that works for the promotion of classical mechanics to quantum mechanics, but this time for Schrödinger theory. The correspondences are summarized in Table 17.1. The key ingredient is promotion of the “classical” variables x or (x, t) to operators through “canonical (anti-)commutation relations”, as outlined in the last two lines of Table 17.1. This procedure of promoting classical variables to operators by imposing canonical commutation or anti-commutation relations is called canonical quantization. Canonical quantization of fields is denoted as field quantization. Since the fields are often wave functions (like the Schrödinger wave function) which arose from the quantization of x and p, field quantization is sometimes also called second quantization. A quantum theory that involves quantized fields is denoted as a quantum field theory. Indeed, quantum field theory is essentially as old as Schrödinger’s wave mechanics, because it was clear right after the inception of quantum mechanics that the formalism was not yet capable of the description of quantum effects for photons. This led to the rapid invention of field quantization in several steps between 1925 and 1928. Key advancements [19, 39, 91] were the formulation of a quantum field © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 R. Dick, Advanced Quantum Mechanics, Graduate Texts in Physics, https://doi.org/10.1007/978-3-030-57870-1_17

367

368

17 Non-relativistic Quantum Field Theory

Table 17.1 Correspondence between first and second quantization Classical mechanics Independent variable t Dependent variables x(t)

Schrödinger’s wave mechanics Independent variables x, t Dependent variables (x, t),  + (x, t)

Newton’s equation

Schrödinger’s equation

mx¨ = −∇V (x)

ih¯ ∂t∂  = −

Lagrangian

Lagrangian % L = i2h¯  + ·

L=

m 2 ˙ 2x

− V (x)

− Conjugate momenta pi (t) = ∂L/∂ x˙i (t) = mx˙i (t)

h¯ 2 + 2m ∇

h¯ 2 2m 

−

∂ + ∂t 

·

&

· ∇ −  + · V · 

Conjugate momenta ˙ " (x, t) = ∂ L/∂ (x, t) = "+  (x, t) = −

Canonical commutators [xi (t), pj (t)] = ih¯ δij , [xi (t), xj (t)] = 0, [pi (t), pj (t)] = 0

∂ ∂t 

+V

ih¯ + 2  (x, t),

ih¯ 2 (x, t)

Canonical (anti-)commutators [(x, t),  + (x  , t)]∓ = δ(x − x  ), [(x, t), (x  , t)]∓ = 0

as a superposition of infinitely many oscillation operators by Born, Heisenberg and Jordan in 1926, the application of infinitely many oscillation operators by Dirac in 1927 for photon emission and absorption, and the introduction of anticommutation relations for fermionic field operators by Jordan and Wigner in 1928. Path integration over fields was introduced by Feynman in the 1940s.

17.1 Quantization of the Schrödinger Field We will now start to perform the program of canonical quantization of Schrödinger’s wave mechanics. First steps will involve the promotion of wave functions like (x, t) and  + (x, t) to field operators or quantum fields through the proposition of canonical commutation or anti-commutation relations, and the identification of related composite field operators like the Hamiltonian, momentum and charge operators. The composite operators will then help us to reveal the physical meaning of the Schrödinger quantum fields (x, t) and  + (x, t) as annihilation and creation operators for particles. The Lagrange density (16.10) yields the canonically conjugate momenta " =

∂L ih¯ =  +, ˙ 2 ∂

" + =

∂L ih¯ = − , ˙+ 2 ∂

(17.2)

17.1 Quantization of the Schrödinger Field

369

and the canonical commutation relations1 translate for fermions (with the upper signs corresponding to anti-commutators) and bosons (with the lower signs corresponding to commutators) into [(x, t),  + (x  , t)]± ≡ (x, t) + (x  , t) ±  + (x  , t)(x, t) = δ(x − x  ), [(x, t), (x  , t)]± = 0,

[ + (x, t),  + (x  , t)]± = 0.

(17.3) (17.4)

Whether the quantum field for a particle should be quantized using commutation or anti-commutation relations depends on the spin of the particle, i.e. on the transformation properties of the field under rotations, see Chap. 8. Bosons have integer spin and are quantized through commutation relations while fermions have half-integer spin and are quantized through anti-commutation relations. Therefore we should include spin labels (which were denoted as ms or a in Chap. 8) with the quantum fields, e.g. ms (x, t), ms ∈ {−s, −s + 1, . . . , s}, for a field describing particles of spin s and spin projection ms . We will explicitly include spin labels in Sect. 17.5, but for now we will not clutter the equations any more than necessary, since spin labels can usually be ignored as long as dipole approximation λ  a0 applies. Here a0 is the Bohr radius and λ is the wavelength of photons which might interact with the Schrödinger field. Spin-flipping transitions are suppressed roughly by a factor a02 /λ2 relative to spin-preserving transitions in dipole approximation. See the remarks after Eq. (18.158). The commutation relations (17.3) in the bosonic case are like the commutation relations [ai , aj+ ] = δij etc. for oscillator operators. Therefore we can think of the field operators (x, t) and  + (x  , t) as annihilation and creation operators for each point in spacetime. We will explicitly confirm this interpretation below by showing that the corresponding Fourier transformed operators a(k) and a + (k) (in the Schrödinger picture) annihilate or create particles of momentum hk, ¯ respectively. We will also see how linear superpositions of the operators ψ + (x) =  + (x, 0) act on the vacuum to generate e.g. states |n, , m  which correspond to hydrogen eigenstates. Note that (x, t) and  + (x, t) are now time-dependent operators and their time evolution is determined by the full dynamics of the system. Therefore they are operators in the Heisenberg picture of the second quantized theory, i.e. what had been representations of states in the Schrödinger picture of the first quantized theory has become field operators in the Heisenberg picture of the second quantized theory. The elevation of wave functions to operators implies that functions or functionals of the wave functions that we had encountered in quantum mechanics now also the canonical commutation relations [xi (t), pj (t)] = ih¯ δij , [xi (t), xj (t)] = 0, [pi (t), pj (t)] = 0 in the Heisenberg picture of quantum mechanics. It is customary to dismiss a factor of 2 in the (anti-)commutation relations (17.3), which otherwise would simply reappear in different places of the quantized Schrödinger theory.

1 Recall

370

17 Non-relativistic Quantum Field Theory

become operators. Particularly important cases of functionals of wave functions include expectation values for observables like energy, momentum, and charge, and these will all become operators in the second quantized theory. E.g. the Hamiltonian densityis related to the Lagrange density through a Legendre transformation (cf. ˙ ˙+ H = i pi q˙i − L in3 mechanics), H = "  +  " + − L. This yields the Hamiltonian H = d x H in the form 

 H =

3

d x

 h¯ 2 + + ∇ (x, t) · ∇(x, t) +  (x, t)V (x)(x, t) . 2m

(17.5)

We have also found the Hamiltonian density in Eq. (16.35) from the energymomentum tensor of the Schrödinger field, which in addition gave us the momentum  P (t) =

d 3 x P(x, t) 

=

d 3x

& h¯ % +  (x, t) · ∇(x, t) − ∇ + (x, t) · (x, t) . 2i

(17.6)

We can just as well use the equivalent expressions2  H =



h¯ 2 +  (x, t)(x, t) +  + (x, t) · V (x) · (x, t) d x − 2m 3

 (17.7)

and P (t) = − ih¯ d 3 x  + (x, t)∇(x, t), which can be motivated from the corresponding equations for the energy and momentum expectation values in the first quantized Schrödinger theory. Other frequently used composite operators3 include the number and charge operators N and Q, cf. (16.40),  N=

2 The

 d 3 x (x, t) =

d 3 x  + (x, t)(x, t) =

1 Q. q

(17.8)

Hamiltonians (17.5) and (17.7) are equivalent if the calculations of their matrix elements convert the field operators (x, t) and  + (x, t) into normalizable wave functions. However, they are also equivalent in evaluations where their matrix elements yield δ functions. We will encounter many examples of matrix elements of second quantized operators in applications of the formalism. 3 For another composite operator we can also define an integrated current density through I (t) = q 3 d x j q (x, t) = qP (t)/m, where the last equation follows from (16.41). However, recall that j q (x, t) is a current density, but it is not a current per volume, and therefore I q (t) is not an electric current but comes in units of e.g. Ampère meter. It is related to charge transport like momentum P (t) is related to mass transport.

17.1 Quantization of the Schrödinger Field

371

Before we continue with the demonstration that (x, t) and  + (x  , t) are annihilation and creation operators, we should confirm our suspicion that they are indeed operators in the Heisenberg picture of quantum field theory. We will do this next.

Time Evolution of the Field Operators Very useful identities for commutators involving products of operators are [AB, C] = ABC − CAB = ABC + ACB − ACB − CAB = A[B, C]± − [C, A]± B,

(17.9)

[A, BC] = ABC − BCA = ABC + BAC − BAC − BCA = [A, B]± C − B[C, A]± .

(17.10)

These relations and the canonical (anti-)commutation relations between the field operators imply that both bosonic and fermionic field operators (x, t) satisfy the Heisenberg evolution equations, ih¯ i ∂ i (x, t) = (x, t) − V (x)(x, t) = [H, (x, t)], (17.11) ∂t 2m h¯ h¯ h¯ ∂ + i i  (x, t) =  + (x, t) +  + (x, t)V (x) = [H,  + (x, t)]. h¯ h¯ ∂t 2im (17.12) However, then we also get (note that here the time-independence of V (x) is important) i d H = [H, H ] = 0, dt h¯

(17.13)

which was already anticipated in the notation by writing H rather than H (t). The relations (17.11) and (17.12) confirm the Heisenberg picture interpretation of the Schrödinger field operators (x, t) and  + (x, t).

k-Space Representation of Quantized Schrödinger Theory In quantum mechanics, we used wave functions in k-space both for scattering theory and for the calculation of the time evolution of free wave packets. The kspace representation becomes even more important in quantum field theory because ensembles of particles have additive quantum numbers like total momentum and

372

17 Non-relativistic Quantum Field Theory

total kinetic energy which depend on the wave vector k of a particle, and this will help us to reveal the meaning of the Schrödinger field operators. The mode expansion in the Heisenberg picture  1 (x, t) = √ 3 d 3 k a(k, t) exp(ik · x), 2π  1 a(k, t) = √ 3 d 3 x (x, t) exp(− ik · x), 2π

(17.14) (17.15)

implies with (17.3) the (anti-)commutation relations for the field operators in kspace, [a(k, t), a + (k  , t)]± = δ(k − k  ), [a(k, t), a(k  , t)]± = 0,

[a + (k, t), a + (k  , t)]± = 0.

(17.16) (17.17)

Furthermore, substitution of Eq. (17.14) into the charge, momentum and energy operators yields  Q = qN = q 

d 3 k a + (k, t)a(k, t),

+ d 3 k hk ¯ a (k, t)a(k, t)

P (t) =

(17.18)

(17.19)

and H = H0 (t) + V (t),

(17.20)

with the kinetic and potential operators 

h¯ 2 k 2 + a (k, t)a(k, t) 2m

(17.21)

d 3 q a + (k + q, t)V (q)a(k, t).

(17.22)

H0 (t) =

d 3k

and 

 V (t) =

3

d k

Here we used the following normalization for the Fourier transform of singleparticle potentials,  V (x) =

d 3 q V (q) exp(iq · x),

(17.23)

17.1 Quantization of the Schrödinger Field

V (q) = V + (− q) =

1 (2π )3

373

 d 3 x V (x) exp(− iq · x).

(17.24)

Field Operators in the Schrödinger Picture and the Fock Space for the Schrödinger Field The relations in the Heisenberg picture i ∂ (x, t) = [H, (x, t)], ∂t h¯

i ∂ a(k, t) = [H, a(k, t)], ∂t h¯

dH =0 dt (17.25)

imply 

   i i (x, t) = exp H t ψ(x) exp − H t , h¯ h¯     i i a(k, t) = exp H t a(k) exp − H t . h¯ h¯

(17.26) (17.27)

The time-independent operators ψ(x) = (x, 0), a(k) = a(k, 0) are the corresponding operators in the Schrödinger picture of the quantum field theory.4 Having time-independent operators in the Schrödinger picture comes at the expense of time-dependent states   i |(t) = exp − H t |(0), h¯

(17.28)

to preserve the time dependence of matrix elements and observables. Here we use a boldface bra-ket notation | and | for states in the second quantized theory to distinguish them from the states | and | in the first quantized theory. The canonical (anti-)commutation relations for the Heisenberg picture operators imply canonical (anti-)commutation relations for the Schrödinger picture operators, [ψ(x), ψ + (x  )]± = δ(x − x  ), [ψ(x), ψ(x  )]± = 0,

[ψ + (x), ψ + (x  )]± = 0,

[a(k), a + (k  )]± = δ(k − k  ), [a(k), a(k  )]± = 0,

4 For

[a + (k), a + (k  )]± = 0.

convenience, we have chosen the time when both pictures coincide as t0 = 0.

(17.29) (17.30) (17.31) (17.32)

374

17 Non-relativistic Quantum Field Theory

These are oscillator-like commutation or anti-commutation relations, and to figure out what they mean we will look at all the composite operators of the Schrödinger field that we had constructed before. Time-independence of the full Hamiltonian implies that we can express H in terms of the field operators (x, t) in the Heisenberg picture or the field operators ψ(x) in the Schrödinger picture, 

 h¯ 2 + + ∇ (x, t) · ∇(x, t) +  (x, t) · V (x) · (x, t) H = d x 2m    h¯ 2 3 + + ∇ψ (x) · ∇ψ(x) + ψ (x) · V (x) · ψ(x) = d x 2m    h¯ 2 k 2 + = d 3k a (k)a(k) + d 3 k d 3 q a + (k + q)V (q)a(k). (17.33) 2m 

3

However, the free Hamiltonians in the Heisenberg picture and in the Schrödinger picture depend in the same way on the respective field operators, but they are different operators if V = 0,    i i H0 = exp − H t H0 (t) exp H t h¯ h¯     i i ¯2 3 h + ∇ (x, t) · ∇(x, t) exp H t = exp − H t d x 2m h¯ h¯   h¯ 2 h¯ 2 k 2 + ∇ψ + (x) · ∇ψ(x) = d 3 k a (k)a(k). (17.34) = d 3x 2m 2m 

The number and charge operators in the Schrödinger picture are  N=

 d 3 x (x) =

d 3 x ψ + (x)ψ(x) =



d 3 k a + (k)a(k) =

1 Q, q

(17.35)

and the momentum operator is  P =

d 3x 

=

& h¯ % + ψ (x) · ∇ψ(x) − ∇ψ + (x) · ψ(x) 2i

d 3 k h¯ k a + (k)a(k).

(17.36)

The momentum operator P (t) in the Heisenberg picture (17.19) is related to the momentum operator P in the Schrödinger picture through the standard transformation between Schrödinger picture and Heisenberg picture,

17.1 Quantization of the Schrödinger Field

375



   i i P (t) = exp H t P exp − H t , h¯ h¯

(17.37)

and the same similarity transformation applies to all the other operators. However, we did not write N(t) or Q(t) in Eqs. (17.8), (17.18), because [H, N] = 0 for the single-particle Hamiltonian (17.33). We are now fully prepared to identify the meaning of the operators a(k) and a + (k). The commutation relations [H0 , a(k)] = −

h¯ 2 k 2 a(k), 2m

[H0 , a + (k)] =

h¯ 2 k 2 + a (k), 2m

(17.38)

+ (k), [P , a + (k)] = hka ¯

(17.39)

[Q, a(k)] = − qa(k),

[Q, a + (k)] = qa + (k),

(17.40)

[N, a(k)] = − a(k),

[N, a + (k)] = a + (k),

(17.41)

[P , a(k)] = − hka(k), ¯

imply that a(k) annihilates a particle with energy h¯ 2 k 2 /2m, momentum hk, ¯ mass m and charge q, while a + (k) generates such a particle. This follows exactly in the same way as the corresponding proof for energy annihilation and creation for the harmonic oscillator (6.14)–(6.16). Suppose e.g. that |K is an eigenstate of the momentum operator, P |K = h¯ K|K.

(17.42)

The commutation relation (17.39) then implies + P a + (k)|K = a + (k) (P + hk) ¯ |K = h¯ (K + k) a (k)|K,

(17.43)

i.e. a + (k)|K ∝ |K + k,

(17.44)

while (17.38) implies for an energy eigenstate |E, H |E = E|E, the relation a + (k)|E ∝ |E + (h¯ 2 k 2 /2m).

(17.45)

The Hamilton operator (17.34) therefore corresponds to an infinite number of 2 harmonic oscillators with frequencies ω(k) = hk ¯ /2m, and there must exist a lowest energy state |0 which must be annihilated by the lowering operators, a(k)|0 = 0.

(17.46)

The general state then corresponds to linear superpositions of states of the form

376

17 Non-relativistic Quantum Field Theory

|{nk } =

 a + (k)nk |0. √ nk ! k

(17.47)

This vector space of states is denoted as a Fock space. The particle annihilation and creation interpretation of a(k) and a + (k) then also implies that the Fourier component V (q) in the potential term of the full Hamiltonian (17.33) shifts the momentum of a particle by p = hq ¯ by replacing a particle with momentum h¯ k with a particle of momentum hk ¯ + hq. ¯

Time-Dependence of H0 The free Hamiltonian H0 (17.34) is time-independent in the Schrödinger picture (and also in the Dirac picture introduced below), but not in the Heisenberg picture if [H0 , H ] = 0. The transformation from the Schrödinger picture into the Heisenberg picture,  H0 (t) =

    h¯ 2 i i + ∇ (x, t) · ∇(x, t) = exp H t H0 exp − H t , d x 2m h¯ h¯ 3

implies the evolution equation dH0 (t) i i = [H, H0 (t)] = [V (t), H0 (t)] dt h¯ h¯     i i i = exp H t [V , H0 ] exp − H t . h¯ h¯ h¯

(17.48)

The operator  V (t) =

d 3 x  + (x, t)V (x)(x, t) = exp



   i i H t V exp − H t h¯ h¯

is the potential operator in the Heisenberg picture, while the potential operator in the Schrödinger picture is  V =

d 3 x ψ + (x)V (x)ψ(x) =



 d 3k

d 3 q a + (k + q)V (q)a(k).

(17.49)

The commutator in the Schrödinger picture follows from the canonical commutators or anti-commutators of the field operators as  [V , H0 ] =

d 3x

& h¯ 2 % + ψ (x) · ∇ψ(x) − ∇ψ + (x) · ψ(x) · ∇V (x) (17.50) 2m

17.2 Time Evolution for Time-Dependent Hamiltonians

 =−

 d 3k

d 3q

 h¯ 2 2 q + 2k · q a + (k + q)V (q)a(k). 2m

377

(17.51)

The integral in Eq. (17.50) contains the current density (1.69), (16.40) of the Schrödinger field. The commutator can therefore be written as  [V , H0 ] = ih¯

d 3 x j (x) · ∇V (x),

(17.52)

and substitution into the Heisenberg picture evolution equations for H0 (t) (17.48) yields d H0 (t) = − dt

 d 3 x j (x, t) · ∇V (x).

(17.53)

However, we have also identified j (x, t) as a velocity density operator for the Schrödinger field, cf. (16.41). The classical analog of Eq. (17.53) is therefore the equation for the change of the kinetic energy of a classical non-relativistic particle moving under the influence of the force F (x) = −∇V (x), d K(t) = − v(t) · ∇V (x). dt

(17.54)

17.2 Time Evolution for Time-Dependent Hamiltonians The generic case in quantum field theory are time-independent Hamilton operators in the Heisenberg and Schrödinger pictures. We will see the reason for this below, after discussing the general case of a Heisenberg picture Hamiltonian H (t) ≡ HH (t) which could depend on time. Integration of Eq. (17.11) yields in the general case of time-dependent H (t) (t) = (t0 ) +

i h¯



t

dτ [H (τ ), (τ )] = U˜ (t, t0 )(t0 )U˜ + (t, t0 ),

(17.55)

t0

with the unitary operator   t  i ˜ ˜ U (t, t0 ) = T exp dτ H (τ ) . h¯ t0

(17.56)

Here T˜ locates the Hamiltonians near the upper time integration boundary leftmost, but for the factor +i in front of the integral, see also Eq. (13.28). Recall that in the Heisenberg picture, we have all time dependence in the operators, but time-independent states. To convert to the Schrödinger picture, we

378

17 Non-relativistic Quantum Field Theory

remove the time dependence from the operators and cast it onto the states such that matrix elements remain the same, (t0 )|(t)|(t0 ) = (t)|(t0 )|(t). The time evolution of the states in the Schrödinger picture is therefore given by |(t) = U˜ (t0 , t)|(t0 ).

(17.57)

This implies a Schrödinger equation ih¯

d |(t) = U˜ (t0 , t)HH (t)|(t0 ) = U˜ (t0 , t)HH (t)U˜ (t, t0 )|(t) dt = HS (t)|(t). (17.58)

Therefore we also have    i t dτ HS (τ ) |(t0 ), |(t) = U (t, t0 )|(t0 ) = T exp − h¯ t0

(17.59)

i.e.   t0  i ˜ ˜ U (t0 , t) = T exp dτ HH (τ ) = U (t, t0 ) h¯ t    i t = T exp − dτ HS (τ ) , h¯ t0

(17.60)

where HS (t) = U˜ (t0 , t)HH (t)U˜ (t, t0 ),

HH (t) = U (t0 , t)HS (t)U (t, t0 ).

(17.61)

The Hamiltonian in the Schrödinger picture depends only on the t-independent field operators (t0 ), i.e. any time dependence of HS can only result from an explicit time dependence of any parameter, e.g. if a coupling constant or mass would somehow depend on time. If such a time dependence through a parameter is not there, then U (t, t0 ) = exp[− iHS (t − t0 )/h] ¯ and HH (t) = HS , i.e. HS is time-independent if and only if HH is time-independent, and then HS = HH . This explains why time-independent Hamiltonians HS = HH are the generic case in quantum field theory. Usually, if we would discover any kind of time dependence in any parameter λ = λ(t) in HS , we would suspect that there must be a dynamical explanation in terms of a corresponding field, i.e. we would promote λ(t) to a full dynamical field operator besides all the other field operators in HS , including a kinetic term for λ(t), and then the new Hamiltonian would again be time-independent. Occasionally, we might prefer to treat a dynamical field as a given timedependent parameter, e.g. include electric fields in a semi-classical approximation instead of dealing with the quantized photon operators. This is standard practice

17.3 The Connection Between First and Second Quantized Theory

379

in the “first quantized” theory, and therefore time dependence of the Schrödinger and Heisenberg Hamiltonians plays a prominent role there. However, once we go through the hassle of field quantization, we may just as well do the same for all the fields in the theory, including electromagnetic fields, and therefore semiclassical approximations and ensuing time dependence through parameters is not as important in the second quantized theory.

17.3 The Connection Between First and Second Quantized Theory For a single particle first and second quantized theory should yield the same expectation values, i.e. matrix elements in the 1-particle sector should agree: | = |.

(17.62)

For the states |x = ψ + (x)|0,

|k = a + (k)|0,

(17.63)

Eq. (17.62) is fulfilled due to the standard Fourier transformation relation between the operators in x-space and k-space. The relations ψ + (x) =



d 3 k a + (k)k|x, ψ(x) =



+

a (k) =

d 3 k x|ka(k),

(17.64)

d 3 x k|xψ(x),

(17.65)



+

d x ψ (x)x|k, a(k) = 3



yield x|k = 0|ψ(x)a + (k)|0 =  =



d 3 k  x|k  0|a(k  )a + (k)|0

1 d 3 k  x|k  0|[a(k  ), a + (k)]± |0 = x|k = √ 3 exp(ik · x). 2π

To explore this connection further, we will use superscripts (1) and (2) to designate operators in first and second quantized theory. E.g. the 1-particle Hamiltonians in first and second quantized theory can be written as  H

(1)

=

 h¯ 2  + V (x) x|, d x |x − 2m 

3

(17.66)

380

17 Non-relativistic Quantum Field Theory

 H

(2)

=



 h¯ 2 d x ψ (x) −  + V (x) ψ(x). 2m +

3

(17.67)

We can rewrite H (2) as  2 h ¯ = d 3 x  d 3 x  d 3 x ψ + (x  )δ(x  − x  ) −  + V (x  ) 2m    3  ×δ(x − x)ψ(x) = d x d 3 x ψ + (x  )x  |H (1) |xψ(x), (17.68) 

H (2)







and again we have exact correspondence between 1-particle matrix elements in the first and second quantized theory, x  |H (1) |x = x  |H (2) |x.

(17.69)

This works in general. For an operator K (1) from first quantized theory, the requirement of equality of 1-particle matrix elements k  |K (2) |k = k  |K (1) |k,

x  |K (2) |x = x  |K (1) |x

(17.70)

can be solved by  K (2) =  =

d 3k d 3x

 

d 3 k a + (k  )k  |K (1) |ka(k) d 3 x ψ + (x  )x  |K (1) |xψ(x).

(17.71)

General 1-Particle States and Corresponding Annihilation and Creation Operators in Second Quantized Theory The equivalence of first and second quantized theory in the single-particle sector also allows us to derive the equations for 1-particle states and corresponding annihilation and creation operators in second quantization. Suppose |m and |n are two states of the first quantized theory. The corresponding matrix element of the Hamiltonian in the first quantized theory is  m|H

(1)

|n =

 h¯ 2  + V (x) x|n d x m|x − 2m 

3

17.3 The Connection Between First and Second Quantized Theory

 =

 3

d x

3 



d x



h¯ 2 d x m|x δ(x − x) −  + V (x) 2m 3 



381





× δ(x − x  )x  |n      h¯ 2 3 3  3    +  + V (x) = d x d x d x m|x 0|ψ(x )ψ (x) − 2m × ψ(x)ψ + (x  )|0x  |n   = d 3 x  d 3 x  m|x  0|ψ(x  )H (2) ψ + (x  )|0x  |n,

(17.72)

where we used the identity δ(x  − x)δ(x − x  ) = 0|ψ(x  )ψ + (x)ψ(x)ψ + (x  )|0

(17.73)

to write the matrix element of the 1st quantized theory as a matrix element of the second quantized theory. We can interpret the result (17.72) as equality of single-particle matrix elements, m|H (1) |n = m|H (2) |n

(17.74)

if we define the 1-particle states of the second quantized theory as  |n =



+

d x ψ (x)|0x|n = 3

d 3 x |xx|n.

(17.75)

This also motivates the definition of corresponding creation and annihilation operators an+

≡

ψn+

 = 

an ≡ ψn =



+

d x ψ (x)x|n = 3

d 3 k a + (k)k|n,

(17.76)

 d 3 x ψ(x)n|x =

d 3 k a(k)n|k.

(17.77)

E.g. the operator + + ≡ ψn,,m = an,,m



d 3 x ψ + (x)x|n, , m =



d 3 k a + (k)k|n, , m

will create an electron (or more precisely, the corresponding quasiparticle for relative motion of the electron and the proton) in the |n, , m state of hydrogen. The canonical relations for the field operators ψ(x) or a(k) imply that the operators for multiplets of quantum numbers n also satisfy (anti-)commutation relations

382

17 Non-relativistic Quantum Field Theory

[ψm , ψn+ ]± = δm,n ,

[ψm , ψn ]± = 0,

+ [ψm , ψn+ ]± = 0.

(17.78)

Substituting x|n = x|n in Eq. (17.75) also shows the completeness relation in the single-particle sector of the Fock space,  d 3 x |xx| = 1.

(17.79)

Time Evolution of 1-Particle States in Second Quantized Theory According to our previous observations, a state in the Schrödinger picture evolves according to 

|(t) = exp − iH (2) t/h¯ |(0).

(17.80)

On the other hand, according to Eq. (17.75), a single particle state at time t = 0 should be given in terms of the corresponding first quantized state |(0),  |(0) =

d 3 x ψ + (x)|0x|(0).

(17.81)

Here we wish to show that this relation is preserved under time evolution. We find from Eqs. (17.80), (17.79) and (17.69) 

 d 3 x |xx| exp − iH (2) t/h¯ |(0)



 d 3 x ψ + (x)|0x| exp − iH (1) t/h¯ |(0)

|(t) = =  =

d 3 x ψ + (x)|0x|(t),

(17.82)

i.e. Eq. (17.75) is indeed preserved under time evolution of the states. We can write the Schrödinger state |(t) also in the form |(t) = + (t)|0

(17.83)

with the creation operator of the particle in the first quantized state |(t), +

 (t) =



+

d x ψ (x)x|(t) = 3



d 3 k a + (k)k|(t).

(17.84)

17.3 The Connection Between First and Second Quantized Theory

383

Note that + (t) is an operator in the Schrödinger picture of the theory. The time-dependence arises only because it is a superposition of Schrödinger picture operators with time-dependent amplitudes. The corresponding Heisenberg picture operator is given in Eq. (17.90) below. Other equivalent forms of the representation of states in the Schrödinger picture involve linear combinations of Heisenberg picture field operators, e.g. 

|(t) = exp − iH (2) t/h¯ |(0) 

 = d 3 x exp − iH (2) t/h¯ ψ + (x)|0x|(0)  =

d 3 x  + (x, − t)|0x|(0)

(17.85)

and

 x|(t) = x| exp − iH (2) t/h¯ |(0) = x, t|(0),

(17.86)

with moving base kets 

   i (2) i (2) |x, t = exp H t |x = exp H t ψ + (x)|0 =  + (x, t)|0, h¯ h¯ |k, t = a + (k, t)|0.

(17.87)

At first sight, the time-dependence of the creation operator in (17.85) may not be what one naively might have expected, but as we have seen it is implied by the correspondence of single-particle matrix elements between the second and first quantized theory. In a slightly different way, the correctness of the time-dependence in (17.85) can also be confirmed by verifying that it is exactly the time-dependence which ensures that the Heisenberg evolution equation (17.12) is equivalent to the Schrödinger equation on the single-particle wave function, see Problem 17.14. The Heisenberg picture state corresponding to |(t) is

 |H  = exp iH (2) t/h¯ |(t) = |(0)  = d 3 x ψ + (x)|0x|(0).

(17.88)

Note that substitution of (17.82) into the first equation in (17.88) implies that we can write this state also in the form  |H  = d 3 x  + (x, t)|0x|(t) = + (17.89) H (t)|0

384

17 Non-relativistic Quantum Field Theory

with the Heisenberg picture operator



 (2) + (2) + ¯  (t) exp − iH t/h¯ H (t) = exp iH t/h  = d 3 x  + (x, t)x|(t).

(17.90)

The time-independence of the Heisenberg picture state |H  is manifest in (17.88) but appears rather suspicious in (17.89). However, the representation (17.89) directly leads back to |(0) (17.88) if we use the correspondence of single-particle matrix elements,

 x|(t) = x|(t) = x| exp − iH (2) t/h¯ |(0), (17.91) and the completeness relation in the single-particle sector, 



d 3 x |xx| exp − iH (2) t/h¯ = 1. exp iH (2) t/h¯

(17.92)

There is a subtle point underlying the discussion in this section that students who go through their first iteration of learning quantum field theory would not notice, because we have not yet discussed interacting quantum field theories. However, I should point out that the equivalence of first and second quantization in the single-particle sector holds if the single-particle states cannot spontaneously absorb another particle or decay into two or more particles. This property also holds in interacting quantum field theories like quantum electronics or quantum electrodynamics, because conservation laws prevent e.g. single charged particles from spontaneously absorbing or radiating photons. These theories require at least two particles in both the initial and final states (or semi-classical inclusion of a second particle in the form of an external potential) for particle number changing processes. However, quantum field theory can also describe inherently unstable particles which decay into two or more particles, and in these cases we cannot expect equivalence of first and second quantized single-particle matrix elements.

17.4 The Dirac Picture in Quantum Field Theory Although our Hamiltonians in the Heisenberg and Schrödinger pictures are usually time-independent in quantum field theory, time-dependent perturbation theory is still used for the calculation of transition rates even with time-independent perturbations V . This will lead again to the calculation of scattering matrix elements Sf i = f |UD (∞, −∞)|i of the time-evolution operator in the interaction picture. Therefore we will automatically encounter field operators in the Dirac picture, which are gotten from the time-independent field operators of the Schrödinger

17.4 The Dirac Picture in Quantum Field Theory

385

picture through application of an unperturbed Hamiltonian H0 = H − V . In many cases this will be the free Schrödinger picture Hamiltonian  H0 =

h¯ 2 ∇ψ + (x) · ∇ψ(x) = d x 2m



3

d 3k

h¯ 2 k 2 + a (k)a(k). 2m

(17.93)

Please note that the free Hamilton operator in the Heisenberg picture (we set again t0 = 0 for the time when the two pictures coincide) H0,H (t) = exp(iH t/h) ¯ H0 exp(− iH t/h) ¯  2 h¯ ∇ + (x, t) · ∇(x, t) = d 3x 2m  h¯ 2 k 2 + a (k, t)a(k, t) = d 3k 2m

(17.94)

usually differs from H0 , because generically [H, H0 ] = [V , H0 ] = 0.

(17.95)

Transformation of the basic field operators from the Schrödinger picture into the Dirac picture yields 

     i i ih¯ 2 k t H0 t a(k) exp − H0 t = a(k) exp − 2m h¯ h¯  1 (17.96) = √ 3 d 3 x ψ(x, t) exp(− ik · x), 2π     i i ψ(x, t) = exp H0 t ψ(x) exp − H0 t h¯ h¯  1 = √ 3 d 3 k aD (k, t) exp(ik · x) 2π    1 ih¯ 2 = √ 3 d 3 k a(k) exp ik · x − k t . (17.97) 2m 2π

aD (k, t) = exp

Due to the simple relation (17.96) aD (k, t) is always substituted with a(k) in applications of the Dirac picture. We summarize the conventions for the notation for basic field operators in Schrödinger field theory in Table 17.2. The Hamiltonian and the corresponding time evolution operator on the states, as well as the transition amplitudes are derived in exactly the same way as in the first quantized theory. However, these topics are important enough to warrant repetition in the framework of the second quantized theory. This time we can limit

386

17 Non-relativistic Quantum Field Theory

Table 17.2 Conventions for basic field operators in different pictures of Schrödinger field theory

Heisenberg picture (x, t)  + (x, t) a(k, t)

Schrödinger picture ψ(x) ψ + (x) a(k)

Dirac picture ψ(x, t) ψ + (x, t) aD (k, t)

a + (k, t)

a + (k)

+ aD (k, t)

the discussion to the simpler case of time-independent Hamiltonians H and H0 in the Schrödinger picture. The states in the Schrödinger picture of quantum field theory satisfy the Schrödinger equation ih¯

d |(t) = H |(t), dt

(17.98)

which implies   i |(t) = exp − H (t − t  ) |(t  ). h¯

(17.99)

The transformation (17.97) ψ(x) → ψ(x, t) into the Dirac picture implies for the states the transformation   i |(t) → |D (t) = exp H0 t |(t). (17.100) h¯ The time evolution of the states in the Schrödinger picture then determines the time evolution of the states in the Dirac picture  |D (t) = exp

     i i i H0 t exp − H (t − t  ) exp − H0 t  |D (t  ) h¯ h¯ h¯

= UD (t, t  )|D (t  )

(17.101)

with the time evolution operator on the states5      i i i   UD (t, t ) = exp H0 t exp − H (t − t ) exp − H0 t . h¯ h¯ h¯ 



(17.102)

This operator satisfies the initial condition UD (t  , t  ) = 1 and the differential equations

5 Recall

that there are two time evolution operators in the Dirac picture. The free time evolution operator U0 (t − t  ) evolves the operators ψ(x, t) = U0+ (t − t  )ψ(x, t  )U0 (t − t  ), while UD (t, t  ) evolves the states.

17.4 The Dirac Picture in Quantum Field Theory

387

      i ∂ i i    ih¯ UD (t, t ) = exp H0 t (H − H0 ) exp − H (t − t ) exp − H0 t ∂t h¯ h¯ h¯     i i = exp H0 t V exp − H0 t UD (t, t  ) = HD (t)UD (t, t  ), h¯ h¯ ∂ ih¯  UD (t, t  ) = − UD (t, t  )HD (t  ), (17.103) ∂t and can therefore also be written as    i t  dτ HD (τ ) . UD (t, t ) = T exp − h¯ t 

(17.104)

The states in the Dirac picture therefore satisfy the Schrödinger equation ih¯

d |D (t) = HD (t)|D (t) dt

(17.105)

with the Hamiltonian 

   i i HD (t) ≡ VD (t) = exp H0 t V exp − H0 t . h¯ h¯

(17.106)

The transition amplitude from an initial unperturbed state |i (t  ) at time t  to a final state |f (t) at time t is 

 i  Sf i (t, t ) = f (t)|i (t) = f (t)| exp − H (t − t ) |i (t  ) h¯       i i i   = f (0)| exp H0 t exp − H (t − t ) exp − H0 t |i (0), h¯ h¯ h¯ 

or with |f  ≡ |f (0)    i t dτ HD (τ ) |i. Sf i (t, t ) = f |UD (t, t )|i = f |T exp − h¯ t  



(17.107)

The scattering matrix Sf i = f |UD (∞, −∞)|i contains information about all processes which take a physical system e.g. from an initial state |i with ni particles to a final state |f  with nf particles. This includes in particular also processes where the interactions in HD (t) generate virtual intermediate particles which do not couple to any of the external particles. These vacuum processes need to be subtracted from the scattering matrix in each order of perturbation theory, which amounts to simply neglecting them in the evaluation of the scattering matrix. The vacuum processes also appear in the vacuum to vacuum amplitude, and the subtraction in each order

388

17 Non-relativistic Quantum Field Theory

of perturbation theory can also be understood as dividing the vacuum to vacuum amplitude out of the scattering matrix, Sf i =

f |UD (∞, −∞)|i . 0|UD (∞, −∞)|0

(17.108)

However, unitarity of the time evolution operator UD (∞, −∞) implies unitarity of the scattering matrix Sf i = f |UD (∞, −∞)|i as defined earlier,

+ Sf i Sif  =

i



Sf i Sf∗  i =

i

=





f |UD (∞, −∞)|if  |UD (∞, −∞)|i∗

i

f |UD (∞, −∞)|ii|UD+ (∞, −∞)|f  

i

= f |UD (∞, −∞)UD+ (∞, −∞)|f   = δff  .

(17.109)

Therefore division by the vacuum to vacuum matrix element 0|UD (∞, −∞)|0 in the alternative definition (17.108) can only yield a unitary scattering matrix if the amplitude 0|UD (∞, −∞)|0 is a phase factor. We can understand this in the following way. Conservation laws prevent spontaneous decay of the vacuum into any excited states |N, N = 0|UD (∞, −∞)|0 = 0.

(17.110)

The completeness relation |00| +



|NN| = 1

(17.111)

N =0

and unitarity of the time evolution operator then imply |0|UD (∞, −∞)|0|2 = |0|UD (∞, −∞)|0|2 +



|N |UD (∞, −∞)|0|2

N =0

= 0|UD+ (∞, −∞)UD (∞, −∞)|0 = 1,

(17.112)

thus confirming that the vacuum to vacuum amplitude is a phase factor. We will continue to use the simpler notation Sf i = f |UD (∞, −∞)|i for the scattering matrix with the understanding that we can neglect vacuum processes.

17.5 Inclusion of Spin

389

17.5 Inclusion of Spin The wave functions of particles with spin s have 2s + 1 components σ (x, t) ≡ x, σ |(t), σ ≡ ms ∈ {− s, − s + 1, . . . , s}, and the normalization condition is s 

d 3 x |σ (x, t)|2 = 1,

(17.113)

σ =−s

see e.g. Sect. 8.1. The Schrödinger equation with spin-dependent local interaction potentials between the different components, ih¯

∂ h¯ 2 Vσ σ  (x)σ  (x, t) σ (x, t) = − σ (x, t) + ∂t 2m 

(17.114)

σ

follows from a Lagrange density L=

  ih¯ h¯ 2 ∂σ+ ∂σ σ+ · − · σ − ∇σ+ · ∇σ 2 σ ∂t ∂t 2m σ − σ+ · Vσ σ  · σ  . (17.115) σ,σ 

Canonical quantization then yields the (anti-)commutation relations for the Heisenberg picture field operators, [σ (x, t), σ+ (x  , t)]± = δσ σ  δ(x − x  ), [σ (x, t), σ  (x  , t)]± = 0,

[σ+ (x, t), σ+ (x  , t)]± = 0,

(17.116) (17.117)

with commutators for bosons (integer spin) and anti-commutators for fermions (half-integer spin). The charge, number, Hamiltonian and momentum operators for particles with spin follow from (17.115) using the methods of Sects. 16.2 and 16.3,  Q = qN = q

d 3x



σ+ (x, t)σ (x, t),

(17.118)

σ

H = H0 (t) + V (t),  H0 (t) =

d 3x

h¯ 2 ∇σ+ (x, t) · ∇σ (x, t), 2m σ

(17.119) (17.120)

390

17 Non-relativistic Quantum Field Theory

 V (t) =

d 3x



σ+ (x, t)Vσ σ  (x)σ  (x, t),

(17.121)

σ,σ 

and  P (t) =

d 3x

h¯ % & σ+ (x, t) · ∇σ (x, t) − ∇σ+ (x, t) · σ (x, t) . 2i σ

The transition to the Schrödinger picture field operators then proceeds in the standard way,     i i ψσ (x) = exp − H t σ (x, t) exp H t , h¯ h¯

(17.122)

[ψσ (x), ψσ+ (x  )]± = δσ σ  δ(x − x  ),

(17.123)

[ψσ (x), ψσ  (x  )]± = 0,

[ψσ+ (x), ψσ+ (x  )]± = 0.

(17.124)

The most common case of non-vanishing spin in non-relativistic quantum mechanics is s = 1/2, and then the common conventions for assigning values for the spin label σ are 1/2, +, ↑ for sz = h/2, and − 1/2, −, ↓ for sz = − h/2. Higher spin ¯ ¯ values can arise in nuclei, atoms and molecules. Since we now use σ (x, t) to denote quantum fields in the Heisenberg picture, we will denote the time-dependent states in the Schrödinger picture and the corresponding wave function components by σ (x, t). Within the framework of the “first quantized theory”, a single-particle state for a particle with spin s is then given by |(t) =

s 

d 3 x |x, σ x, σ |(t),

(17.125)

σ =−s

such that |x, σ |(t)|2 = |σ (x, t)|2 is the probability density to find the particle with spin projection hσ ¯ in the location x at time t, and the normalization condition is s 

d 3 x |x, σ |(t)|2 = 1.

(17.126)

σ =−s

The Fock space creation and annihilation operators for particles in first quantized particle states |(t) are then in direct generalization of (17.84) + (t) =

 σ

d 3 x ψσ+ (x)x, σ |(t)

17.5 Inclusion of Spin

391

=



d 3 k aσ+ (k)k, σ |(t)

(17.127)

σ

and (t) =



d 3 x ψσ (x)(t)|x, σ  =

σ



d 3 k aσ (k)(t)|k, σ .

σ

(17.128) A single-particle wave function with a set n of orbital quantum numbers and definite spin projection σ is e.g. x, σ  |n,σ (t) = x|n (t)δσ σ  , and the corresponding single-particle state in the quantized field theory is |n,σ (t) = + n,σ (t)|0 =



d 3 x ψσ+ (x)|0x|n (t).

(17.129)

A general single-particle state can involve a superposition of different spin projections σ ∈ {− s, . . . , s} and have different orbital quantum number nσ for each projection. E.g. a general single-electron state in the Schrödinger picture can be expressed in the form |n↑ ,n↓ (t) =

+ n↑ ,n↓ (t)|0



=

d 3 x ψs+ (x)|0x|ns (t),

(17.130)

s∈{↑,↓}



d 3 x |x|ns (t)|2 = 1.

(17.131)

s∈{↑,↓}

Here we switched from the notation σ ∈ {− s, . . . , s} for the spin projections to s ∈ {↑, ↓} ≡ {1/2, − 1/2}. The Pauli operators of the second quantized theory are with the index mapping s → a(s) = (3/2) − s, see Eq. (8.9), σ = i

 s,s 

i d 3 x ψs+ (x)σa(s),a(s  ) ψs  (x).

(17.132)

The Pauli spinor notation is only relevant when we return to first quantization through projections x, s|n↑ ,n↓ (t), which yields two orthogonal components, 

x, ↑ |n↑ ,n↓ (t) x, ↓ |n↑ ,n↓ (t)



 =

x|n↑ (t) x|n↓ (t)



= x|n↑ (t)| ↑ + x|n↓ (t)| ↓.

(17.133)

It is appropriate to add a remark on notation for the general single-particle field operators. Following the conventions of Eq. (17.76), we like to write the creation

392

17 Non-relativistic Quantum Field Theory

operator for a particle with a set n of orbital quantum numbers and spin projection + (t) or a + (t), but if no quantum numbers are specified and we just talk hσ ¯ as ψn,σ n,σ about an abstract single-particle state |(t), the notation + (t) from Eq. (17.127) is more suitable. The three notations for given sets of orbital quantum numbers and spin projection are therefore + + + n,σ (t) = ψn,σ (t) = an,σ (t) =

 =



d 3 x ψσ+ (x)x|n (t)

d 3 k aσ+ (k)k|n (t).

(17.134)

Note that these time-dependent operators are operators in the Schrödinger picture of the theory. Their time-dependence arises only because they are time-dependent superpositions of the Schrödinger picture operators ψσ+ (x) or aσ+ (k). We can easily verify that the state (17.129) is a Schrödinger picture state by using the correspondence of single-particle matrix elements between first and second quantized theory, x|n (t)δσ σ  = x, σ  |n,σ (t) = x, σ  | exp(− iH (1) t/h)| ¯ n,σ (0) = x, σ  | exp(− iH t/h)| ¯ n,σ (0),

(17.135)

since this implies |n,σ (t) =

 σ

=



d 3 x ψσ+ (x)|0x, σ  |n,σ (t) d 3 x |x, σ  x, σ  | exp(− iH t/h) ¯ |n,σ (0)

σ

= exp(− iH t/h) ¯ |n,σ (0).

(17.136)

The operators n,σ (t) and + n,σ (t) satisfy canonical commutation or anti-commutation relations as a consequence of the corresponding relations for ψσ (x) and ψσ+ (x), see Problem 17.6. The Heisenberg picture operators for a state with quantum numbers n and σ are + H,n,σ (t)



   i i + = exp H t n,σ (t) exp − H t h¯ h¯  = d 3 x σ+ (x, t)x|n (t).

(17.137)

The Schrödinger picture state (17.129) yields the correct expectation value for the kinetic energy of the particle if evaluated with the time-independent kinetic Hamiltonian H0 in the Schrödinger picture, but the Heisenberg picture state

17.5 Inclusion of Spin

393

+ |n,σ  = exp(iH t/h) ¯ |n,σ (t) = H,n,σ (t)|0  = d 3 x σ+ (x, t)|0x|n (t)

 = |n,σ (0) =

d 3 x ψσ+ (x)|0x|n (0).

(17.138)

has to be evaluated with the generically time-dependent kinetic Hamiltonian H0 (t) in the Heisenberg picture to yield the kinetic energy of the particle. The actual time-independence of d 3 x σ+ (x, t)|0x|n (t) follows already from the first line of Eq. (17.138), but we can also verify it again from the correspondence (17.135) of single particle matrix elements,   i Ht d 3 x |x, σ  x, σ  | h¯ σ    i × exp − H t |n,σ (0) = |n,σ (0) = d 3 x ψσ+ (x)|0x|n (0). h¯



d 3 x σ+ (x, t)|0x|n (t) = exp



We will mostly use Schrödinger picture states and operators in the remainder of this chapter. A general two-particle state in the Schrödinger picture with particle species a and a  (e.g. an electron and a proton, or two electrons) will have the form |a,a  (t) = √

  1 + d 3 x d 3 x  a,σ (x, − t)a+ ,σ  (x  , − t)|0 1 + δaa   σ,σ



×x, σ ; x , σ  |a,a  (0)   1 3 + = √ (x)ψa+ ,σ  (x  )|0 d x d 3 x  ψa,σ 1 + δaa   σ,σ



×x, σ ; x , σ  |a,a  (t).

(17.139)

For identical particles it makes sense to require the symmetry property x, σ ; x  , σ  |a,a (t) = ∓ x  , σ  ; x, σ |a,a (t),

(17.140)

with the upper sign applying to fermions and the lower sign for bosons. Any part of the two-particle wave function for identical particles a = a  and opposite symmetry + (x), ψ + (x  )] = 0. is projected out in (17.139) due to [ψa,σ ± a,σ  The equivalence of the two representations of |a,a  (t) in terms of operators + (x, − t) + (x  , − t) and ψ + (x)ψ + (x  ) generalizes the corresponding a,σ a,σ a  ,σ  a  ,σ  equivalence between (17.82) and (17.85) for single-particle states to multi-particle states and is a consequence of the fact that the second quantized Schrödinger

394

17 Non-relativistic Quantum Field Theory

equation for multi-particle states is equivalent to a corresponding first quantized Schrödinger equation in systems where particle numbers are preserved, see (17.164), (17.166). In the ideal case of a completely normalizable system (e.g. two particles trapped in an oscillator potential or a box), the quantity |x, σ ; x  , σ  |a,a  (t)|2 is a probability density for finding one particle at x with spin projection σ and the second particle at x  with spin projection σ  , and we should have 

 d 3x

d 3 x  |x, σ ; x  , σ  |a,a  (t)|2 = 1,

(17.141)

σ,σ 

if we know that there is exactly one particle of kind a and one particle of kind a  in the system. It then follows with (17.140) that the state (17.139) is also normalized, a,a  (t)|a,a  (t) = 1.

(17.142)

For an example, we consider a stationary state where a particle of type a has orbital quantum numbers n and spin projection σ , and a particle of type a  has orbital quantum numbers n and spin projection σ  . The two-particle amplitude (again with the upper sign for fermions) x, ρ; x  , ρ  |a,n,σ ;a  ,n ,σ   = x, ρ; x  , ρ  |a,n,σ , a  ,n ,σ   δρσ δρ  σ  x|a,n x  |a  ,n   ≡ √ 1 + δaa  1 ∓ δaa  δσ σ  |cnn |2 ∓ δaa 

δρ  σ δρσ  x  |a,n x|a  ,n   2(1 ∓ δσ σ  |cnn |2 )

(17.143)

yields the tensor product state  |a,n,σ , a  ,n ,σ   =

 d 3x

+ d 3 x  ψa,σ (x)ψa+ ,σ  (x  )|0

x|a,n x  |a  ,n  × . 1 ∓ δaa  δσ σ  |cnn |2

(17.144)

d 3 x a,n |xx|a,n 

(17.145)

Here the parameter  cnn =

occurs only for particles of the same kind a = a  . It satisfies |cnn | ≤ 1. For identical fermions we must have δσ σ  |cnn | < 1 because otherwise the state would vanish. For the normalization of the state (17.144) we note that the following equations hold with upper signs for fermions (this uses the same tricks as Eq. (17.163) below),

17.5 Inclusion of Spin

395

+ 0|ψa  ,σ  (y  )ψa,σ (y)ψa,σ (x)ψa+ ,σ  (x  )|0 = δ(y − x)δ(y  − x  )

∓ δaa  δσ σ  δ(y − x  )δ(y  − x)

(17.146)

and therefore     3 3  3 + (x)ψa+ ,σ  (x  )|0 d y d y d x d 3 x  0|ψa  ,σ  (y  )ψa,σ (y)ψa,σ × a  ,n |y  a,n |yx|a,n x  |a  ,n    = d 3 x |x|a,n |2 d 3 x  |x  |a  ,n |2 

 ∓ δaa  δσ σ 

d 3 x a  ,n |xx|a,n 

= 1 ∓ δaa  δσ σ  |cnn |2 ,

d 3 x  a,n |x  x  |a  ,n  (17.147)

i.e. the two-particle state (17.144) is properly normalized. It may appear surprising that we did not impose orthonormality cnn = δnn on the single-particle orbitals of the two particles. We can always construct a basis of two-particle states from an orthonormal basis of single-particle states, and if we were only interested in those basis states for the two-particle system, we could have imposed cnn = δnn . However, that is not how N -particle states are practically used. In a many-electron problem for the electron levels in a molecule, different electrons with the same spin projection will occupy mutually orthogonal orbital energy eigenstates of the molecular Hamiltonian. If we would know those orbitals, we would use the corresponding quantum numbers for n and n and the orbitals would automatically be orthonormal. However, generically we will not know those exact molecular orbitals, but rather we might try to find approximate solutions in the factorized form (17.144) using e.g. atomic orbitals with quantum numbers n from one atom in the molecule and combine them with atomic orbitals with quantum numbers n from another atom in the molecule in a first ansatz for the approximate construction of molecular orbitals. Those orbitals will generically not be orthogonal. N -particle product states are extensively used in quantum chemistry and condensed matter physics in approximate calculations of N-particle quantum states. There are very good practical reasons for this, but I would be remiss if I would not point out that the reliance on product ansätze for N-particle states is a principal limitation of N-particle calculations. Two-particle states provide the perfect opportunity to explain this. A two-particle state will generically not have the factorized form (17.144) because this form is incompatible with interactions between the two particles. We can explain this with the case of a system of two different particles that we have solved in Chap. 7. A stationary two-particle state of a proton with quantum numbers N and an electron with quantum numbers n and definite spin projections of the two particles could be written in the form

396

17 Non-relativistic Quantum Field Theory





|φn,σ , N,  =

d 3x

+ + d 3 x  ψe,σ (x)ψp, (x  )|0x|φn x  |N ,

but we had seen in Chap. 7 that no such electron-proton state is compatible with the Coulomb interaction of the two particles. There is no solution of the Schrödinger equation for the two particles which factorizes into the product of an electron wave function with a proton wave function. We did find factorized solutions of the form x, x  |K,n,,m  = R|K r|n,,m ,

(17.148)

where the first factor R|K  = √

1 2π

3

exp(iK · R)

(17.149)

describes center of mass motion, and the second factor r|n,,m  = r|n, , m

(17.150)

describes relative motion. Therefore we can write down a two-particle state for the electron-proton system in the form  |K,n,,m;σ,  =

 d 3x

+ + d 3 x  ψe,σ (x)ψp, (x  )|0x − x  |n,,m 

×(me x/M) + (mp x  /M)|K .

(17.151)

I am emphasizing this to caution the reader. We frequently calculate scattering matrix elements and expectation values for many-particle states which are products of independent single-particle states like (17.144). However, we should keep in mind that two-particle systems with an interaction potential V (x − x  ) (and without external fields affecting both particles differently, see Sect. 18.4 below) always allow for separation of the center of mass motion, and the states can be written in the form (17.151). However, the general form is (17.139). These remarks immediately generalize to N-particle states. Those states will be characterized by amplitudes x 1 , σ1 ; . . . ; x N , σN |(t) with appropriate (anti-)symmetry properties for identical particles. In spite of all those cautionary remarks about the limitations of tensor product states of the form (17.144) in the actual description of many-particle systems, we will continue to use those states because they will help us to understand important aspects of expectation values in many-particle systems in Sects. 17.6 and 17.7. Approximating N-particle states as products of N single-particle orbitals is also indispensable for actual calculations in quantum chemistry and condensed matter physics.

17.6 Two-Particle Interaction Potentials and Equations of Motion

397

We can form singlet states and triplet states for two spin 1/2 fermions with singleparticle orbital quantum numbers n and n . The triplet states are |n,n ;1,±1  = |n,±1/2 , n ,±1/2  = − |n ,n;1,±1 , |n,1/2 , n ,−1/2  + |n,−1/2 , n ,1/2  √ 2 = − |n ,n;1,0 ,

(17.152)

|n,n ;1,0  =

(17.153)

and the singlet state is |n,n ;0,0  =

|n,1/2 , n ,−1/2  − |n,−1/2 , n ,1/2  √ 2(1 + δnn )

= |n ,n;0,0 .

(17.154)

Here we assumed orthonormal single-particle orbitals, cnn = δnn , for simplicity.

17.6 Two-Particle Interaction Potentials and Equations of Motion It is only a small step now to describe particle interactions as exchange of virtual particles between particles. We will take this step in Sect. 20.7 for exchange of nonrelativistic virtual particles, and in Chap. 23 for photon exchange between charged particles. However, a description of interactions through two-particle interaction potentials V is often sufficient. Furthermore, interaction potentials also appear in quantum electrodynamics in the Coulomb gauge.6 The Hamiltonian with time-independent particle-particle interaction potentials Va,a  (x) has the same form in the Schrödinger picture and in the Heisenberg picture, H =

1 2



 d 3x

d 3x

a,a  σ,σ 

+ ψa,σ (x)ψa+ ,σ  (x  )Va,a  (x − x  )ψa  ,σ  (x  )ψa,σ (x)



h¯ 2 + ∇ψa,σ (x) · ∇ψa,σ (x) 2m a a,σ   1 + a,σ (x, t)a+ ,σ  (x  , t)Va,a  (x − x  ) = d 3x d 3x 2   +

d 3x

(17.155)

a,a σ,σ

6 We will derive this for non-relativistic charged particles in Sect. 18.5, see Eq. (18.100), and for relativistic charged particles in Sect. 22.4, see Eq. (22.155).

398

17 Non-relativistic Quantum Field Theory

×a  ,σ  (x  , t)a,σ (x, t)  h¯ 2 + ∇a,σ (x, t) · ∇a,σ (x, t). + d 3x 2m a a,σ

(17.156)

+ If the operators ψ1,σ (x) describe electrons, we would include e.g. the repulsive Coulomb potential V11 (x − x  ) = e2 /(4π 0 |x − x  |) between pairs of electrons. The ordering of annihilation and creation operators in the potential term in Eq. (17.155) is determined by the requirement that the expectation value of the interaction potential for the vacuum |0 and for single-particle states |n (t) + (x)ψ + (x  )ψ   (x  )ψ vanishes. The nested structure ψa,σ a,σ (x) of the operators a ,σ a  ,σ  ensures the correct sign for the interaction energy of two-fermion states. It is also instructive to write the Hamiltonian (17.155) in wavevector space. We use the following conventions for the Fourier transformation of the potential,

 V (x) =

d 3 q V (q) exp(iq · x),

(17.157)

with inversion 1 V (q) = V (− q) = (2π )3 +

 d 3 x V (x) exp(− iq · x),

(17.158)

to avoid extra factors of 2π in the particle interaction terms in wavevector space. This yields the representations H =

1 2



 d 3k

d 3k

 d 3q

σ,σ 



aσ+ (k + q)aσ+ (k  − q)V (q)aσ  (k  )aσ (k)

h¯ 2 k 2

a + (k)aσ (k) 2m σ    1 aσ+ (k + q, t)aσ+ (k  − q, t)V (q) = d 3k d 3k d 3q 2  +

d 3k

σ

σ,σ

×aσ  (k  , t)aσ (k, t) +

 d 3k

h¯ 2 k 2 σ

2m

aσ+ (k, t)aσ (k, t),

(17.159)

where the labels a and a  for the particle species are suppressed. The representations in momentum space imply that the Fourier component V (q) of the two-particle interaction potential describes exchange of momentum hq ¯ between the two interacting particles. Note that symmetric interaction potentials V (x) = V (− x) are also symmetric in wavevector space V (q) = V (− q). For example, we can infer from the derivation of the energy-dependent Green’s function (11.12), (11.24) that the Coulomb potential is dominated by small momentum exchange,

17.6 Two-Particle Interaction Potentials and Equations of Motion

V (q) =

1 e2 . 3 2 8π 0 q − i

399

(17.160)

The Hamiltonians in the Dirac picture are  H0 =

d 3x

 h¯ 2 h¯ 2 k 2 ∇ψσ+ (x) · ∇ψσ (x) = d 3 k aσ+ (k)aσ (k) 2m 2m σ σ

d 3x

h¯ 2 ∇ψσ+ (x, t) · ∇ψσ (x, t) 2m σ

 =  =

d 3k

h¯ 2 k 2 σ

2m

+ aD,σ (k, t)aD,σ (k, t)

(17.161)

and 

   i i HD (t) = exp H0 t (H − H0 ) exp − H0 t h¯ h¯   1 = ψσ+ (x, t)ψσ+ (x  , t)V (x − x  )ψσ  (x  , t)ψσ (x, t) d 3x d 3x 2 σ,σ     1 + +  = aD,σ (k + q, t)aD,σ d 3k d 3k d 3q  (k − q, t)V (q) 2  σ,σ

×aD,σ  (k  , t)aD,σ (k, t),

(17.162)

with the time-dependent field operators in the Dirac picture. Recall that H0 determines the time evolution of the operators, while HD (t) determines the time evolution of the states in the Dirac picture.

Equation of Motion The derivation of the equation of motion for the Schrödinger picture state (17.139) with the Schrödinger picture Hamiltonian (17.155) is easily done with the relation ψρ  (y  )ψρ (y)ψσ+ (x)ψσ+ (x  )|0 = [ψρ  (y  ), [ψρ (y), ψσ+ (x)ψσ+ (x  )]− ]± |0 = δρσ δ(x − y)δρ  σ  δ(x  − y  )|0 ∓ δρσ  δ(x  − y)δρ  σ δ(x − y  )|0, (17.163) with the upper signs for fermions. This yields both for bosons and for fermions the equation

400

17 Non-relativistic Quantum Field Theory

d |a,a  (t) = H |a,a  (t) dt   1 + ψa,σ (x)ψa+ ,σ  (x  )|0 d 3x d 3x =√ 1 + δaa  σ,σ    h¯ 2 h¯ 2   × − −  + Va,a  (x − x ) x, σ ; x  , σ  |a,a  (t). 2ma 2ma 

ih¯

(17.164)

Here we used  ≡ ∂ 2 /∂x 2 ,  ≡ ∂ 2 /∂x 2 and symmetry of the potential: Va,a  (x − x  ) = Va  ,a (x  − x). + (x)ψ + (x  )|0 (or equivalently appliLinear independence of the states ψa,σ a  ,σ   cation of the projector 0|ψa  ,ρ  (y )ψa,ρ (y) and the symmetry property (17.140)) implies that Eq. (17.164) is equivalent to the two-particle Schrödinger equation   ∂ h¯ 2 h¯ 2     ih¯ x, σ ; x , σ |a,a  (t) = − −  + Va,a  (x − x ) ∂t 2ma 2ma  ×x, σ ; x  , σ  |a,a  (t).

(17.165)

The second quantized two-particle Schrödinger equation (17.164) also implies |a,a  (t) = exp(− iH t/h)| ¯ a,a  (0),

(17.166)

and this implies the equivalence of the two representations of the state |a,a  (t) in Eq. (17.139). Note that it is important for these representations that particle numbers do not change during the time-evolution of the system. Time-independence of the Hamiltonian implies that we can also write the Schrödinger equation in the time-independent form 

h¯ 2 h¯ 2  Ex, σ ; x , σ |a,a   = − −  + Va,a  (x − x  ) 2ma 2ma  





×x, σ ; x  , σ  |a,a  .

(17.167)

These are exactly the two-particle Schrödinger equations that we would have expected for a wave function which describes two particles interacting with a potential V . Indeed, we have used this expectation already in Chap. 7 to formulate the equation of motion for the electron-proton system that constitutes a hydrogen atom. The not entirely trivial observation at this point is that these two-particle Schrödinger equations also hold for identical particles. The only manifestation of statistics of the particles is the symmetry property (17.140) of the two-particle wave function.

17.6 Two-Particle Interaction Potentials and Equations of Motion

401

The amplitude for a general N -particle state with particle species ai for the i-th particle would satisfy ih¯

∂ x 1 , σ1 ; . . . ; x N , σN |a1 ,...,aN (t) = ∂t

+

N −1

 −

N h¯ 2 i 2mai i=1

 Vai ,aj (x i − x j ) x 1 , σ1 ; . . . ; x N , σN |a1 ,...,aN (t),

N

(17.168)

i=1 j =i+1

and after transformation into the energy domain for time-independent potentials,  Ex 1 , σ1 ; . . . ; x N , σN |a1 ,...,aN (E) =

−

N h¯ 2 i 2mai i=1

+

N −1

N

 Vai ,aj (x i − x j ) x 1 , σ1 ; . . . ; x N , σN |a1 ,...,aN (E).

(17.169)

i=1 j =i+1

The derivation of the two-particle Eq. (17.165) or the general N-particle Eq. (17.168) crucially relies on the fact that the Hamiltonian (17.156) preserves the number of particles for each species a. This is a simple consequence of the fact that each individual term does not have an excess of annihilation or creation operators for any particular species a. Otherwise we  could not use linear independence of states within a particular subsector of ( a na = N)-particle states to read off the first quantized evolution equations (17.168) for the time-dependent N -particle amplitudes x 1 , σ1 ; x 2 , σ2 ; . . . |a1 ,a2 ,... (t). If the Hamiltonian would not preserve the numbers na of particles for each species, we could have derived coupled systems of equations for wave functions with different particle numbers na and possibly also different total numbers N of particles. However, the first quantized formalism becomes unwieldy if we have to include sets of coupled many-particle Schrödinger equations with different particle numbers, and one rather calculates everything in the second quantized formalism then.

Relation to Other Equations of Motion In N -particle mechanics, the two-particle Schrödinger equations (17.165) and (17.167) and their extensions to N > 2 correspond to the equation for the total energy of the particle system. However, in mechanics we are used to deal with separate equations of motion for each particle in terms of the forces acting on the particle. Sometimes the experience from mechanics leads to the suspicion that there should be a separate Schrödinger type equation for each particle. We can infer that this naive expectation is not correct from the observation that the

402

17 Non-relativistic Quantum Field Theory

equations for individual particles in a classical particle system are equations for forces, not energies. We can easily recover the separate N-particle equations of classical mechanics from the N-particle Eq. (17.168), because this equation implies the N-particle version of the Ehrenfest theorem, e.g. for two particles 





 ∂+ ¯ ∂ a,σ ;a  ,σ  (x, x , t) h a,σ ;a  ,σ  (x, x  , t) ∂t i ∂x  h¯ ∂ ∂a,σ ;a  ,σ  (x, x  , t)  (x, x , t) · + + a,σ ;a  ,σ  i ∂x ∂t   ∂Va,a  (x − x  )  a,σ ;a  ,σ  (x, x  , t), = − d 3 x d 3 x  + a,σ ;a  ,σ  (x, x , t) ∂x

dp a (t) = dt

3

d x

3 

d x

i.e. d p (t) = − ∇Va,a  (x − x  )(t), dt a d d p  (t) = − ∇  Va,a  (x − x  )(t) = − p a (t). dt a dt

(17.170) (17.171)

Another reason why one might incorrectly suspect that there should be N separate Schrödinger type equations for an N-particle system is that each single-particle field operator a,σ (x, t) in the Heisenberg picture still satisfies its own evolution equation, ih¯

∂ a,σ (x, t) = − [H, a,σ (x, t)] ∂t =−

h¯ 2 a,σ (x, t) + Va (x, t)a,σ (x, t), (17.172) 2ma

with Va (x, t) =

 a

σ

d 3 x  a+ ,σ  (x  , t)Va,a  (x − x  )a  ,σ  (x  , t).

(17.173)

However, note that this nonlinear Schrödinger equation is an operator equation which holds for the operators annihilating and creating the particle species a, but not separately for a wave function for each single particle of type a. Indeed, the second quantized many particle Schrödinger equation ih¯

d |ν1 ,ν2 ,... (t) = H |ν1 ,ν2 ,... (t) dt

(17.174)

17.6 Two-Particle Interaction Potentials and Equations of Motion

403

and the corresponding first quantized many particle Schrödinger equation (17.168) can be derived from the single-particle operator equations (17.172), see Problem 17.15. Yet another way to find sets of coupled differential equations which resemble nonlinear Schrödinger equations is to substitute products of N single-particle orbitals into the N-particle Schrödinger equation (17.169) and then project onto the different factors. As a pedagogical example for the general N-particle case, we explain this in a two-particle system in the interaction potential V (x − x  ), using products of orbitals of the two particles. However, as explained in Chap. 7, if we would really want to solve the two-particle system we would use factorized orbitals in terms of center of mass and relative coordinates. The factorized two-particle state (17.144) for identical particles a = a  satisfies (Eq. (17.163) is very useful in these calculations), H0 |n,σ , n ,σ   =  

1





d 3 x  ψσ+ (x)ψσ+ (x  )|0

d 3x

1 ∓ δσ σ  |cnn |2

 h¯ 2 h¯ 2    × − n (x)n (x ) − n (x) n (x ) , 2m 2m

(17.175)

while the potential operator in the Schrödinger picture, V =

1 2



 d 3x

d 3x

σ,σ 

ψσ+ (x)ψσ+ (x  )V (x − x  )ψσ  (x  )ψσ (x)

(17.176)

satisfies with V (x − x  ) = V (x  − x) both for fermions and for bosons the equation 1 V |n,σ , n ,σ   =  1 ∓ δσ σ  |cnn |2



 d 3x

d 3 x  ψσ+ (x)ψσ+ (x  )|0

×V (x − x  )n (x)n (x  ).

(17.177)

Projection of the Schrödinger equation E|n,σ , n ,σ   = (H0 + V )|n,σ , n ,σ   onto 0|ψσ  (y  )ψσ (y) then yields the equation " ! E n (x)n (x  ) ∓ δσ σ  n (x  )n (x) =−

" h¯ 2 ! n (x)n (x  ) + n (x) n (x  ) 2m

± δσ σ 

" h¯ 2 !   n (x  )n (x) + n (x  )n (x) 2m

(17.178)

404

17 Non-relativistic Quantum Field Theory

! " + V (x − x  ) n (x)n (x  ) ∓ δσ σ  n (x  )n (x) .

(17.179)

We can now try to project onto single-orbital equations through projection with the dual orbitals ˜ n (x) = 

n (x) − n (x)cn n , 1 − |cnn |2

˜ n (x) = 

n (x) − n (x)cnn . 1 − |cnn |2

if we use orthonormal orbitals cnn In 3particular,  ) . . . yields d x  + (x  n En (x)[1 ∓ δσ σ  δnn ] = − h¯ 2 − 2m ±δ

σσ

 +



(17.180)

= δnn , projection with

h¯ 2 n (x)[1 ∓ δσ σ  δnn ] 2m

    d 3 x  + n (x ) n (x )n (x)

h¯ 2 2m



    d 3 x  + n (x ) n (x )n (x)

 2 d 3 x  V (x − x  ) n (x  ) n (x) 

∓ δσ σ 

   d 3 x  V (x − x  )+ n (x )n (x )n (x).

(17.181)

Coupled single-orbital equations as approximations for N-particle systems are particularly relevant for electrons. Using δσ σ  δnn = 0 for fermions (since otherwise the two-fermion state would vanish and Eq. (17.181) reduces to 0 = 0) yields  h¯ 2 h¯ 2     n (x) − d 3 x  + n (x ) n (x )n (x) 2m 2m  h¯ 2     + δσ σ  d 3 x  + n (x ) n (x )n (x) 2m   2 + d 3 x  V (x − x  ) n (x  ) n (x)

En (x) = −

 − δσ σ 

   d 3 x  V (x − x  )+ n (x )n (x )n (x).

(17.182)

This is actually an example of a properly orthogonalized Hartree-Fock equation for fermions, see Problem 17.11, where Eq. (17.245) provides another example of this in a three-particle system. The last term in (17.182) is an example of an exchange term, and we will see in Sect. 17.7 that these exchange terms appear also in energy expectation values. Contrary to the potential exchange term, the kinetic exchange term in (17.182)

17.7 Expectation Values and Exchange Terms

405

does not show up in energy expectation values. We can also confirm these facts by projecting Eq. (17.181) with d 3 x + n (x) . . .. This yields  ! " h¯ 2 + E=− d 3 x + n (x)n (x) + n (x)n (x) 2m   2  + d 3 x d 3 x  V (x − x  ) |n (x)|2 n (x  ) 

 − δσ σ 

d 3x

+   d 3 x  V (x−x  )+ n (x)n (x)n (x )n (x ). (17.183)

The first three terms are the sum of kinetic plus potential energy of the two-particle system, which we would also have expected from classical physics. The last term is the exchange energy of the two identical particles in the interaction potential V . It is a direct consequence of quantum statistics and shows up in macroscopic physics through magnetism, as will be shown in Sect. 17.7.

17.7 Expectation Values and Exchange Terms It is not difficult to discuss expectation values for the general two-particle state (17.139). However, it is more instructive to do this for the tensor product of singleparticle states (17.144) with a = a  . Equation (17.175) yields the expectation value of the kinetic energy operator in the two-particle state (17.144), H0  = n,σ , n ,σ  |H0 |n,σ , n ,σ   = −  ×

 d 3x

h¯ 2 2m(1 ∓ δσ σ  |cnn |2 )

 +   d 3 x  + n (x )n (x)n (x)n (x )

 +    + + n (x )n (x)n (x) n (x ) +    ∓ δσ σ   + n (x)n (x )n (x)n (x )

 +     (x ) . (x) (x ) (x)  ∓ δσ σ  +  n n n n

(17.184)

In the following we assume that the single particle orbitals are not only normalized, but orthonormal,  cnn ≡ d 3 x + (17.185) n (x)n (x) = δnn . This yields

406

17 Non-relativistic Quantum Field Theory

H0  = −

h¯ 2 2m



! " + d 3 x + n (x)n (x) + n (x)n (x) .

(17.186)

Since δσ σ  δnn = 0 for the fermions, it is convenient to write 1 ∓ δσ σ  δnn = 1 + δb δσ σ  δnn in the expectation value for the potential operator, where δb = 1 for bosons and δb = 0 otherwise. Equation (17.177) then yields (again with upper signs for fermions) the result  V  = n,σ , n ,σ  |V |n,σ , n ,σ   =

 d 3x

d 3x

V (x − x  ) 1 + δb δσ σ  δnn

! " +  +  +      × + n (x )n (x)n (x)n (x ) ∓ δσ σ n (x)n (x )n (x)n (x ) , i.e. the expectation value for the potential energy becomes V  =

Cnn ∓ Jnn δσ σ  , 1 + δb δσ σ  δnn

with the Coulomb term7   +    Cnn = d 3 x d 3 x  + n (x)n (x )V (x − x )n (x )n (x)

(17.187)

(17.188)

and the exchange integral [73, 74]  Jnn =

 d 3x

+     d 3 x  + n (x)n (x )V (x − x )n (x )n (x).

(17.189)

The Coulomb term is what we would have expected for the energy of the interaction of two particles with quantum numbers n and n . The exchange interaction, on the other hand, is a pure quantum effect which only exists as a consequence of the canonical (anti-)commutation relations for bosonic or fermionic operators. In the first quantized theory it appears as a consequence of symmetrized boson wave functions and anti-symmetrized fermion wave functions. For electrons with aligned spins (17.152) and also for the m = 0 triplet state (17.153) we must have n = n , and the result (17.187) implies a shift of the ordinary Coulomb term Cnn by the exchange term Jnn , Cnn → Cnn − Jnn . For the m = 0 triplet state the exchange integral arises from the cross multiplication terms in the evaluation of the expectation value. By the same token, the Coulomb term for the singlet state (17.154) gets shifted to (Cnn + Jnn )/(1 + δnn ) due to the cross

7 As

derived, this result applies to every two-particle interaction potential. The most often studied case in atomic, molecular and condensed matter physics is the Coulomb interaction between electrons, and therefore the standard (non-exchange) interaction term is simply denoted as the Coulomb term.

17.7 Expectation Values and Exchange Terms

407

multiplication terms. The exchange interaction therefore splits the potential energy levels for n = n according to  Cnn →

Cnn + Jnn Cnn − Jnn

singlet state, anti-aligned spins, triplet state, aligned spins.

(17.190)

This implies that a positive exchange integral Jnn > 0 favors ferromagnetism through alignment of electron spins, whereas Jnn < 0 favors anti-ferromagnetism. The impact of the exchange interaction on the potential energies of different spin configurations can be used to replace the potential energy operator (17.176) with an effective spin interaction Hamiltonian [41] (with dimensionless spins: S/h¯ → S)   1 Cnn + Jnn − Jnn (S + S  )2 1 + δnn   1 1  Cnn − Jnn (1 + 4S · S ) , = 1 + δnn 2

Hnn =

(17.191)

where S and S  are the dimensionless spin operators for two electrons. Equation (17.191) gives the correct shifts by ∓Jnn because (S + S  )2 = 2, S · S  = 1/4, in the triplet state and (S + S  )2 = 0, S · S  = − 3/4, in the singlet state. Note that Cnn = Jnn and therefore (Cnn − Jnn )/(1 + δnn ) = Cnn − Jnn . The Hamiltonian (17.191) without the constant terms is the spin-spin coupling Hamiltonian which is also known as the Heisenberg Hamiltonian.8 Equation (17.191) shows that the Coulomb interaction between electrons, through the exchange integral, effectively generates an interaction of the same form as the magnetic spin-spin interaction. However, note that this is only an effective description of the exchange interaction through its impact on different spin configurations. Contrary to the usually much weaker magnetic dipole interaction between spins, (17.191) is not a genuine spin-spin interaction. The underlying interaction is still the Coulomb interaction, which generates exchange interaction through indistinguishability of particles of the same kind. The exchange interaction usually dominates over the magnetic spin-spin interaction in materials. For example, in atoms or molecules Jnn will be of order of a few eV, whereas the energy of the genuine magnetic dipole-dipole interaction will only be of order meV or smaller. Exchange interaction with Jnn > 0 can therefore align electron spins to generate ferromagnetism,9 but the actual magnetic dipole interaction between spins will certainly not accomplish this at room temperature.

8 Heisenberg had introduced exchange integrals in 1926, and he had also noticed their relevance for

the understanding of the magnetic properties of materials [76]. However, the effective Hamiltonian (17.191) was introduced by Dirac in the previously mentioned reference in 1929. 9 Ferromagnetism or anti-ferromagnetism in magnetic materials usually requires indirect exchange interactions, see e.g. [23, 61, 95, 166].

408

17 Non-relativistic Quantum Field Theory

17.8 From Many Particle Theory to Second Quantization Second quantization (or field quantization) of the Schrödinger field is relevant for condensed matter physics and statistical physics, but it is usually not introduced through quantization of the corresponding Lagrangian field theory. An alternative approach proceeds through the observation that field quantization yields the same matrix elements as symmetrized wave functions (for bosons) or anti-symmetrized wave functions (for fermions) in first quantized theory with a fixed number N of particles. In short this reasoning goes as follows. We assume a finite volume V = L3 of our system. Then we can restrict attention to discrete momenta k=

2π n, n ∈ N3 , L

(17.192)

and the N-particle momentum eigenstates are generated by states of the form |k 1 , . . . k N  = |k 1  . . . |k N .

(17.193)

This state needs to be symmetrized for indistinguishable bosons by summing over all N! permutations P of the N momenta,

P |k 1 , . . . k N 

(17.194)

P ∈SN

However, this state is not generically normalized. If the momentum k is realized nk times in the state, then 2  2        N!  P |k 1 , . . . k N  = ' nk ! = N! nk !,   P ∈SN k nk ! k k

(17.195)

' since there are N !/ k nk ! different ' distinguishable states in the symmetrized state, and each of these states occurs k nk ! times. Therefore the correctly normalized Bose states are |{nk } = 

N!

1 ' k

nk ! P ∈S

P |k 1 , . . . k N .

(17.196)

N

The action of the operator |k  k| on this state is for k = k  |k  k| . . . , nk , . . . , nk  , . . . = nk

nk + 1 | . . . , nk − 1, . . . , nk  + 1, . . . nk

17.9 Problems

409

=

 nk (nk + 1)| . . . , nk − 1, . . . , nk  + 1, . . .

= a + (k  )a(k)| . . . , nk , . . . , nk  , . . .,

(17.197)

|kk| . . . , nk , . . . = nk | . . . , nk , . . . = a + (k)a(k)| . . . , nk , . . ..

(17.198)

and for k = k  ,

i.e. we find that for 1-particle operators, the operator  K (1) =

d 3k



d 3 k |k  k  |K (1) |kk|

(17.199)

has the same effect in the first quantized theory as  K (2) =

d 3k



d 3 k k  |K (1) |ka + (k  )a(k)

(17.200)

has in the second quantized theory. E.g. the first quantized 1-particle Hamiltonian H (1) =

p2 = 2m

 d 3 k |k

h¯ 2 k 2 k| 2m

(17.201)

becomes  H

(2)

=

d 3k

h¯ 2 k 2 + a (k)a(k). 2m

(17.202)

Once the beasts a + (k) and a(k) are let loose, it is easy to recognize from their commutation or anti-commutation relations that they create and annihilate particles, and the whole theory can be developed from there. The approach through quantization of Lagrangian field theories is preferred in this book because it also yields an elegant formalism for the identification of conservation laws and generalizes more naturally to the relativistic case.

17.9 Problems 17.1 Propagator and Green’s function as vacuum expectation values 17.1a Suppose the Hamiltonian of the second quantized theory is given by Eq. (17.5). Use agreement of the evolution equations and initial conditions for the first-quantized and second-quantized matrix elements below to argue that the xrepresentations of the first-quantized time evolution operator U (t) = exp(− iH t/h) ¯

410

17 Non-relativistic Quantum Field Theory

and the retarded Green’s function G(t) can also be expressed as the vacuum matrix elements of Schrödinger field operators in the Heisenberg picture, 0|(x, t) + (x  , t  )|0 = x|U (t − t  )|x  ,

(17.203)

− t  )|x  . 0|Tc (x, t) + (x  , t  )|0 = ihx|G(t ¯

(17.204)

Here Tc is a chronological time ordering operator, Tc A(t)B(t  ) = (t − t  )A(t)B(t  ) ± (t  − t)B(t  )A(t).

(17.205)

The lower sign applies if both A(t) and B(t  ) are fermionic operators. 17.1b Verify the relations (17.203), (17.204) explicitly for free Schrödinger fields. 17.2 Calculate the evolution equations dN(t)/dt and dP (t)/dt for the number and momentum operators in the Heisenberg picture if the Hamiltonian is given by Eq. (17.5). Show in particular that the evolution equation for dP (t)/dt can be written in the form dP (t) i i = [H, P (t)] = [V (t), P (t)] dt h¯ h¯  = − d 3 x  + (x, t) · ∇V (x) · (x, t),

(17.206)

where, similar to the equivalence of (17.5) and (17.7), we also assume vanishing boundary terms in partial integrations. See also footnote 2. Equation (17.206) is the formulation of the Ehrenfest theorem in second quantization. 17.3 The relation p0 = − E/c in relativity motivates the identification of the Hamiltonian H with a timelike momentum operator P0 = − H /c. Assume that the particles described by the Heisenberg picture field operator (x, t) are moving in a potential V (x, t). Show that the field operators satisfy the commutation relations [Pμ (t), (x, t)] = ih∂ ¯ μ (x, t).

(17.207)

Remark Time-dependence has been included here in the notation for the 4momentum components Pμ (t) because the presence of the potential V (x, t) implies that none of the 4-momentum components may be preserved, e.g. the Hamiltonian becomes time-dependent, 

 H (t) =

3

d x

 h¯ 2 + + ∇ (x, t) · ∇(x, t) +  (x, t)V (x, t)(x, t) . 2m

17.9 Problems

411

17.4 Time-ordered vacuum expectation values from the scattering matrix 17.4a Suppose the Heisenberg picture, the Schrödinger picture and the Dirac picture agree at time t0 . Show that the Heisenberg and Dirac picture field operators are related through the time evolution operator on Dirac picture states, H (t) = UD+ (t, t0 )D (t)UD (t, t0 ).

(17.208)

Note that for time-independent H and H0 this can also be written in the form H (t) = (t)D (t)+ (t),

(17.209)

with the Møller operator for time t0 of coincidence of the pictures of quantum dynamics, 

   i i (t) = exp H (t − t0 ) exp − H0 (t − t0 ) = UD+ (t, t0 ), h¯ h¯ UD (t, t  ) = + (t)(t  ),

(17.210) (17.211)

see also Eq. (13.61). 17.4b Suppose that both H and H0 are time-independent. We add a classical current J (x, t) to the Hamiltonian as a driving force for the quantum field H (x, t),  (17.212) H → H [J ](t) = H − d 3 x J (x, t)H (x, t). Show that this changes the interaction picture Hamiltonian HD (t) according to  HD (t) → HD [J ](t) = HD (t) −

d 3 x J (x, t)D (x, t).

(17.213)

Also show that this changes the scattering matrix elements (0)

(0)

Sf i (tf , ti ) = ψf |UD (tf , ti )|ψi 

(17.214)

for tf > t > ti (or for tf < t < ti ) such that  δSf i [J ](tf , ti )  i = f {tf }|H (x, t)|i {ti },  δJ (x, t) h¯ J =0

(17.215)

where |{t} = (t)|ψ (0) , cf. Eq. (13.63). More generally, the following equation holds if all the times t1 , t2 , . . . tn are between ti and tf ,

412

17 Non-relativistic Quantum Field Theory

  n  δ n Sf i [J ](tf , ti ) i  = f {tf }|TH (x 1 , t1 ) . . .  δJ (x 1 , t1 ) . . . δJ (x n , tn ) J =0 h¯ ×H (x n , tn )|i {ti }.

(17.216)

This means that the vacuum-to-vacuum scattering matrix in the presence of the current J (x, t),   Z[J ] ≡ Sf i [J ] (0) = 0|UD [J ](∞, −∞)|0, (17.217) (0) |ψf =|ψi =|0

generates all the time-ordered vacuum expectation values10 of the Heisenberg picture operators. 17.5 Canonical commutation relations in the first-quantized theory from second-quantized theory 17.5a Calculate the expectation value of the operator  x=

d 3x



+ ψa,σ (x)xψa,σ (x)

(17.218)

a,σ

for the two-particle state (17.144). + (k). 17.5b Express the operator x in terms of k space operators aa,σ (k) and aa,σ

17.5c Show that the operators x, P and N satisfy the operator algebra [x ⊗, P ] = ihN1, ¯

[x, N] = 0,

[P , N] = 0,

(17.219)

i.e. the canonical commutation relations of the first quantized single particle theory are a direct reflection of the corresponding relations in the second quantized theory. 17.6 Creation operators for general single-particle states 17.6a Suppose n is a set of orbital quantum numbers and σ is a spin quantum number such that H (1) |n, σ  = En |n, σ  and the completeness relations

|n, σ n, σ | = 1,

n, σ |n , σ   = δnn δσ σ 

(17.220)

n,σ

hold. The range of n may contain both discrete and continuous components such that the sum over n may also contain integrations over the continuous components and δnn = δ(n − n ) for n and n in the continuous components. Show that the

10 These vacuum

expectation values are also known as n-point functions or correlation functions or (generalized) Green’s functions.

17.9 Problems

413

creation and annihilation operators + (t) = ψn,σ



d 3 x ψσ+ (x)x, σ  |n, σ  exp(− iEn t/h) ¯

σ

 =

d 3 x ψσ+ (x)x|n exp(− iEn t/h) ¯

(17.221)

d 3 x ψσ (x)n|x exp(iEn t/h) ¯

(17.222)

and  ψn,σ (t) =

satisfy canonical (anti-)commutation relations + [ψn,σ (t), ψn+ ,σ  (t)]± = 0,

[ψn,σ (t), ψn ,σ  (t)]± = 0,

[ψn,σ (t), ψn+ ,σ  (t)]± = δnn δσ σ  .

(17.223)

17.6b Suppose the second quantized Hamiltonian is H =

 σ

 3

d x

 h¯ 2 + + ∇ψσ (x) · ∇ψσ (x) + ψσ (x)V (x)ψσ (x) . 2m

(17.224)

+ (t) and ψ Express H in terms of the operators ψn,σ n,σ (t). Show that + (t)|0 |ψn,σ (t) = ψn,σ

(17.225)

is an eigenstate of H , and that ψn ,σ  (t)|ψn,σ (t) = δnn δσ σ  |0.

(17.226)

17.7 Expectation values of density operators in two-particle states 17.7a Show that the expectation value (t)|ns (x)|(t) of the spin-polarized density operator ns (x) = ψs+ (x)ψs (x) for the two-particle state (17.139), (17.140) (with identical particles, a = a  ) is  (t)|ns (x)|(t) = 2

d 3x

  x, s; x  , s  |(t)2 .

(17.227)

s

17.7b Show also that the expectation value for two identical particles in the factorized state (17.144) is

414

17 Non-relativistic Quantum Field Theory

1 ∓ δσ σ  δn,n 1 + δσ σ  δn,n  |n (x, t)|2 + δs,σ  |n (x, t)|2 .

n,σ ;n ,σ  (t)|ns (x)|n,σ ;n ,σ  (t) =

× δs,σ

(17.228)

17.8 Expectation values of two-particle states 17.8a Calculate the expectation value H0  for kinetic energy in the two-particle state (17.139) with identical particles a = a  . 17.8b Calculate the expectation value V  for the single-particle potential energy operator V =



d 3 x ψσ+ (x)V (x)ψσ (x)

(17.229)

σ

in the two-particle state (17.139) with identical particles. 17.8c Calculate the expectation value V  for the two-particle interaction operator V =

1 2 



 d 3x

d 3 x  ψσ+ (x)ψσ+ (x  )V (x − x  )ψσ  (x  )ψσ (x)

(17.230)

σ,σ

in the two-particle state (17.139) with identical particles. 17.8d Which results do you get in Problems 17.8a-c if you substitute the particular two-particle state (17.144) with a = a  ? 17.9 Potential energy for singlet states 17.9a Calculate the matrix element n,σ (t), n ,σ  (t)|V |n,τ (t), n ,τ  (t) of the two-particle interaction operator (17.230) between two different two-particle states with identical pairs of orbital quantum numbers but possibly different spin configurations, where  |n,σ (t), n ,σ  (t) =

 d 3x

×

d 3 x  ψσ+ (x)ψσ+ (x  )|0

x|n (t)x  |n (t) . √ 1 + δσ σ  δnn

(17.231)

17.9b Use the result from Problem 17.9a to calculate the expectation value of the two-particle interaction operator (17.230) for the singlet state  1

|n,n ;0,0 (t) = √ |n,↑ (t), n ,↓ (t) − |n,↓ (t), n ,↑ (t) 2 of two electrons with orbital quantum numbers n and n .

(17.232)

17.9 Problems

415

17.10 Show that for pairs of spin-1 bosons, the interaction energy for states (17.144) V  = n,σ (t), n ,σ  (t)|V |n,σ (t), n ,σ  (t) =

Cnn + Jnn δσ σ  1 + δσ σ  δnn

(17.233)

corresponds to the following values for the interaction energy in the singlet, triplet, and quintuplet states, (0) (2) Enn  = Enn =

Cnn + Jnn , 1 + δnn

(1) Enn  =

Cnn − Jnn = Cnn − Jnn . 1 + δnn

(17.234)

Show also that these energies can be reproduced with an effective spin-spin interaction Hamiltonian 

 1 1 Hnn = Cnn + Jnn 4 − 6(S + S  )2 + (S + S  )4 1 + δnn 4

 1 Cnn − Jnn + Jnn S · S  + Jnn (S · S  )2 . (17.235) = 1 + δnn 17.11 Hartree-Fock equations 17.11a Calculate the expectation value of the Hamiltonian (17.155) for a threeparticle state which is a tensor product of normalized single-particle factors for a helium nucleus and two electrons,    + +  + | = d 3 x d 3 x  d 3 y ψe,σ (x)ψe,σ  (x )ψα (y)|0 ×x|φn x  |φn y|φN .

(17.236)

Here n and n are the orbital quantum numbers of the electrons and N are the orbital quantum numbers of the helium nucleus (why is this clear from Eq. (17.236)?). The state (17.236) is normalized for δσ σ  δnn = 0, and it would vanish otherwise. 17.11b Show that the requirement of stationary energy expectation value |H | under the constraints of normalized single-particle wave functions yields a set of three nonlinear coupled equations for the single-particle wave functions. You have to use Lagrange multipliers to include the normalization constraints, i.e. you have to calculate the variational derivatives of the functional   3 2 F [φn , φn , φN ] = |H | − n d x |x|φn | − 1  − n 

   d 3 x |x|φn |2 − 1 − N d 3 x |x|φN |2 − 1 ,

(17.237)

416

17 Non-relativistic Quantum Field Theory

e.g. δ

F [φn , φn δφn+ (x)



, φN ] = 0.

(17.238)

This yields nonlinearly coupled equations which resemble time-independent Schrödinger equations, e.g.  |φn (x  )|2 h¯ 2 e2 φn (x) φn (x) + d 3x 2me 4π 0 |x − x  |  φ + (x  )φn (x  ) e2 φn (x) d 3x n − δσ σ  4π 0 |x − x  |  |φN (x  )|2 e2 d 3x − φn (x). 2π 0 |x − x  |

n φn (x) = −

(17.239)

These equations are examples of Hartree-Fock equations.11 The equations for the electrons contain exchange terms due to the presence of identical particles, and Hartree-Fock type equations have been successfully applied to calculate electronic configurations in atoms, molecules and solids.12 17.11c Show that the Lagrange multipliers i add up to the sum of the kinetic energy plus twice the potential energy of the system. Show also that n is the energy required to remove the particle with wave function φn (x) from the system if this removal does not change the other wave functions (Koopmans’ theorem [101]). 17.11d Suppose σ = σ  . The two electron orbitals must then necessarily be different, n = n , for nonvanishing atomic state |(t). What condition do you get if you project Eq. (17.239) onto the other electron orbital φn (x)? Remark You find that consistency of the Hartree-Fock equations with orthonormal electron spinors φn (x)χσ and φn (x)χσ  requires h¯ 2 me

11 Very



d 3 x φn+ (x)φn (x) +

e2 π 0



 d 3x

d 3x

|φN (x  )|2 + φ  (x)φn (x) = 0. |x − x  | n

good textbook discussions of Hartree-Fock equations can be found in [110, 153, 154], and a comprehensive discussion of the uses of Hartree-Fock type equations in chemistry and materials physics is contained in [59]. 12 The limitation of the variation of the N -particle states to tensor product states is a principal limitation of the Hartree-Fock method and also of other practical implementations of manyparticle calculations. The lowly hydrogen atom already told us that translation invariant interaction potentials V (x − x  ) entangle two-particle states in such a way that the energy eigenstates of the coupled system cannot be written as tensor products of single-particle states.

17.9 Problems

417

This condition will be approximately fulfilled for highly localized helium nucleus |φN (x)|2  δ(x) if we choose our electron orbitals as hydrogen-type wave functions for charge 2e, i.e. as orbitals of the He+ ion: En(0) φn (x) = −

h¯ 2 e2 φn (x), φn (x) + 2me 2π 0 |x|

(17.240)

due to orthogonality of different eigenfunctions of the He+ Hamiltonian. Resubstitution into (17.239) then yields En(1) φn (x)

 

|φn (x  )|2 e2 (0) ≡ n − En φn (x)  d 3x φn (x) 4π 0 |x − x  |  φ + (x  )φn (x  ) e2 φn (x), (17.241) d 3x n − δσ σ  4π 0 |x − x  |

which simply determines n − En as the first order shift of the He+ energy levels due to the electron-electron interactions, i.e. we have only reproduced an equation of first order perturbation theory for energy levels (although, sadly, for a system where the perturbation cannot be considered as small compared to the solved part of the potential). To construct atomic states (17.236) with mutually orthogonal single-electron orbitals without encountering the problem (17.240), (17.241), we need to add another set of Lagrange multiplier terms (0)

 F [φn , φn ] = − δσ σ  n n  − δσ σ  nn

d 3 x φn+ (x)φn (x) d 3 x φn+ (x)φn (x)

(17.242)

to ensure that variation of the orbitals maintains orthogonality. This modifies the Hartree-Fock equation for the electron orbital φn (x) by shifting the left-hand side n φn (x) → n φn (x) + δσ σ  nn φn (x)

(17.243)

and the Lagrange multiplier becomes nn = − −

h¯ 2 2me



e2 2π 0

d 3 x φn+ (x)φn (x) 

 d 3x

d 3x

|φN (x  )|2 + φ  (x)φn (x). |x − x  | n

(17.244)

We therefore find that the proper Hartree-Fock equations for orthonormal electron orbitals are

418

17 Non-relativistic Quantum Field Theory

 h¯ 2 h¯ 2 φn (x) + δσ σ  d 3 x  φn+ (x  )φn (x  )φn (x) 2me 2me  |φn (x  )|2 e2 d 3x + φn (x) 4π 0 |x − x  |  φ + (x  )φn (x  ) e2 − δσ σ  φn (x) d 3x n 4π 0 |x − x  |   e2 |φN (x  )|2 +  3  φ  (x )φn (x  )φn (x) d x d 3 x   + δσ σ  2π 0 |x − x  | n  |φN (x  )|2 e2 (17.245) d 3x − φn (x). 2π 0 |x − x  |

n φn (x) = −

These equations must actually hold for the 2-spinors φn (x)χσ , φn (x)χσ  of the two electrons. That is why the orthogonalization terms ∝ δσ σ  are only needed if both electrons have the same spin projection. Note that the additional terms in the orthogonalized Hartree-Fock equations do not change the result n + n + N = K + 2V , where V contains both the Coulomb and the exchange contributions to the potential energy. 17.12 We consider field operators for spin-1/2 fermions, {ψσ (x), ψσ+ (x  )} = δσ σ  δ(x − x  ), {ψσ (x), ψσ  (x  )} = 0,

{ψσ+ (x), ψσ+ (x  )} = 0.

(17.246) (17.247)

17.12a The spin-polarized particle density operator is nσ (x) = ψσ+ (x)ψσ (x). Show that the single-particle state |x1 , σ1  = ψσ+1 (x1 )|0 and the two-particle state |x1 , σ1 ; x2 , σ2  = ψσ+1 (x1 )ψσ+2 (x2 )|0 are eigenstates of nσ (x) in the sense that relations of the kind nσ (x)|x1 , σ1  = λσ,σ1 (x, x1 )|x1 , σ1 , nσ (x)|x1 , σ1 ; x2 , σ2  = λσ,σ1 ,σ2 (x, x1 , x2 )|x1 , σ1 ; x2 , σ2 

(17.248) (17.249)

hold. Calculate the “eigenvalues” λσ,σ1 (x, x1 ) and λσ,σ1 ,σ2 (x, x1 , x2 ). 17.12b We can define a density-density correlation operator Gσ σ  (x, x  ) = nσ (x)nσ  (x  ).

(17.250)

Evaluate the matrix elements x 1 , σ1 |Gσ σ  (x, x  )|x2 , σ2  of this operator in 1particle states.

17.9 Problems

419

Solution 17.12a For the 1-particle state we can write nσ (x)|x1 , σ1  = ψσ+ (x){ψσ (x), ψσ+1 (x 1 )}|0 = δσ σ1 δ(x − x 1 )ψσ+ (x)|0 = δσ σ1 δ(x − x 1 )|x1 , σ1 .

(17.251)

For the two-particle state we use the relation [A, BC] = {A, B}C − B{C, A} in nσ (x)|x1 , σ1 ; x2 , σ2  = ψσ+ (x)[ψσ (x), ψσ+1 (x 1 )ψσ+2 (x 2 )]|0 = δσ σ1 δ(x − x 1 )ψσ+ (x)ψσ+2 (x 2 )|0 − δσ σ2 δ(x − x 2 )ψσ+ (x)ψσ+1 (x 1 )|0 & % (17.252) = δσ σ1 δ(x − x 1 ) + δσ σ2 δ(x − x 2 ) |x1 , σ1 ; x2 , σ2 . 17.12b From the previous results and the orthogonality of the single-particle states (following from the anti-commutation relation between ψσ1 (x 1 ) and ψσ+2 (x 2 )) we find x 1 , σ1 |Gσ σ  (x, x  )|x2 , σ2  = x 1 , σ1 |nσ (x)nσ  (x  )|x2 , σ2  = δσ σ1 δ(x − x 1 )δσ  σ2 δ(x  − x 2 )δσ1 σ2 δ(x 1 − x 2 ).

(17.253)

17.13 Pair correlations in the Fermi gas We know that the Pauli principle excludes two fermions from being in the same state, i.e. they cannot have the same quantum numbers. But what does that mean for continuous quantum numbers like the location x of a particle? What exactly does the statement mean: “Two electrons of equal spin cannot be in the same place”? How far apart do two electrons of equal spin have to be to satisfy this constraint? We will figure this out in this problem. In the previous problem we have found that the density-density correlation operator Gσ σ  (x, x  ) = nσ (x)nσ  (x  ) has non-vanishing matrix elements for 1particle states. Therefore we define the pair correlation operator gˆ σ σ  (x, x  ) = Gσ σ  (x, x  ) − δσ σ  δ(x − x  )nσ (x) = ψσ+ (x)ψσ+ (x  )ψσ  (x  )ψσ (x)

(17.254)

as a measure for the probability to find a fermion with spin orientation σ at the point x, when we know that there is another fermion with spin orientation σ  at the point x  . This ordering of operators eliminates the 1-particle matrix elements. 17.13a Show that the corresponding combination of classical electron densities divided by 2, 1 1 1 g˜ σ σ  (x, x  ) = nσ (x)nσ  (x  ) − δσ σ  δ(x − x  )nσ (x), 2 2 2

(17.255)

420

17 Non-relativistic Quantum Field Theory

can be interpreted as a probability density normalized to the number of fermion pairs to find a fermion with spin projection hσ ¯ in x and a fermion with spin projection hσ ¯  in x  . Hint: There are N↑ + N↓ = N fermions in the volume V . What do you get from Eq. (17.255) by integrating over the volume? 17.13b The ground state of a free fermion gas is 

| =

+ + a1/2 (k)a−1/2 (k)|0,

(17.256)

k,|k|≤kF

where kF is the Fermi wave number (12.51). Calculate the pair distribution function gσ σ  (x, x  ) = |gˆ σ σ  (x, x  )|

(17.257)

of the free fermion gas in the ground state. Hints for 17.13b: With discrete momenta k=

2π n L

(17.258)

the anti-commutation relation for fermionic creation and annihilation operators becomes {aσ (k), aσ+ (k  )} = δσ σ  δk,k  .

(17.259)

The corresponding mode expansion for the annihilation operator in x-space becomes 1 ψσ (x) = √ aσ (k) exp(ik · x), V k

(17.260)

and the inversion is 1 aσ (k) = √ V

 d 3 x ψσ (x) exp(− ik · x).

(17.261)

V

Substitute the mode expansions for ψ and ψ + into the operator gˆ σ σ  (x, x  ). This yields a fourfold sum over momenta |

k,k  ,q,q 

In the next step, you can use that

. . . |.

(17.262)

17.9 Problems

421

|aσ+ (q) = (kF − q)|aσ+ (q), (17.263) because e.g. in the first equation, if k > kF , aσ (k) would simply anti-commute through all the creation operators in | and yield zero through action on the vacuum |0. This reduces the four sums to sums over momenta inside the Fermi sphere, aσ (k)| = (kF − k)aσ (k)|,



|

. . . |.

(17.264)

{k,k  ,q,q  }≤kF

For the following steps, you can use that for fermionic operators (aσ+ (k))2 = 0 and therefore k ≤ kF :

aσ+ (k)| = 0.

(17.265)

This observation can be used to replace operator products with commutators or anticommutators, e.g. q  ≤ kF :

aσ+ (q  )aσ  (k  )aσ (k)| = [aσ+ (q  ), aσ  (k  )aσ (k)]|,

(17.266)

aσ+ (q)aσ  (k  )| = {aσ+ (q), aσ  (k  )}|.

(17.267)

q ≤ kF :

This helps to get rid of all the operators in gσ σ  (x, x  ). For the last steps, you have to figure out what the sum over all momenta inside a Fermi sphere is, k,|k|≤kF 1 (you know this sum because you know that there are N Fermions in the system). For another term, you have to use that for N  1 1 V

k,|k|≤kF

f (k) 

1 (2π )3

 d 3 k f (k).

(17.268)

k≤kF

Solution 17.13a Integration of Eq. (17.255) over x and x  yields for identical spins e.g. 1 2



 3

d x

d 3 x  g˜ ↑↑ (x, x  ) =

1 N↑ (N↑ − 1) = N↑↑ , 2

(17.269)

which is the number of independent fermion pairs with both fermions having spin up, and we also find 1 2



 d 3x

d 3 x  g˜ ↑↓ (x, x  ) =

1 1 N↑ N↓ = N↑↓ , 2 2

(17.270)

which is the number of independent fermion pairs with opposite spins if we take into account that we want e.g. spin up in the location x and spin down in x  . Note also that summation over spin polarizations then yields

422

17 Non-relativistic Quantum Field Theory

 σ,σ 

 d 3x

1 1 1 d 3 x  g˜ σ σ  (x, x  ) = N↑ (N↑ − 1) + N↓ (N↓ − 1) + N↑ N↓ 2 2 2 =

1 N(N − 1) = N , 2

(17.271)

i.e. the total number of independent fermions pairs. 17.13b The discussion below is a modification of the discussion given by Schwabl [154]. Other derivations of the exchange hole of the effective charge density experienced by a Hartree-Fock electron in a metal can be found in [59, 86]. We have gσ σ  (x, x  ) =

1 V2



! " exp i(k · x + k  · x  − q · x − q  · x  )

k,k  ,q,q 

×|aσ+ (q)aσ+ (q  )aσ  (k  )aσ (k)|.

(17.272)

The observation (17.263) limits the sums over wave numbers, gσ σ  (x, x  ) =

1 V2



! " exp i(k · x + k  · x  − q · x − q  · x  )

{k,k  ,q,q  }≤kF

×|aσ+ (q)aσ+ (q  )aσ  (k  )aσ (k)|.

(17.273)

For the next step we use (17.265), (kF − q  )aσ+ (q  )| = 0, to replace operator products with commutators or anti-commutators: gσ σ  (x, x  ) =

1 V2



! " exp i(k · x + k  · x  − q · x − q  · x  )

{k,k  ,q,q  }≤kF

×|aσ+ (q)[aσ+ (q  ), aσ  (k  )aσ (k)]| 1 exp [i(k − q) · x] |aσ+ (q)aσ (k)| = 2 V  {k,k ,q}≤kF

− δσ σ 

1 V2



! " exp i(k · x + k  · x  − q · x − k · x  )

{k,k  ,q}≤kF

×|aσ+ (q)aσ  (k  )| 1 1 1 − δσ σ  2 = 2 V V  {k,k }≤kF



! " exp i(k − k  ) · (x − x  ) .

{k,k  }≤kF

Here the notation ≤ kF under the summation indicates that e.g. the summation over k is over all vectors k inside the Fermi sphere: |k| ≤ kF .

17.9 Problems

423

For the further evaluation we note that there are two fermions per momentum inside the Fermi sphere, and therefore

1 V

1=

k,|k|≤kF

n N = . 2V 2

(17.274)

This yields   2 1 n gσ σ  (x, x  ) = − δσ σ   4 V

k,|k|≤kF

2   ! " exp ik · (x − x  )  

    1 ! "2 n2 3   ≈ d k exp ik · (x − x )  − δσ σ   4 (2π )3 k≤kF    kF  1  1 ! "2 n2 2   − δσ σ   = dk dξ k exp ik|x − x |ξ  4 (2π )2 0 −1  k   F % &2 1 n2    − δσ σ  4 = dk k sin k|x − x |  4 4π |x − x  |2  0 ! % & % &"2 sin kF |x − x  | − kF |x − x  | cos kF |x − x  | n2 − δσ σ  = . 4 4π 4 |x − x  |6 In particular, the result for equal spin orientation can be written as & % &"2 ! % sin kF |x − x  | − kF |x − x  | cos kF |x − x  | 4  g (x, x ) = 1 − 9 , σ,σ n2 (kF |x − x  |)6 where n = kF3 /3π 2 (12.51) was used. This means that up to a distance of order

π 1 π λF 3 = = 2 kF 3n

(17.275)

the probability to find a fermion of like spin is significantly reduced: The Pauli principle prevents two fermions of like spin to be in the same place, even if there is no interaction between the fermions. The function n42 gσ,σ (x, x  ) is plotted in Fig. 17.1. The first maximum n42 gσ,σ (x, x  ) = 1 is reached at kF |x − x  | ≈ 4.4934,

(17.276)

i.e. depending on the maximal momentum in the fermion gas the minimal distance between two fermions of the same spin orientation is given by the minimal

424

17 Non-relativistic Quantum Field Theory

Fig. 17.1 The scaled pair correlation function 4gσ,σ (r)/n2 for 0 ≤ kF r ≤ 8 (upper panel) and 4 ≤ kF r ≤ 16 (lower panel)

wavelength in the gas: rhole ≈ 0.7λF .

(17.277)

On the other hand, the density of fermions with equal spin is n/2. Equal separation between those fermions would correspond to a distance

17.9 Problems

425

 1 2 3 a= , n

(17.278)

and inserting the result for the Fermi momentum

1 3 kF = 3π 2 n ,

 λF =

8π 3n

1 3

(17.279)

yields  a=

3 4π

1 3

λF  0.62λF .

(17.280)

Comparison with rhole (17.277) shows that the Pauli principle effectively repels fermions of equal spin such that they try to fill the available volume uniformly. Note that the existence of this exchange hole in the pair correlation between identical fermions of like spin has nothing to do e.g. with any electromagnetic interaction between the fermions. It is only a consequence of avoidance due to the Pauli principle. Stated differently (and presumably in the simplest possible way): The Pauli principle implies that free fermions of the same spin orientation try to occupy a volume as uniformly as possible to avoid contact. If we want to add free fermions to a fermion gas of constant volume we have to increase the energy in the gas, thereby increasing the maximal momentum in the gas, to squeeze more fermions into the volume and reduce the mean distance between fermions of like spin. The presence of the exchange hole implies a local reduction of the charge density of the other electrons seen by an electron in a metal (a “Hartree-Fock electron in a jellium model” [59, 86, 109]),   n 9 n 2 ρe (r) = − e − e 1− [sin (kF r) − kF r cos (kF r)] 2 2 (kF r)6   9 2 (17.281) = − en 1 − [sin (kF r) − kF r cos (kF r)] . 2 (kF r)6 This effective electron charge density is plotted in Fig. 17.2. A fermion gas where the fermions fill the lowest possible energy states under the constraint of the Pauli principle is denoted as a degenerate fermion gas. Addition of a fermion to a degenerate free fermion gas or compression of the free fermion gas costs energy according to (12.51), and effectively this amounts to a repulsive force between the fermions. It is easy to calculate the corresponding degeneracy pressure for the nonrelativistic degenerate fermion gas. The total energy of the degenerate free fermion gas is

426

17 Non-relativistic Quantum Field Theory

Fig. 17.2 The effective scaled electron charge density ρe (r)/(−ne) in a metal for 0 ≤ kF r ≤ 8

 E = 2V

|k|≤kF

h¯ 2 kF5 d 3 k h¯ 2 k 2 ¯ 2 (3N)5/3 4/3 h = V = π . 10m V 2/3 (2π )3 2m 10π 2 m

(17.282)

The degeneracy pressure is therefore p=−

∂E h¯ 2 5/3 = (3π 2 )2/3 n . ∂V 5m

(17.283)

The corresponding chemical potential is the Fermi energy, of course, μ=

h¯ 2 ∂E h¯ 2 2 = (3π 2 n)2/3 = k , ∂N 2m 2m F

(17.284)

and the average energy per particle is E/N = 0.6μ. 17.14 Show that the time-dependence of the Heisenberg picture operator in (17.85) implies equivalence of the Heisenberg evolution Eq. (17.12) with the Schrödinger equation for the single-particle wave function x|(t). Solution We have found the following representations of a second quantized singleparticle state associated with a general wave function x|(t),  |(t) =

d 3 x ψ + (x)|0x|(t)

17.9 Problems

427

 =

d 3 x  + (x, − t)|0x|(0).

(17.285)

The first representation (cf. (17.82)) for the Schrödinger picture state implies d ih¯ |(t) = dt



d 3 x ψ + (x)|0ih¯

∂ x|(t). ∂t

(17.286)

On the other hand, the second representation (17.85) and the Heisenberg evolution equation (17.12) imply d ih¯ |(t) = dt





 h¯ 2 + + d x −  (x, − t) + V (x) (x, − t) |0x|(0). 2m 3

Partial integration of the kinetic term and the correspondence (17.285) together with the linear independence of the states |x = d 3 x ψ + (x)|0 then imply that the single-particle wave function x|(t) must satisfy the Schrödinger equation   ∂ h¯ 2 ih¯ x|(t) = −  + V (x) x|(t). ∂t 2m

(17.287)

17.15 The two-particle state (17.139) in the Schrödinger picture can also be written in terms of Heisenberg picture field operators, cf. (17.85), |

a,a 

  i (t) = exp − H t |a,a  (0) h¯   1 + =  (x, − t)a+ ,σ  (x  , − t)|0 d 3 x d 3 x  a,σ 1 + δa,a  σ,σ  ×x, σ ; x  , σ  |a,a  (0).

(17.288)

Show that the Heisenberg evolution equations (17.172) with the Hamiltonian (17.156) yield again the two-particle Schrödinger equations (17.164) and (17.165). Show also that this derivation works for the general N -particle state and yields the N -particle Eq. (17.168). For the potential terms you can use the product rule for commutators, [A, B1 B2 . . . BN ] = [A, B1 ]B2 . . . BN +

N −2

B1 . . . BI [A, BI +1 ]BI +2 . . . BN

I =1

+ B1 . . . BN −1 [A, BN ].

(17.289)

428

17 Non-relativistic Quantum Field Theory

17.16 Charges as generators of transformations Note that the charges (16.27) with the current densities (16.26) can be written in the form (specified to three spatial dimensions) 



0 H + (δφ +  · ∇φ) · "φ Q=− d x c  & % = d 3 x  μ Pμ − δφ · "φ ,



3

(17.290)

with the Hamiltonian density and canonically conjugate momentum fields H = ˙ respectively. Since we have quantized φ˙ · "φ − L = − cP0 and "φ = ∂L/∂ φ, the fields, these “charges” are now operators which yield the charges as expectation values. Following standard terminology, we will address the operators as charge operators, or sometimes (sloppily) as charges. The observation (17.290), the Heisenberg evolution equation ih¯ φ˙ = [φ, H ], and the canonical (anti-)commutation relations [φ(x, t), "φ (x  , t)]± = ihδ(x − x) ¯ motivate the expectation 

[φ, Q] = − ih¯ δφ +  · ∇φ +  0 ∂0 φ .

(17.291)

Verify this for the charge operators for the Schrödinger field. The charge operators therefore generate the symmetry transformations on the field operators φ(x) ≡ φ(x, t) in the form  φ  (x  ) = exp(− iQ/h)φ(x ) exp(iQ/h) ¯ ¯ = #δ · φ(x),

(17.292)

where #δ is the matrix representation for the internal symmetry transformations or the spin representation for the Lorentz transformations, respectively. Remarks 1. Suppose the internal symmetry group is a matrix group with generators X, φ  (x) = #(ϕ) · φ(x) = exp(iqϕ · X/h) ¯ · φ(x),

(17.293)

where the inessential constants q and h¯ are only included for conventional reasons, because that is how these kinds of transformations appear in generalizations of electrodynamics. The charge operators then take the form Q(ϕ) = qϕ · ,

=−

i h¯

 d 3 x "φ · X · φ,

(17.294)

with independent charge operators $i which satisfy [$i , φa (x)] = −Xia b φb (x)

(17.295)

17.9 Problems

429

and exp(iqϕ · /h) φ  (x) = exp(− iqϕ · /h)φ(x) ¯ ¯ = exp(iqϕ · X/h) ¯ · φ(x).

(17.296)

The composition of two transformations is then       q q q φ (x) = exp − i ϕ 2 · exp − i ϕ 1 · φ(x) exp i ϕ 1 · h¯ h¯ h¯   q × exp i ϕ 2 · h¯       q q q = exp i ϕ 1 · X · exp − i ϕ 2 · φ(x) exp i ϕ 2 · h¯ h¯ h¯     q q (17.297) = exp i ϕ 1 · X · exp i ϕ 2 · X · φ(x). h¯ h¯ 

This shows that the matrix representation X and the operator representation

of the symmetry generators must satisfy the same commutation relations [Xi , Xj ] = ifij k Xk (see e.g. (7.50) or (7.51) for the commutation relations for the generators of SO(3)), since the commutation relations of the generators of a continuous group determine the composition law of two elements of the group, see Appendix E. The relation [$i , $j ] = ifij k $k can be directly verified from (17.294). 2. The relations for momentum imply in particular (see also (17.207)) [Pμ , φ(x)] = ih∂ ¯ μ φ(x).

(17.298)

This is the Heisenberg evolution equation for μ = 0. The equations for the spatial momentum operators together with the relations for the first quantized momentum operator p imply the following realizations of the action of translations x 0 → x = x 0 −  on the Heisenberg picture field operators if we use “first quantized bra-ket notation” (x) ≡ (x, t) = x|  (t) for the field operators, x|  (t) = x + |(t) = x| exp(i · p/h)|(t) ¯ = exp(− i · P /h)x|(t) exp(i · P /h). ¯ ¯

(17.299)

We used the “special” case of time-independent 4-momentum operators Pμ in the formulation of equations (17.298), (17.299), because this is actually the generic case in quantum field theory, where interactions are expressed through multiplication terms of different field operators instead of potentials. However, if we have external potentials V (x) in the Hamiltonian, we need to write (17.298) in the form (17.207) and include the time-dependence in (17.299),

430

17 Non-relativistic Quantum Field Theory

x|  (t) = x + |(t) = x| exp(i · p/h)|(t) ¯ = exp[− i · P (t)/h]x|(t) exp[i · P (t)/h]. (17.300) ¯ ¯ Charge operators still generate their associated transformations even if those transformations are not symmetries of the system and the charges are not conserved.

Chapter 18

Quantization of the Maxwell Field: Photons

We will now start to quantize the Maxwell field Aμ (x) = {− (x)/c, A(x)} similar to the quantization of the Schrödinger field. The fact that electromagnetism has a gauge invariance implies that there are more components than actual dynamical degrees of freedom in the Maxwell field. This will make quantization a little more challenging than for the Schrödinger field, but we will overcome those difficulties. Electromagnetic field theory is implicitly relativistic, and quantized Maxwell theory therefore also provides us with a first example of a relativistic quantum field theory. Appendix B provides an introduction to 4-vector and tensor notation in electromagnetic theory.

18.1 Lagrange Density and Mode Expansion for the Maxwell Field The equations of motion for the Maxwell field are the inhomogeneous Maxwell equations,1 % & ∂μ F μν = ∂μ ∂ μ Aν − ∂ ν Aμ = − μ0 j ν .

(18.1)

These equations can be written as jν +

% & ∂L ∂L 1 ∂μ ∂ μ Aν − ∂ ν Aμ = − ∂μ =0 μ0 ∂Aν ∂(∂μ Aν )

(18.2)

1 Recall

from electrodynamics that the homogeneous Maxwell equations, viz. Gauss’ law of absence of magnetic monopoles and Faraday’s law of induction, were solved through the introduction of the potentials Aμ .

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 R. Dick, Advanced Quantum Mechanics, Graduate Texts in Physics, https://doi.org/10.1007/978-3-030-57870-1_18

431

432

18 Quantization of the Maxwell Field: Photons

if we use the Lagrange density L = j ν Aν −

1 0 1 2 Fμν F μν = E 2 − B + j · A − . 4μ0 2 2μ0

(18.3)

This Lagrangian provides us with the canonically conjugate momentum for the vector potential A: ˙ =  0 (A ˙ + ∇) = − 0 E, A = ∂L/∂ A

(18.4)

˙ =0 " = ∂L/∂ 

(18.5)

but

vanishes identically! Therefore we cannot simply impose canonical commutation relations between the four components Aμ of the 4-vector potential and four conjugate momenta "ν . To circumvent this problem we revisit the pertinent Maxwell equations (18.1), i.e. Coulomb’s law, ˙ =−  + ∇ · A

1 , 0

(18.6)

and Ampère’s law, ∇(∇ · A) − A +

1 ∂ 1 ∂2 A + 2 ∇ = μ0 j . 2 2 c ∂t c ∂t

(18.7)

One way to solve the problem with " = 0 is to eliminate ∇ · A from the equations of motion through the gauge freedom (x, t) → f (x, t) = (x, t) − f˙(x, t),

(18.8)

A(x, t) → Af (x, t) = A(x, t) + ∇f (x, t),

(18.9)

i.e. we impose the gauge condition ∇ · Af = 0. The equation f (x, t) = − ∇ · A(x, t)

(18.10)

can be solved with the Green’s function G(r) = (4π r)−1 for the Laplace operator, 

1 = − δ(x − x  ), 4π |x − x  |

see Eqs. (11.14) and (11.24) for E = 0. This yields

(18.11)

18.1 Lagrange Density and Mode Expansion for the Maxwell Field

f (x, t) =

1 4π



d 3x

1 ∇ · A(x  , t). |x − x  |

433

(18.12)

This gauge is denoted as Coulomb gauge. We denote the gauge transformed fields again with  and A, i.e. we have ∇ · A(x, t) = 0.

(18.13)

and  = −

1 , 0

(18.14)

1 ∂2 1 ∂ A − A + 2 ∇ = μ0 j . c2 ∂t 2 c ∂t

(18.15)

We can now get rid of  by solving (18.14) again with the Green’s function for the Laplace operator, 1 (x, t) = 4π 0



d 3x

(x  , t) . |x − x  |

(18.16)

The resulting equation for A is 

   ∂ 1 ∂2 μ0 3  x−x (x  , t) −  A(x, t) = μ j (x, t) + x d 0 4π c2 ∂t 2 |x − x  |3 ∂t  μ0 1 ∇  · j (x  , t) = μ0 j (x, t) + d 3x ∇ 4π |x − x  |  μ0 1 = μ0 j (x, t) + d 3 x  j (x  , t) · ∇ ⊗ ∇ 4π |x − x  | = μ0 J (x, t).

(18.17)

We can also evaluate the derivatives in the integral using ∇⊗∇

er 1 4π 1 − 3er ⊗ er = −∇ ⊗ 2 = − 1δ(x) − . r 3 r r3

(18.18)

This yields J (x, t) =

2 j (x, t) 3  3(x − x  ) ⊗ (x − x  ) − |x − x  |2 1 + d 3x · j (x  , t). (18.19) 4π |x − x  |5

434

18 Quantization of the Maxwell Field: Photons

The new current density J satisfies ∇ · J (x, t) = 0.

(18.20)

This follows from the definition of J in the first line of (18.17) and charge conservation, or directly from the second or third line of (18.17), which can be considered as projections of the vector j onto its divergence-free part. We also require localization of charges and currents in the sense of lim |x|j (x, t) = 0.

(18.21)

|x|→∞

Equation (18.17) can be solved e.g. with the retarded Green’s function, cf. Eq. (J.115), 

 1 ∂2 −  G(x, t) = δ(x)δ(t), c2 ∂t 2

G(x, t) =

r 1

δ t− , 4π r c

(18.22)

in the form 

3 

AJ (x, t) = μ0 =

μ0 4π



d x 

d 3x

dt  G(x − x  , t − t  )J (x  , t  )

  1 |x − x  |  , t − J x . |x − x  | c

(18.23)

This satisfies ∇ · AJ (x, t) = 0 due to (18.20), (18.21). The vector field is only a special solution of the inhomogeneous equation (18.17), and the general solution will be a superposition A(x, t) = AJ (x, t) + AD (x, t)

(18.24)

of the special inhomogeneous solution with the general solution of the homogeneous equations 

 1 ∂2 −  AD (x, t) = 0. c2 ∂t 2

(18.25)

The homogeneous solution still has to satisfy the gauge condition ∇ · AD = 0, because the total vector potential A has to satisfy this condition. Fourier decomposition 1 AD (x, t) = 4π 2



 3

d k

dω AD (k, ω) exp[i(k · x − ωt)],

(18.26)

18.1 Lagrange Density and Mode Expansion for the Maxwell Field

AD (k, ω) =

1 4π 2

435





dt AD (x, t) exp[− i(k · x − ωt)]

d 3x

(18.27)

transforms the condition ∇ · AD (x, t) = 0 and Eq. (18.25) into k · AD (k, ω) = 0

(18.28)

  ω2 2 k − 2 AD (k, ω) = 0. c

(18.29)

and

Equation (18.28) is the statement that photons are transverse, whereas Eq. (18.29) implies that AD (k, ω) can be written as 

2

π h¯ μ0 c AD (k, ω) =  α (k) aα (k)δ(ω − ck) k α=1  + + aα (− k)δ(ω + ck) ,

(18.30)

where the prefactor so far is a matter of convention and the two vectors  α (k) are a Cartesian basis in the plane orthogonal to k:  α (k) ·  β (k) = δαβ ,

k ·  α (k) = 0.

(18.31)

Substitution of Eq. (18.30) into Eq. (18.26) yields 

2

d 3k AD (x, t) =  α (k) aα (k) exp[i(k · x − ckt)] √ 2k α=1  + aα+ (k) exp[− i(k · x − ckt)] , (18.32)

hμ ¯ 0c (2π )3

and for the fields ∂ AD (x, t) ∂t  2 

k h¯ μ0 c3 3 =i k  (k) aα (k) exp[i(k · x − ckt)] d α 2 (2π )3 α=1  (18.33) − aα+ (k) exp[− i(k · x − ckt)] ,

E D (x, t) = −

B D (x, t) = ∇ × AD (x, t)

436

18 Quantization of the Maxwell Field: Photons



2

d 3k k ×  α (k) aα (k) exp[i(k · x − ckt)] √ 2k α=1  − aα+ (k) exp[− i(k · x − ckt)] . (18.34)

h¯ μ0 c =i (2π )3

Inversion of Eqs. (18.32), (18.33) yields  aα (k) =

  √ d 3x i ˙  kAD (x, t) + √ A  α (k) · D (x, t) c k (2π )3 2μ0 hc ¯

× exp[− i(k · x − ckt)],

aα+ (k) =



(18.35)

  √ d 3x i ˙  kAD (x, t) − √ A  α (k) · D (x, t) c k (2π )3 2μ0 hc ¯

× exp[i(k · x − ckt)].

(18.36)

We can think of the vector potential (18.32) as a state |AD (t) with components  k, α|AD (t) =

& hμ ¯ 0c % aα (k) exp(− ickt) + aα+ (− k) exp(ickt) 2k

(18.37)

in wave vector space, and x, i|AD (t) =



2

d 3k i α (k) aα (k) exp[i(k · x − ckt)] √ 2k α=1  + aα+ (k) exp[− i(k · x − ckt)] (18.38)

hμ ¯ 0c (2π )3

in x space. This corresponds to transformation matrices x, i|k, α = √

1 2π

 i (k) exp(ik 3 α

· x),

(18.39)

and we can easily check the completeness relations k, α|k  , β =

 d 3x



k, α|x, ix, i|k  , β = δ(k − k  ) α (k) ·  β (k)

i

= δ(k − k  )δαβ ,

(18.40)

18.1 Lagrange Density and Mode Expansion for the Maxwell Field

x, i|x  , j  =

 d 3k

437

x, i|k, αk, α|x  , j  α

=

1 (2π )3

1 = (2π )3 1 = (2π )3

  

d 3 k exp[ik · (x − x  )]



αi (k)αj (k)

α

d 3 k exp[ik · (x − x  )]P⊥ (k) ij



ki kj d k exp[ik · (x − x )] δ − 2 k 

3



ij

= δ⊥ (x − x  ). ij

(18.41)

Here the equation 2

 α (k) ⊗  α (k) = 1 − kˆ ⊗ kˆ

(18.42)

α=1

has been used, cf. the decomposition of unity (4.30). Equation (18.41) defines the transverse δ function.

Energy-Momentum Tensor for the Free Maxwell Field The Lagrange density for the free Maxwell field, L=−

1 Fμν F μν , 4μ0

(18.43)

yields a canonical energy-momentum tensor μ ν = ημ ν L − ∂μ Aλ

1 ∂L = ∂(∂ν Aλ ) μ0



1 ∂μ Aλ · F νλ − ημ ν Fκλ F κλ 4

 (18.44)

which is not gauge invariant. However, the free equation ∂ν F νλ = 0 implies a trivial conservation law −

% & 1 ∂ν ∂λ Aμ · F νλ = 0 μ0

(18.45)

which can be added to the conservation law for the free fields, ∂ν μ ν = 0. In this way we can improve the energy-momentum tensor μ ν to a gauge invariant energymomentum tensor

438

18 Quantization of the Maxwell Field: Photons

T μ = μ ν

ν

1 1 − ∂λ Aμ · F νλ = μ0 μ0

  1 ν νλ κλ Fμλ F − ημ Fκλ F . 4

(18.46)

The corresponding energy-momentum density vector Pμ = Tμ 0 /c yields the well known expressions for the energy and momentum densities of electromagnetic fields, H = − cP0 = −T0 0 =

0 2 1 2 E + B , 2 2μ0

P = 0 E × B.

(18.47) (18.48)

The components of the energy current density (the Poynting vector) are given by the components −cT0 i (because −∂t T0 0 = −c∂0 T0 0 = c∂i T0 i ): S=

1 E × B = c2 P. μ0

(18.49)

We are interested in the energy and momentum densities for the free fields AD (x, t), because those will become the freely evolving field operators in the Dirac picture.

18.2 Photons In the previous section we got rid of  and even of the longitudinal component of A. Now we might be tempted to impose canonical commutation relations [Ai (x, t), "j (x  , t)] ∼ ihδ ¯ ij δ(x − x  ). However, this would be inconsistent, since Eq. (18.13) implies that application of ∂/∂xi and summation over i on the left-hand side would yield zero, but on the right hand side would not yield zero! This problem arises irrespective of whether we wish to quantize the full vector potential A or only the free vector potential AD . Therefore we have to invoke the transverse δ-function (18.41) to formulate the canonical commutation relations for the Maxwell field. We will use these relations primarily for the Dirac picture operators, but we omit the index D from now on,    ki kj ih¯  3 ˙ [Ai (x, t), Aj (x , t)] = d k δij − 2 exp[ik · (x − x  )] 0 (2π )3 k  2 ih¯ 3 = k α,i (k)α,j (k) exp[ik · (x − x  )], d 0 (2π )3 α=1

or in short form

18.2 Photons

439

[Ai (x, t), A˙ j (x  , t)] =

ih¯ ⊥ δ (x − x  ). 0 ij

(18.50)

This equation can also be written using the zero energy Green’s function G(x − x  ) = x|G(0)|x   (cf. Eqs. (J.28), (J.30) and (J.51)),  ih¯

δij δ(x − x  ) + ∂i ∂j G(x − x  ) , [Ai (x, t), A˙ j (x  , t)] = 0

(18.51)

and using (18.18) we find ˙  , t)] = [A(x, t) ⊗, A(x

ih¯ 0 +



2 1 δ(x − x  ) 3

 3(x − x  ) ⊗ (x − x  ) − |x − x  |2 1 . (18.52) 4π |x − x  |5

The remaining commutation relations are [Ai (x, t), Aj (x  , t)] = 0,

[A˙ i (x, t), A˙ j (x  , t)] = 0.

(18.53)

The relations (18.50), (18.53) yield for the operators aα (k), aβ+ (k  ) harmonic oscillator relations, [aα (k), aβ (k  )] = 0,

[aα+ (k), aβ+ (k  )] = 0,

[aα (k), aβ+ (k  )] = δαβ δ(k − k  ).

(18.54)

The prefactor in (18.30) was chosen such that no extra factor appears in the commutation relation of aα (k) and aβ+ (k  ). The energy and momentum densities (18.47) and (18.48) yield energy and momentum operators 

 H = =

d 3x  α

P = 0

0 2 1 2 E (x, t) + B (x, t) 2 2μ0



+ d 3 k hck ¯ aα (k)aα (k),

 d x E(x, t) × B(x, t) = 3

(18.55) 

d 3 k h¯ k aα+ (k)aα (k).

(18.56)

α

From these expressions we can infer by the meanwhile standard methods that aα+ (k) creates a photon of momentum hk, ¯ energy h¯ ck and polarization  α (k), while aα (k) annihilates such a photon. In particular,

440

18 Quantization of the Maxwell Field: Photons

|k, α = aα+ (k)|0

(18.57)

is a single photon state with momentum hk, ¯ energy h¯ ck and polarization  α (k). In writing the k space integrals for H and P we have used the prescription of normal ordering, i.e. writing the creation operators on the left side of the annihilation operators. This ensures that vacuum expectation values of charges and currents vanish. Many authors like to explicitly indicate normal ordering for x space representations of charges or currents through double colons, e.g. for Eq. (18.55) this would read     0 2 1 2 + H = : d 3x E + B := (18.58) d 3 k hck ¯ aα (k)aα (k). 2 2μ0 α We will not use the double colon notation and instead use the implicit convention that charges and currents have to be normal ordered in terms of creation and annihilation operators. If we want to construct a creation operator aα+ (x) in x space (corresponding to the operator ψ + (x) in Schrödinger theory) we find 1 aα+ (x) = √ 3 2π 1 = (2π )3 − 



d 3 k aα+ (k) exp(− ik · x) 

 d 3k i

2kμ0 c3 h¯

 d 3 x   α (k) ·

k A(x  , t) 2μ0 ch¯

  ˙ A(x , t) exp[ik · (x  − x) − ickt].

(18.59)

The expression on the right-hand side is time-independent and can just as well be written in terms of the Schrödinger picture operators A(x) = A(x, 0) and ˙ ˙ A(x) = A(x, 0). However, the important observation is that contrary to Schrödinger theory, the original operator in x space, A(x, t), is not a pure annihilation or creation operator any more, but instead is a superposition of annihilation and creation operators. This is a generic feature of relativistic field operators. The property a + (x) = ψ + (x) is a special feature of the non-relativistic Schrödinger field. It is because of this feature of the Schrödinger field that we did not have to use an explicit double colon notation or an implicit agreement to use normal ordering in Schrödinger field theory. Normal ordered expressions in x space were automatically normal ordered in k space. The time evolution of the free photon operators in k space is given by the standard Heisenberg evolution equations,  aDα (k, t) = aα (k) exp(− ickt) = exp

   i i H t aα (k) exp − H t , h¯ h¯

(18.60)

18.3 Coherent States of the Electromagnetic Field

∂ i aDα (k, t) = [H, aDα (k, t)], ∂t h¯

441

(18.61)

and therefore we also have for the field operators in x space    i i A(x, t) = exp H t A(x) exp − H t , h¯ h¯ 

(18.62)

but to recover the evolution equation (18.25) in x space we have to use iterated Heisenberg evolution equations, ∂ i A(x, t) = [H, A(x, t)] = − E(x, t), h¯ ∂t ∂2 1 i A(x, t) = − [H, E(x, t)] = − 2 [H, [H, A(x, t)]]. h¯ ∂t 2 h¯

(18.63)

(18.64)

This is a general property of bosonic relativistic fields.

18.3 Coherent States of the Electromagnetic Field We can directly apply what we have learned about coherent oscillator states to construct a quantum state with the property that the operators E(x, t) (18.33) and B(x, t) (18.34) yield expectation values which correspond to a classical electromagnetic wave, ζ |E(x, t)|ζ  = E(x, t)





2

k  α (k) ζα (k) exp[i(k · x − ckt)] 2 α=1  − ζα+ (k) exp[− i(k · x − ckt)] , (18.65)

hμ ¯ 0 c3 =i (2π )3

3

d k

ζ |B(x, t)|ζ  = B(x, t) 

2

d 3k k ×  α (k) ζα (k) exp[i(k · x − ckt)] √ 2k α=1  − ζα+ (k) exp[− i(k · x − ckt)] . (18.66)

=i

hμ ¯ 0c (2π )3

The results of Sect. 6.5 imply that the state |ζ  can be unitarily generated out of the vacuum [62],

442

18 Quantization of the Maxwell Field: Photons

 |ζ  = exp

 2 % & + + ζα (k)aα (k) − ζα (k)aα (k) |0 d k 3

 = exp

α=1

 2  1 ζα (k)aα+ (k) − |ζα (k)|2 d 3k |0. 2

(18.67)

α=1

The corresponding equations in the Schrödinger picture are ζ (t)|E(x)|ζ (t) = E(x, t),  |ζ (t) = exp

ζ (t)|B(x)|ζ (t) = B(x, t),

 2 % & + + ζα (k, t)aα (k) − ζα (k, t)aα (k) |0, d k 3

(18.68) (18.69)

α=1

with ζα (k, t) = ζα (k) exp(− ickt).

(18.70)

The expectation values for photon number, momentum and energy in the electromagnetic wave are2  n = ζ |

d 3k

2

aα+ (k)aα (k)|ζ  =

 d 3k

α=1

2

|ζα (k)|2 ,

(18.71)

α=1

 ζ |P |ζ  =

d 3k

2

h¯ k |ζα (k)|2 ,

(18.72)

2 hck ¯ |ζα (k)| ,

(18.73)

α=1

 ζ |H |ζ  =

d 3k

2 α=1

and we find with  3

d k

2 α=1

2 aα+ (k)aα (k)

 =

 3

d k

3 

d k

2

aα+ (k)aα+ (k  )aα  (k  )aα (k)

α,α  =1

that the photons satisfy E = cp, but the electromagnetic wave will generically satisfy  ζ |H |ζ   c|ζ |P |ζ | only if 2α=1 |ζα (k)|2 is strongly peaked at a particular wave number, i.e. if it is dominated by one particular kind of photons.

2 Note

18.4 Photon Coupling to Relative Motion

443

 +

d 3k

2

aα+ (k)aα (k)

(18.74)

α=1

the relations  n  = 2

3

d k

2

2 |ζα (k)|

2

 +

3

d k

α=1

n =



 n =

3

d k

2

|ζα (k)|2 ,

(18.75)

n 1 =√ . n n

(18.76)

α=1 2 α=1

1/2 |ζα (k)|

2

,

Every free quantum field has an expansion in terms of oscillator operators, and therefore each quantum field has coherent states which yield classical expectation values for the field. They are particularly important for electromagnetic fields because classical electromagnetic waves are so abundant, readily available, and of technical relevance. This is a consequence of boson statistics and of the vanishing mass and charge of photons. Generating and packing together huge numbers of photons is very inexpensive in terms of energy.

18.4 Photon Coupling to Relative Motion The discussion of photon interactions with atoms or molecules usually does not involve discussions of photon interactions with the individual constituent electrons and nuclei, but assumes either an effective coupling to the quasiparticles which describe the relative motion between nuclei and electrons, or otherwise assumes coupling of the photons to only one kind of particle in a many particle system. A shortcut justification e.g. for assuming that photons should primarily couple to electrons rather than nuclei in atoms is that physical intuition would indicate that an electromagnetic wave should shake a lighter particle more easily than a heavier particle. Indeed, when we calculate the cross section for photon scattering off free charged particles in Sect. 23.3, we will find that scattering of low energy photons is suppressed with the mass of the scattering particle like m−2 , and for high energy photons like m−1 . Furthermore, atomic matrix elements for optical dipole transitions scale like m−1 which also indicates a preference for coupling to the lighter charged components in a composite system. However, the intuition can be misleading. A simple counterexample is provided e.g. by the absorption or emission of infrared photons by molecules. The dominant degrees of freedom which undergo transitions in these cases are molecular vibrations or rotations, but not electronic transitions, i.e. the dominant photon-matter interaction for infrared photons concerns coupling to clusters of ions or atomic nuclei. What this means

444

18 Quantization of the Maxwell Field: Photons

is that we have to make judicious calls on which terms in a quantum electronics or quantum electrodynamics Hamiltonian will make the dominant contributions to photon interactions, depending on the photon energy range that we are interested in and the available atomic or molecular transitions. However, if we use standard atomic orbitals to model the states of the unperturbed matter system, then this does imply an approximation of photon coupling to the quasiparticle which describes relative motion, or to the electrons as the lightest charges. It is therefore instructive to revisit the problem of separation of center of mass motion and relative motion in the presence of electromagnetic fields. The two-particle Hamiltonian (7.1) with the electromagnetic vector potentials included takes the form H =

1 1 (p1 − q1 A(x1 , t))2 + (p2 − q2 A(x2 , t))2 2m1 2m2 +V (|x1 − x2 |).

(18.77)

Substitution of the single-particle momenta with the total momentum and the effective momentum in the relative motion (7.6) yields H =

1 (P − q1 A(x1 , t) − q2 A(x2 , t))2 2M   m2 q1 A(x1 , t) − m1 q2 A(x2 , t) 2 1 p− + + V (|x1 − x2 |). (18.78) 2μ M

Now we assume that the electromagnetic potentials vary weakly over the extension of the two-particle system, A(x 1 , t)  A(x 2 , t)  A(R, t).

(18.79)

This yields an effective Hamiltonian H =

1 1 (P − QA(R, t))2 + (p − qA(R, t))2 + V (|r|), 2M 2μ

(18.80)

with the total charge Q = q1 + q2 in the kinetic term for center of mass motion, and a reduced charge in the quasiparticle kinetic term, q=

m2 q1 − m1 q2 , m1 + m2

(18.81)

with inversions q1 =

m1 Q + q, M

q2 =

m2 Q − q. M

(18.82)

18.4 Photon Coupling to Relative Motion

445

The equations of motion of the classical two-particle system are in the approximations B(x 1 , t)  B(x 2 , t)  B(R, t), (same for E) given by ¨ = (QR ˙ + q r˙ ) × B + QE, MR √   ˙ + μQ + M(M − 4μ)q r˙ × B + qE − ∂V . μ¨r = q R M ∂r

(18.83) (18.84)

Even the simplifying assumption (18.79) does not allow for separation of the center of mass motion any more, and the equations do not separate in terms of center of mass and relative coordinates, nor in total and reduced masses or charges. However, equations (18.83), (18.84) show that the coupling of photons to the center of mass motion is suppressed with inverse total mass. Therefore the impact of the photons on the relative motion dominates over the impact on center of mass motion, and in leading order we are left with an effective single-particle Hamiltonian for the relative motion in the center of mass frame, H =

1 (p − qA(t))2 + V (|r|). 2μ

(18.85)

The corresponding statement at the classical level (18.83), (18.84) is that in leading order of μ/M, the center of mass frame is preserved and we have an effective singleparticle problem for relative motion, μ¨r = q r˙ × B + qE −

∂V . ∂r

(18.86)

Equation (18.81) for the effective charge q yields q = q1 if q2 = −q1 , and m2  m1 also implies q  q1 . This entails that in atoms or molecules, we can think of photons as effectively coupling to the electrons if the photon wavelengths hc/Eγ are large compared to the size of the atoms or molecules. An alternative justification of using effective single-particle Hamiltonians like (18.85) for photon interactions with bound systems is therefore also to discard the contribution for the heavier particle with mass m2 in the original two-particle Hamiltonian (18.77), but still assume the bound states |n, , m  which were derived for the relative motion r = x 1 −x 2 to hold for the coordinate x 1 of the lighter particle. This is an equivalent approximation up to correction terms of the same order m1 /m2  μ/M. However, the approximation clearly becomes invalid if transitions for the lighter particles are prohibited either by selection rules or by absence of suitable energy levels. The derivation of (18.85) required negligible spatial variation of the photon terms over the extension of the unperturbed wave functions for relative motion, to justify minimal photon coupling into the Hamiltonian for relative motion. Dipole approximation and minimal electromagnetic coupling to the Hamiltonian for relative motion in a bound system therefore use the same premise.

446

18 Quantization of the Maxwell Field: Photons

18.5 Energy-Momentum Densities and Time Evolution in Quantum Optics Further discussions of photon-matter interactions and of time evolution of the quantized Maxwell field require the Hamiltonian and momentum operators for coupled electromagnetic and matter fields. The study of electromagnetic interactions with non-relativistic matter fields is the domain of quantum optics or quantum electronics. A Lagrange density for coupled electromagnetic and non-relativistic matter fields is   ih¯  h¯ 2 ∂a ∂a+ L= a+ · ∇a+ · ∇a − · a − qa a+ a − 2 ∂t ∂t 2m a a    ↔ q2 1 qa h¯ −i A · a+ ∇ a − a a+ A2 a − Fμν F μν . (18.87) 2ma 2ma 4μ0 Here  = − cA0 is the electric potential, and we use the definition of an alternating derivative operator ↔

ψ + ∇ ψ ≡ ψ + · ∇ψ − ∇ψ + · ψ.

(18.88)

The summation over a in (18.87) refers to different kinds of non-relativistic particles (e.g. electrons, protons etc.), and a summation over spin labels is implicitly understood. Phase invariance yields the electric charge and current densities = j 0 /c =



qa a+ a ,

(18.89)

a

j =

 qa  ↔ + + ∇  − 2iq  A h ¯ a a a a a . 2ima a

(18.90)

Like the energy-momentum tensor (18.44), the canonical energy-momentum tensor following from the Lagrange density (18.87) according to the general result (16.31), μ = η μ ν

ν

  ∂L ∂L 1 νλ + , + ∂μ a ∂μ a L+ ∂μ Aλ · F − μ0 ∂(∂ν a ) ∂(∂ν a+ ) a

is not gauge invariant. Just like in the case of the free Maxwell field, we can cure this by adding the trivially conserved tensor

18.5 Energy-Momentum Densities and Time Evolution in Quantum Optics

δμ ν = −

1 1 ∂λ (Aμ F νλ ) = − Aμ j ν − ∂λ Aμ ·F νλ , μ0 μ0

∂ν δμ ν ≡ 0.

447

(18.91)

The improved energy-momentum tensor tμ ν = μ ν + δμ ν yields in particular the gauge invariant energy density for quantum optics, H = − t0 0 =

0 2 1 2 1  2 E + h¯ ∇a+ · ∇a B + 2 2μ0 2m a a    ↔ + 2 + 2 ·  + iqa hA ∇  ¯ a + qa a A a , a

(18.92)

and the gauge invariant momentum density, P=

  ↔ 1 0 1 + + ti e i = 0 E × B + h ¯ a ∇ a − 2iqa a Aa . c 2i a

(18.93)

In materials science it is convenient to explicitly disentangle the contributions from Coulomb and photon terms in Coulomb gauge ∇ · A = 0. We split the electric field components in Coulomb gauge according to E  = − ∇

(18.94)

∂A . ∂t

(18.95)

and E⊥ = −

The equation for the electrostatic potential decouples from the vector potential,  = −

1 qa a+ a , 0 a

(18.96)

and is solved by (x, t) =

1 4π 0



d 3x

a

qa  + (x  , t)a (x  , t). |x − x  | a

(18.97)

Furthermore, the two components of the electric field are orthogonal in the Coulomb gauge, 

 d 3 x E  (x, t) · E ⊥ (x, t) =

d 3 k E  (k, t) · E ⊥ (− k, t) 

=−

d 3 x (x, t)

∂ ∇ · A(x, t) = 0, (18.98) ∂t

448

18 Quantization of the Maxwell Field: Photons

and the contribution from E  to the Hamiltonian generates the Coulomb potentials   0 0 d 3 x E 2 (x, t) = − d 3 x (x, t)(x, t) 2 2  1 = d 3 x (x, t) (x, t) 2   a+ (x, t)a+ (x  , t)a  (x  , t)a (x, t) , (18.99) = d 3 x d 3 x  qa qa  8π 0 |x − x  | 

HC =

aa

where the ordering of the field operators was performed to ensure correct expectation values for the interaction energy of two-particle states after second quantization. The summation may also implicitly include spinor indices. The resulting Hamiltonian in Coulomb gauge therefore has the form  1  H = d x h¯ 2 ∇a+ (x, t) · ∇a (x, t) 2m a a    ↔ + + iqa hA(x, t) · a (x, t) ∇ a (x, t) + qa2 a+ (x, t)A2 (x, t)a (x, t) ¯ 

3

 1 2 0 2 E ⊥ (x, t) + B (x, t) 2 2μ0   a+ (x, t)a+ (x  , t)a  (x  , t)a (x, t) . (18.100) d 3 x d 3 x  qa qa  + 8π 0 |x − x  |  +

aa

This Hamiltonian yields the minimally coupled Schrödinger equations for the operators a (x, t) and Eq. (18.17) for the vector potential in Coulomb gauge through the corresponding Heisenberg picture evolution equations, see Problem 18.4. The momentum operator in Coulomb gauge follows from (18.93) and 

 d 3 x 0 E  × B = −

 d 3 x 0 A =

 d 3 x A =

d 3x



qa a+ Aa

a

as 

 P =

3

d x

 h¯ +  ∇a + 0 E ⊥ × B . i a a

(18.101)

Recall that Heisenberg or Schrödinger picture field operators satisfy the same canonical commutation relations as the Dirac picture operators because the quantum pictures are related by unitary transformations. For the vector potential A(x, t) in Coulomb gauge this implies the same commutation relations (18.50), (18.53) as

18.5 Energy-Momentum Densities and Time Evolution in Quantum Optics

449

for the Dirac picture vector potential. The Hamiltonian (18.100) then yields the Schrödinger equations for the matter fields from ih¯

∂ (x, t) = [(x, t), H ], ∂t

(18.102)

and the electromagnetic wave equation in Coulomb gauge (18.17) from ih¯

∂ A(x, t) = [A(x, t), H ], ∂t

1 ∂2 A(x, t) = 2 [H, [A(x, t), H ]]. ∂t 2 h¯

(18.103)

These relations imply that also after quantization of the Maxwell field, field operators in the Heisenberg and Schrödinger pictures are still related according to A(x, t) = exp(iH t/h) ¯ A(x) exp(− iH t/h), ¯

(18.104)

and the derivation of scattering matrix elements with the automatic emergence of the interaction picture proceeds exactly as in the previous cases of quantum mechanics and non-relativistic quantum field theory. Removal of the time-dependence from the operators onto the states implies that the full time evolution operator on the states in the Schrödinger picture is (for time-independent Hamiltonian H , i.e. no explicit time-dependence from time-dependent parameters) U (t, t  ) = exp[− iH (t − t  )/h]. ¯ This yields the scattering matrix elements between unperturbed states |f (t) = exp[− iH0 t/h]|f ¯  and |i(t), Sf i = f (∞)|U (∞, − ∞)|i(− ∞) = f |UD (∞, − ∞)|i,

(18.105)

     i i i H0 t exp − H (t − t  ) exp − H0 t  h¯ h¯ h¯    t i = T exp − dτ HD (τ ) , (18.106) h¯ t      i i HD (t) = exp H0 t V exp − H0 t , (18.107) h¯ h¯

UD (t, t  ) = exp



where the identification of H0 = H − V depends on what part of H we can solve and what part we wish to take into account through perturbation theory. I.e. we find the same basic structure of time-dependent perturbation theory in terms of Hamilton operators also after introduction of the relativistic photon operators. We will see in Chaps. 22 and 23 that this property persists in general in quantum field theory also after introduction of other relativistic field operators.

450

18 Quantization of the Maxwell Field: Photons

18.6 Photon Emission Rates The calculation of transition probabilities between Fock states requires timedependent perturbation theory in the second quantized formalism. The relevant part of the Hamiltonian (18.100) for a coupled system of nonrelativistic charged particles and photons is H = H0 + HI + HI I   h¯ 2 0 ˙ 2 1 = d 3x ∇ψσ+ · ∇ψσ + ψσ+ V ψσ + A + (∇ × A)2 2m σ 2 2μ 0 σ    ↔ q2 + 2 q h¯ + ψσ ∇ ψσ + A· +i ψ A ψσ , (18.108) 2m 2m σ σ σ where V is an intra-atomic or intra-molecular potential and the interaction terms between the charged particles and the photons are  HI =

d 3x

  q h¯ ↔ A · ψσ+ ∇ ψσ i 2m σ

(18.109)

q2 + 2 ψ A ψσ . 2m σ σ

(18.110)

and  HI I =

d 3x

Here we explicitly included the spin summations and wrote the Hamiltonian in terms ˙ ˙ of the Schrödinger picture field operators (A(x) ≡ A(x, 0)). In principle there is also the electrostatic repulsion between the particles, HC =

q2 8π 0



 d 3x

d 3x

σ,σ 

ψσ+ (x)ψσ+ (x  )

1 ψσ  (x  )ψσ (x). |x − x  |

(18.111)

However, we will only study transitions with single matter particles, or single quasiparticles describing relative motion in a bound two-particle system, in the initial state. HC will not contribute to transition matrix elements in these cases. For the following calculations we use hydrogen states as an example to illustrate the method, and we use ψσ (x) and ψσ+ (x) as the Schrödinger picture field operators of the effective quasiparticle which describes relative motion of the proton and electron in the atom,  |n, , m , σ ; t   = |n,,m ,σ (t  ) = d 3 x n,,m (x, t  )ψσ+ (x)|0

18.6 Photon Emission Rates

451

= exp(− iEn, t  /h) ¯  =



d 3 x n,,m (x)ψσ+ (x)|0

+ d 3 x n,,m (x) exp(− iH0 t  /h)ψ ¯ σ (x)|0,

(18.112)

i.e. ψσ (x) and ψσ+ (x) are the Schrödinger picture field operators which arise from quantization of the wave function x, σ |(t) in Schrödinger’s wave mechanics. See Problem 18.14 for the question why the state (18.112) is an eigenstate of H0 . According to our results from Sect. 18.4, the Hamiltonian (18.108) includes an approximation if we use it for coupling the electromagnetic potential to the hydrogen atom, because we introduced the photon operators through minimal coupling into the effective single-particle problem that resulted from separation of the center of mass motion. This is a good approximation if the electromagnetic potentials vary only weakly over the size of the atom, A(x p , t)  A(x e , t). Indeed, it is an excellent approximation for the study of transitions between bound hydrogen states, because in these cases λ > hc/13.6 eV = 91 nm. We wish to calculate the photon emission rate, i.e. the transition rate from the initial state (18.112) into a final state with the electron in another atomic state and a photon with momentum hk ¯ and polarization  α (k), |n ,  , m , σ  ; k, α; t = 



+ + d 3 x n , ,m (x) exp(− iH0 t/h)ψ ¯ σ  (x)aα (k)|0



  En , = exp − i + ck t d 3 x n , ,m (x)ψσ+ (x)aα+ (k)|0. h¯

(18.113)

The relevant transition matrix elements for photon emission between t  and t are Sf i (t, t  ) ≡ Sn , ,m ,σ  ;k,α|n,,m ,σ (t, t  )   i      = n ,  , m , σ ; k, α; t| exp − H (t − t ) |n, , m , σ ; t   h¯    i t     = n ,  , m , σ ; k, α|T exp − dτ HD (τ ) |n, , m , σ , h¯ t  where 

   i i HD (τ ) = exp H0 τ (HI + HI I ) exp − H0 τ h¯ h¯

(18.114)

is the time evolution operator on the states in the interaction picture. The scattering matrix element is in leading order and with the standard choice t → ∞, t  → −∞, Sf i = Sn , ,m ,σ  ;k,α|n,,m ,σ = n ,  , m , σ  ; k, α|UD (∞, −∞)|n, , m , σ ; 0

452

18 Quantization of the Maxwell Field: Photons

 ! " 1 ∞  dt exp i(ωn , ;n, + ck)t n ,  , m , σ  ; k, α| ih¯ −∞    iq h¯ ↔ × d 3x A(x) · ψν+ (x) ∇ ψν (x) |n, , m , σ ; 0 2m ν

(18.115)

with the field operators in the Schrödinger picture. We also took into account that the energy levels are  dependent through fine structure. At this stage we are still using A(x), although our reasoning in Sect. 18.4 already indicated that any x dependence in A(x) must be negligible to justify minimal photon coupling into the effective Hamiltonian for relative motion in the atom. We will return to this point below. Substitution of the mode expansion (18.32) for the photon operator and evaluation of the second quantized matrix element transforms the transition matrix element for photon emission into a matrix element of first quantized theory, Sn , ,m ,σ  ;k,α|n,,m ,σ

iq  2π δ(ωn , ;n, + ck) mh¯



hμ ¯ 0c δσ σ  16π 3 k

×n ,  , m | α (k) · p exp(− ik · x)|n, , m . (18.116) The operators  α (k) · p and k · x commute, whence we do not encounter a normal ordering problem in the first quantized matrix element. Equation (18.116) can be interpreted as a first quantized matrix element of the perturbation operator V (t) = −

q (p · A(x, t) + A(x, t) · p), 2m

(18.117)

which contains a first quantized operator corresponding to a classical transversely polarized plane wave 

Aα(+) (x, t)

hμ ¯ 0c  α (k) exp[− i(k · x − ckt)] 16π 3 k = k, α|AD (x, t)|0, =

(18.118)

where AD (x, t) is the second quantized photon operator (18.32). This classical plane wave apparently represents a single emitted photon of sharp energy hck ¯ and momentum hk, ¯ and second quantization helped us to determine both the proper amplitude for the single photon wave and the k-dependent term in the transition matrix element. The corresponding calculation for absorption of a photon yields a first quantized matrix element of the perturbation operator (18.117) with a single photon vector potential  Aα(−) (x, t)

=

hμ ¯ 0c  α (k) exp[i(k · x − ckt)] 16π 3 k

18.6 Photon Emission Rates

453

= 0|AD (x, t)|k, α,

(18.119)

see Eq. (18.145). We can understand the amplitudes of the single photon wave functions (18.118) and (18.119) also in the following way: The mode expansion (18.32) becomes in finite volume V 

A(x, t) =

2

hμ ¯ 0 c  α (k) aα (k) exp[i(k · x − ckt)] √ V 2k k α=1  + aα+ (k) exp[− i(k · x − ckt)] ,

(18.120)

and the corresponding energy and momentum operators3 are H =

k

α

+ hck ¯ aα (k)aα (k),

P =

k

+ hk ¯ aα (k)aα (k).

(18.121)

α

These equations tell us for a classical amplitude aα (k) that this amplitude would (up to an arbitrary phase ϕ) have to be a Kronecker δ with respect to momentum and polarization to represent a single photon of momentum hk, ¯ energy hck ¯ and polarization  α , and therefore the classical vector potential for the single photon in the continuum limit V → 8π 3 is 

hμ ¯ 0c  (k) exp[i(k · x − ckt + ϕ)] Aγ ,k,α (x, t) = α 16π 3 k  + exp[− i(k · x − ckt + ϕ)]  hμ ¯ 0c  α (k) cos(k · x − ckt + ϕ). =2 (18.122) 16π 3 k Note however that for emission only the plane wave with exp[− i(k · x − ckt + ϕ)] contributes to the transition matrix element, whereas for absorption only the other term contributes. The vector potential in box normalization (18.120) does have the expected units Vs/m, whereas the continuum limit vector potentials (18.32), (18.122) come in units of m3/2 Vs/m. This is related to the fact that their transition matrix elements squared yield transition probability densities per volume unit d 3 k in the photon state space, see e.g. Eq. (18.128) below. It is the same effect that we encountered in scattering theory for momentum eigenstates exp(ik · x)/V 1/2 in box normalization or exp(ik · x)/(2π )3/2 in the continuum limit.

3 Classically

these equations would hold for time averages.

454

18 Quantization of the Maxwell Field: Photons

Evaluation of the Transition Matrix Element in the Dipole Approximation We have already emphasized that the coupling of the electromagnetic potentials to the effective single-particle model for relative motion in atoms assumes a long wavelength approximation in the sense A(x p , t)  A(x e , t), see Eqs. (18.79) and (18.85). Therefore the exponential factor exp(− ik · x) must effectively be constant over the extension of the atomic wave functions and can be replaced by exp(− ik · x)  1. For an estimate of the product |k ·x|, we recall that the energy of the emitted photon from an excited bound state cannot exceed the binding energy of hydrogen, hc e2 hcα < −E1 = = , λ 8π 0 a0 4π a0

(18.123)

and therefore λ>

4π a0  1.72 × 103 a0 , α

ka0
En . Therefore the previously discussed transition n → n involved photon emission, while the process n → n involves photon absorption. Later on we will also compare emission and absorption rates, and it is desirable to make the distinction between emission and absorption rates more visible in the notation. Therefore we will denote absorption ˜ rates with the symbol . We assume that the absorbing state is discrete. This would typically be the case for photon absorption by atoms and molecules, and also applies to X-ray absorption

460

18 Quantization of the Maxwell Field: Photons

by core electrons in solid materials. We will discuss both discrete and continuous final states.

Photon Absorption into Discrete States The leading order scattering matrix element for photon absorption due to a transition from the discrete state |n ,  , m , σ  ; k, α to the discrete state |n, , m , σ ; 0,  ! " q h¯ 1 ∞ dt exp i(ωn,;n , − ck)t i ih¯ −∞ 2m    ↔ A(x) · ψν+ (x) ∇ ψν (x) |n ,  , m , σ  ; k, α × n, , m , σ ; 0| d 3 x

Sn,,m ,σ |n , ,m ,σ  ;k,α =

ν

is just the negative complex conjugate of the emission matrix element (18.115). The resulting scattering matrix element after evaluation of the field operators, Sn,,m ,σ |n , ,m ,σ  ;k,α = 2π δ(ωn,;n ,

iq − ck) mh¯



h¯ μ0 c δσ σ  16π 3 k

× n, , m | α (k) · p exp(ik · x)|n ,  , m , (18.145) therefore has the form of a first quantized scattering matrix element with perturbation (18.117) and vector potential (18.119). The equality of the scattering matrix elements up to a phase factor also implies that the absorption rate per k space volume of the incoming photons has the same value as the corresponding emission rate (18.128) per k space volume of emitted photons, d ˜ (α) (k)n , ,m ,σ  →n,,m ,σ d 3k



c3 e2

μ0 8π 2 h¯

=

 2   Sn,,m ,σ |n , ,m ,σ  ;k,α 

T 2  kδσ σ  n, , m | α (k) · x|n ,  , m  δ(ωn,;n , − ck),

(18.146)

where q = − e was substituted. This yields the differential absorption rate for polarized photons in terms of the angles θα,± between the vectors n, , m |x± |n ,  , m  and the polarization  α (k), d ˜ (α) (k)n , ,m ,σ  →n,,m ,σ d 3k



2  μ0 c3 e2 kδσ σ  n, , m |x|n ,  , m  2 8π h¯ ×

cos2 θα,− + cos2 θα,+ δ(ωn,;n , − ck). 2

18.7 Photon Absorption

461

 ˜ The differential absorption rate for unpolarized photons, d (k) = α d ˜ (α) (k), depends on the angles θ± between the vectors n, , m |x± |n ,  , m  and the incident vector k, ˜ n , ,m ,σ  →n,,m ,σ d (k)   d 3k



  μ0 c3 e2    2  n, , m |x|n ,  , m  kδ σ σ  8π 2 h¯ ×

sin2 θ− + sin2 θ+ δ(ωn,;n , − ck). (18.147) 2

The total absorption rate between the specified states follows as ˜ n , ,m ,σ  →n,,m ,σ =

2  μ0 e2 3 ωn,;n , δσ σ  n, , m |x|n ,  , m  , 3π hc ¯

(18.148)

and the total absorption rate per atom for photons of angular frequency ωn,;n , is 

˜ n , →n, =

 1 2 + 1  m

=

=−

 m =−

˜ n , ,m →n,,m

μ0 e2 2 ω   fn,|n , . 2π mc n,;n ,

(18.149)

This differs from the corresponding spontaneous emission rate (18.142) for photons of angular frequency ωn,;n , only through the different averaging factors for the respective initial states, ˜ n , →n, =

2 + 1 n,→n , . 2 + 1

(18.150)

The number of absorption events will be proportional to the flux of incoming photons, and therefore another observable of interest is the absorption rate per flux of incoming photons, i.e. the absorption cross section. The photon flux or current density of monochromatic photons of momentum hk ¯ can be calculated by dividing their energy current density S(k) by their energy h¯ ck. Equations (18.33), (18.34, (18.49) and (18.122) yield E×B c ˆ S(k) = = k. h¯ ck μ0 h¯ ck (2π )3

(18.151)

This is actually a photon flux dj (k)/d 3 k per k space volume due to the use of the photon wave functions in the continuum limit.5

5 The

ˆ result in box normalization is j (k) = (c/V )k.

462

18 Quantization of the Maxwell Field: Photons

Equations (18.146) and (18.151) yield the polarized photon absorption cross section σ

(α)

(k)n , ,m →n,,m =

d ˜ (α) (k)n , ,m →n,,m dj (k)

2  π μ0 ce2 ωn,;n , n, , m | α (k) · x|n ,  , m  δ(ωn,;n , − ck) h¯ 2  = 4π 2 αS ωn,;n , n, , m | α (k) · x|n ,  , m  δ(ωn,;n , − ck),



(18.152)

where we encounter again Sommerfeld’s fine structure constant αS = μ0 ce2 /4π h¯ (7.149) (not to be confused with the polarization index, of course). To average (18.152) over the angles of the incident photons, we can use the same methods that we applied for the calculation of the total polarized emission rate (18.135), except for an extra factor of (4π )−1 from the averaging over directions. This yields an isotropic cross section for polarized photons σ (α) (k)n , ,m →n,,m 

2  4π 2 αS ωn,;n , n, , m |x|n ,  , m  3 ×δ(ωn,;n , − ck), (18.153)

which also equals the total isotropic cross section σ (k)n , ,m →n,,m since this would average over the polarizations of the incoming photons. The average absorption cross section per atom for photons of angular frequency ck follows then again through averaging over initial states and summation over final states, 

σ (k)n , →n,

 1 =  2 + 1 



m =− m =−

=

σ (k)n , ,m →n,,m

2π 2 h¯ αS fn,|n , δ(ωn,;n , − ck). m

(18.154)

We get a more realistic representation for absorption cross sections if we take into account the representation (2.23) of the δ function, δ(ωn,;n ,

1 − ck) = lim γ →0 2π = lim

γ →0



∞ −∞

dt exp[i(ωn,;n , − ck)t − γ |t|]

γ 1 . π (ωn,;n , − ck)2 + γ 2

(18.155)

Keeping a finite value of γ yields a Lorentzian absorption line shape of half width 2γ ,

18.7 Photon Absorption

463

σ (k)n , →n, =

γ 2π h¯ αS fn,|n , . m (ωn,;n , − ck)2 + γ 2

(18.156)

A finite width of line shapes arises from many sources. A certainly not exhaustive list of mechanisms includes adiabatic switching of perturbations, lifetime broadening, pressure broadening, Doppler broadening, and broadening through chemical shifts. We can write the absorption cross section (18.153), (18.156) with generic labels i and f for the discrete initial and final states in the form σ (k)i→f 

γf i 4π αS ωf i |f |x|i|2 , 3 (ωf i − ck)2 + γf2i

(18.157)

where the notation γf i takes into account that the line width will be different for different absorption lines. We have found  ∝ δσ σ  both for photon emission and absorption, i.e. no spinflips in either process. The same holds in arbitrary order with the Hamiltonian (18.108), since there are no spin flipping terms there. How then can a magnetic field flip spins even for non-relativistic electrons? There is actually a term missing in the Hamiltonian (18.108), the Pauli term:  q HB = − ψσ+ (x)S σ,σ  · (∇ × A(x))ψσ  (x). (18.158) d 3x m  σ,σ

This term induces spin flips through two of the three components of the vector of Pauli matrices S = h¯ σ /2, and it follows from the non-relativistic expansion of the relativistic wave equation for electrons, see Sect. 22.5. We could neglect the Pauli term in the present calculation because a derivative on the vector potential yields a factor k, whereas a derivative on the wave functions amounts approximately to a factor of order 1/a0 . The Pauli term is therefore suppressed when dipole approximation λ  a0 applies. E.g. for transition between bound states in hydrogen, hck < −E1 implies that HB is suppressed relative to HI by approximately ¯ ka0 < αS /2, which translates into a suppression of spin-flipping transitions between bound hydrogen states by about αS2 /4  1.3 × 10−5 . An exception to negligibility of spin-flipping transitions with low energy photons concerns situations where spinpreserving electronic transitions do not exist in the same energy range. This is the case e.g. for the 21 cm transition in hydrogen.

Photon Absorption into Continuous States The spin labels are omitted in the following discussion because HI does not induce spin flips.

464

18 Quantization of the Maxwell Field: Photons

If we have photon absorption due to transition into continuous states, e.g. from |n ,  , m  to |E, , m , we have to take into account the proper measure for the continuous states from the completeness relation. E.g. for hydrogen states we have (7.184)  ∞



=0 m =−

=

∞

 |n, , m n, , m | +



=0 m =−

 dK K |K, , m K, , m | 2

0

n=+1

 ∞

∞

∞



 |n, , m n, , m | +

dE (E) |E, , m E, , m |

n=+1

= 1, and if we directly use the Coulomb wave states for the continuous energy eigenstates without rescaling, |E, , m  = |K, , m , we have in the continuous part of the spectrum (E) = (E)K 2

1 dK = (E) 3 2m3 E. dE h¯

(18.159)

The first order scattering matrix element with the interaction Hamiltonian HI then yields a differential absorption rate for polarized photons d ˜ (α) (k)n , ,m →E,,m d 3 kdE  (E)

e2 ck 8π 2 0

= (E)

 2   SE,,m |n , ,m ;k,α 

T   E, , m | α (k) · x|n ,  , m 2 δ(E − En , − hck), ¯  (18.160)

and integration over the energy E of the ionized state yields d ˜ (α) (k)n , ,m →E,,m d 3k



e2 c  k (E) 8π 2 0 2   × E, , m | α (k) · x|n ,  , m 

E=En , +h¯ ck

.

The photons appear in the initial state and are therefore taken into account by dividing out their current density from the transition rate, thus yielding an absorption cross section, see the general discussion for initial continuous states in Sects. 13.5 and 13.6. However, for photon absorption due to transition from a discrete into a continuous atomic or electronic state we can also calculate a spectral absorption cross

18.7 Photon Absorption

465

section dσ (α) (k)/dEγ since Eγ = hck = E − En , implies dEγ = dE. ¯ This allows us to define a spectral absorption cross section for polarized photons according to dσ (α) (k)n , ,m →E,,m dEγ  (E)

=

d ˜ (α) (k)n , ,m →E,,m dEdj (k)

2 π e2  k E, , m | α (k) · x|n ,  , m  δ(E − En , − hck), ¯ 0

(18.161)

where (18.151) was used. In practical applications of (18.161) the energy preserving δ function could again be replaced by a Lorentzian line shape as in (18.155). The absorption cross section for polarized photons with momentum hk ¯ follows from σ (α) (k) = d ˜ (α) (k)/dj (k) = (8π 3 /c)d ˜ (α) (k)/d 3 k or from (18.161) as σ (α) (k)n , ,m →E,,m 

π e2  k (E) 0 2   × E, , m | α (k) · x|n ,  , m 

E=En , +h¯ ck

,

and averaging over the directions like in (18.153) yields the isotropic absorption cross section σ (k)n , ,m →E,,m 

 2  π e2  k (E) E, , m |x|n ,  , m  . E=En , +h¯ ck 30

We can write this with generic labels i and f for the discrete initial state and continuous final state in the form σ (k)i→f 

  4π 2 αS h¯ ωf i (Ef ) |f |x|i|2 . Ef =Ei +h¯ ck 3

(18.162)

Photon Absorption Coefficients If we illuminate a material with photons of current density j γ , the photon current density jγ (x) at depth x in the material will usually drop exponentially with the penetration depth, jγ (x) = jγ (0) exp(−μx),

djγ (x)/dx = −μjγ (x).

(18.163)

We can understand this in the following way. According to the definition of the absorption cross section σ , every absorption center in the material (e.g. atoms which

466

18 Quantization of the Maxwell Field: Photons

can absorb X-ray photons due to excitation of their core electrons) reduces the number of photons according to dNγ(1) /dt = − σjγ ,

(18.164)

where the superscript reminds us that this is due to a single absorption center. In a time dt, the photons will encounter dN = nAcdt

(18.165)

absorption centers, where n is the volume density of the absorption centers, A is the illuminated area, and c is the speed of light in the material. The photon current N˙ γ (x) = jγ (x)A going through the area A at location depth x and the photon current N˙ γ (x + dx) = N˙ γ (x + cdt) differ by N˙ γ (x + dx) − N˙ γ (x) = N˙ γ (x + cdt) − N˙ γ (x) = dN ·

(1)

dNγ (x) dt

= − nAcdt · σjγ (x) = − nσ N˙ γ (x)dx.

(18.166)

This is Eq. (18.163) with the absorption coefficient or attenuation coefficient μ = nσ.

(18.167)

This will depend on the energy hck ¯ of the photons since the absorption cross section σ depends on photon energy. We can use the results (18.157), (18.162) to calculate the attenuation coefficient μ if photon absorption in the material is dominated by discrete initial states. The absorption cross sections (18.157) due to transitions into discrete final states will then contribute absorption lines and the cross sections (18.162) will contribute a continuous portion of the absorption spectrum, σ = σ + σc ,

(18.168)

with σ =

4π i,f

3

αS ωf i |f |x|i|2

γf i (ωf i − ck)2 + γf2i

(18.169)

and σc =

4π 2 i,f

3

αS ωf i (Ef ) |f |x|i|2 δ(ωf i − ck).

(18.170)

18.8 Stimulated Emission of Photons

467

The sum over final states in the continuous absorption spectrum (18.170) includes an integration over the final state energy Ef and also integrations over any other continuous final state quantum numbers.

18.8 Stimulated Emission of Photons Here we use box normalization in a volume V = L3 , i.e. k = 2π n/L. If we have already nk,α photons of momentum hk ¯ and polarization  α (k) in the initial state,  |n, , m , σ ; nk,α  =

d 3 x ψσ+ (x)

(aα+ (k))nk,α  |0x|n, , m , nk,α !

the basic oscillator relation n + 1|a + |n = scattering matrix elements the relation Sn , ,m ,σ  ;nk,α +1|n,,m ,σ ;nk,α =

√

(18.171)

n + 1 yields for the leading order

 nk,α + 1Sn , ,m ,σ  ;k,α|n,,m ,σ ,

(18.172)

i.e. the emission rate scales with the number of photons of momentum hk, ¯ energy   and fixed polarization like hck = h ω ¯ ¯ n,;n , (α)

(α)

n,;nk,α →n , ;nk,α +1 = (nk,α + 1)n,;0→n , ;1  nk,α + 1 = 2 + 1





m =− m =−

= (nk,α + 1)

(α)

n,,m

 →n

 , ,m 

  μ0 e2 2 ωn,;n , fn , |n,  . 4π mc

(18.173)

The total polarized emission rate in the presence of the nk,α photons therefore differs (α) (α) from the “spontaneous” emission rate n,;0→n , ;1 ≡ n,→n , = n,→n , /2 (cf. Eq. (18.142)) by an additional “stimulated” emission rate (s,α) (α) = nk,α n,;0→n n,;n    , ;1 = nk,α k,α →n , ;nk,α +1

  μ0 e2 2 ωn,;n , fn , |n,  4π mc

which is proportional to the number of photons which are already present in the system. This is sometimes metaphorically explained as a consequence of one of the original photons stimulating the emission by shaking the excited state. However, in the end it is nothing but a combinatorial quantum effect of indistinguishable photon operators. On the other hand, we find for the absorption of a photon in the initial state

468

18 Quantization of the Maxwell Field: Photons





|n , 

, m , σ  ; nk,α 

from n − 1|a|n =

 =

d 3 x ψσ+ (x)

(aα+ (k))nk,α  |0x|n ,  , m , nk,α !

(18.174)

√ n the relation

Sn,,m ,σ ;nk,α −1|n , ,m ,σ  ;nk,α =

√

nk,α Sn,,m ,σ |n , ,m ,σ  ;k,α √ = − nk,α Sn∗ , ,m ,σ  ;k,α|n,,m ,σ . (18.175) 

Therefore the polarized absorption rate in the presence of nk,α photons of momentum hk ¯ and polarization  α (k) is (α) (α) ˜ n , ;nk,α →n,;nk,α −1 = nk,α ˜ n , ;1→n,;0 

 nk,α =  2 + 1 



m =− m =−

(α) ˜ n , ,m →n,,m = nk,α 



μ0 e2 2 ω   fn,|n , . 4π mc n,;n ,

This equals corresponding stimulated and total emission rates up to the different averaging factors for the different initial states which enter into the averaged and summed transition matrix elements, 2 + 1 (α)    2 + 1 n,;nk,α −1→n , ;nk,α 2 + 1 (s,α)  =  .   2 + 1 n,;nk,α →n , ;nk,α +1

(α) ˜ n , ;nk,α →n,;nk,α −1 =

(18.176)

Note that it does not matter that we used the single photon absorption rate and current density in the calculation (18.152) of the polarized photon absorption cross section without explicitly taking into account the number nk,α of available photons. The common factor nk,α cancels in the ratio d ˜ n , ,m ;n (α)

σ

(α)

(k)n , ,m →n,,m =



k,α →n,,m ;nk,α −1

dJ (α) (k)

˜ (α) 8π 3 d n , ,m ;nk,α →n,,m ;nk,α −1 . = nk,α c d 3k

(18.177)

18.9 Photon Scattering

469

18.9 Photon Scattering For the following calculations we switch back to a generic notation |n, ζ  for atomic or molecular states, where the energy levels En depend on the index set n and the index set ζ enumerates the degenerate states. Scattering concerns transitions which involve a photon both in the initial and in the final state: |n, ζ ; k, α → |n , ζ  ; k  , α  . Here we consider scattering of photons by bound non-relativistic systems, i.e. the initial state |n, ζ  and the final state |n , ζ   of the scattering system are discrete, and we use minimal coupling of the photon to effective single-particle models for relative motion in the bound system. We have seen in Sect. 18.4 that photon coupling to the relative motion in materials effectively amounts to photon-electron coupling, and therefore we use photon scattering off bound electrons as the relevant paradigm for the following discussion. To have a non-vanishing matrix element between different 1-photon states in lowest order requires two copies of the photon operator A—one to annihilate the initial photon and one to create the final photon. The relevant interaction Hamiltonian for photon interactions with non-relativistic electrons is  Hint =

    ↔ eh¯ e2 + 2 eh¯ + + A· ψ ∇ ψ + ψ A ψ+ ψ σ · Bψ d x −i 2m 2m 2m 3

= HI + HI I + HB ,

(18.178)

where HB is the Pauli term (18.158). Summations over spinor indices are tacitly understood. We have already substituted q = − e, because we have seen in Sect. 18.4 that the coupling of long wavelength photons to bound systems involving electrons can effectively be considered as coupling of the photons to a charge − e if the charge binding the electron is q2 = e or if the mass m2 of the binding charge is much larger than the electron mass, m2  me . The reduced mass m in the Hamiltonian (18.178) is usually also m  me in excellent approximation.6 We can get two copies of A from HI2 , HI HB , HB HI and HB2 in second order perturbation theory, and from HI I in first order perturbation theory. Among these terms, only those involving the Pauli term can induce spin flips. However, we will focus on photon energies up to the soft X-ray regime, Eγ  1 keV. Due to the suppression of the Pauli term by about a0 /λ the allowed transition matrix elements of HI in the soft X-ray regime are typically at least an order of magnitude larger than the allowed matrix elements of HB , see the discussion after (18.158). This implies that spin preserving scattering probabilities |Sf i |2 of order HI4 will generically be at least two orders of magnitude larger than spin preserving scattering of order HI2 HB2 or spin reversing scattering of order (HI HB )2 .

6 An

exception is positronium with m = me /2.

470

18 Quantization of the Maxwell Field: Photons

Therefore we neglect HB in the following calculations. The relevant scattering matrix elements in order O(e2 ) are then Sn ,ζ  ;k  ,α  |n,ζ ;k,α = n , ζ  ; k  , α  |UD (∞, −∞)|n, ζ ; k, α|e2    i ∞ = n , ζ  ; k  , α  |T exp − dt HD (t) |n, ζ ; k, α|e2 h¯ −∞ (I )

(I I )

= Sn ,ζ  ;k  ,α  |n,ζ ;k,α + Sn ,ζ  ;k  ,α  |n,ζ ;k,α ,

(18.179)

with contributions from HI2 , (I )

Sn ,ζ  ;k  ,α  |n,ζ ;k,α = −

1 h¯ 2





∞ −∞

dt

t

−∞

" ! " ! dt  exp i(ωn + ck  )t exp − i(ωn + ck)t 

  i ×n , ζ  ; k  , α  |HI exp − H0 (t − t  ) HI |n, ζ ; k, α, h¯ and from HI I , (I I ) Sn ,ζ  ;k  ,α  |n,ζ ;k,α

 =

∞

! " dt exp i(ωn ,n + ωk  ,k )t n , ζ  ; k  , α  |HI I |n, ζ ; k, α. i h −∞ ¯

The first order term S (I I ) is the easier one to evaluate. Insertion of the mode expansion (18.32) for the photon field yields Sn(I ,ζI ) ;k  ,α  |n,ζ ;k,α =

=

μ0 ce2 √  α  (k  ) ·  α (k)δ(ωn ,n + ωk  ,k ) 8π 2 im kk   ! " × d 3 x exp i(k − k  ) · x n+ ,ζ  (x)n,ζ (x) μ0 ce2 √  α  (k  ) ·  α (k)δ(ωn ,n + ωk  ,k ) 8π 2 im kk   × d 3 q n+ ,ζ  (q + k − k  )n,ζ (q).

(18.180)

" ! This leaves in dipole approximation exp i(k − k  ) · x  1 the amplitude (I I )

Sn ,ζ  ;k  ,α  |n,ζ ;k,α =

μ0 e2  α  (k  ) ·  α (k)δ(k  − k)δn n δζ  ζ , 8π 2 imk

(18.181)

i.e. only elastic photon scattering, but no Raman scattering from HI I . The term S (I ) splits into amplitudes with zero or two photons in virtual intermediate states,

18.9 Photon Scattering

471

(I )

(I ),0

(I ),2

Sn ,ζ  ;k  ,α  |n,ζ ;k,α = Sn ,ζ  ;k  ,α  |n,ζ ;k,α + Sn ,ζ  ;k  ,α  |n,ζ ;k,α .

(18.182)

We omit the indices in the amplitudes S (I ),0 and S (I ),2 in the following calculations. The amplitude with no photons in the virtual intermediate state is S (I ),0 =

 ∞  t  " ! " ! e2 dt dt  exp i(ωn ,n + ck  )t exp i(ωn ,n − ck)t  2 4m n ,ζ  −∞ −∞    ↔ × d 3 x  n , ζ  ; k  , α  |A(x  ) · ψ + (x  ) ∇ ψ(x  ) |n , ζ  ; 0   ↔ + d x n , ζ ; 0|A(x) · ψ (x) ∇ ψ(x) |n, ζ ; k, α.

 ×



3



(18.183)

 The notation n ,ζ  takes into account that the intermediate states can also be part of the energy continuum of the scattering system. We have already evaluated the time integrals in second order perturbation terms in (13.98), 



∞

−∞

dt

t −∞

= − 2π i

" ! " ! dt  exp i(ωn ,n + ck  )t exp i(ωn ,n − ck)t  + t 

δ(ωn ,n + ωk  ,k ) . ωn ,n − ck − i

(18.184)

Evaluation of the matrix elements of the field operators then yields again in dipole approximation exp(− ik  · x  )  1, exp(ik · x)  1 the result S

(I ),0

 hμ 1 ¯ 0 ce2   = √ δ(ωn ,n + ωk ,k ) 2 2     ω − ck − i 32π im kk n ,n n ,ζ    ↔ × d 3 x   α  (k  ) · n+ ,ζ  (x  ) ∇ n ,ζ  (x  )  ×

  ↔ d 3 x  α (k) · n+ ,ζ  (x) ∇ n,ζ (x) .

(18.185)

We can transform this from velocity into length form using the by now standard trick hp ¯ = im[H0 , x] to find S (I ),0 =

 ωn ,n ωn ,n μ0 ce2 √ δ(ωn ,n + ωk  ,k ) 8π 2 ih¯ kk  n ,ζ  ωn ,n − ck − i ×n , ζ  | α  (k  ) · x|n , ζ  n , ζ  | α (k) · x|n, ζ .

(18.186)

472

18 Quantization of the Maxwell Field: Photons

For the amplitude with two photons in the intermediate state we have to take into account that for two-photon states 1 2



d 3κ 

 d 3κ



|κ  , β  ; κ, βκ  , β  ; κ, β| = 1.

(18.187)

β  ,β

This yields e2 8m2

S (I ),2 =



d 3κ 

 d 3κ





n ,ζ 



∞

β  ,β −∞

dt

t −∞

dt 

" ! " ! × exp i(ωn ,n + ck  − cκ − cκ  )t exp i(ωn ,n + cκ + cκ  − ck)t     ↔ × d 3 x  n , ζ  ; k  , α  |A(x  ) · ψ + (x  ) ∇ ψ(x  ) |n , ζ  ; κ  , β  ; κ, β    ↔ × d 3 xn , ζ  ; κ  , β  ; κ, β|A(x) · ψ + (x) ∇ ψ(x) |n, ζ ; k, α. The matrix elements of the photon operators are given by  



κ , β ; κ, β|A(x)|k, α =

hμ ¯ 0c  β (κ) exp(− iκ · x)δ(κ  − k)δβ  α 16π 3 κ  hμ ¯ 0c +  β  (κ  ) exp(− iκ  · x)δ(κ − k)δβα 16π 3 κ 

and a corresponding conjugate expression. This yields in dipole approximation 

d 3κ 

 d 3κ



! " exp ic(κ + κ  )(t  − t)

β  ,β

× k  , α  |A(x  )|κ  , β  ; κ, βκ  , β  ; κ, β|A(x)|k, α   β (κ) ⊗  β (κ) ! " hμ ¯ 0c   δ(k − k ) exp ic(κ + k)(t  − t) δ d 3κ  αα 3 κ 8π β

+



! " hμ ¯ 0 c  α (k) ⊗  α  (k ) exp ic(k + k  )(t  − t) . √ 3 8π kk 

(18.188)

The first term in (18.188) corresponds to an electron self-energy contribution where the external photon does not interact with the electron, but there are two photons in the intermediate state due to emission and re-absorption of a virtual photon by the electron, see Fig. 18.1. This is an effect which leads to a renormalization of the electron mass in quantum field theory, but does not contribute to photon scattering.

18.9 Photon Scattering

473

Fig. 18.1 A process with two photons in an intermediate state due to emission and re-absorption of a virtual photon. The straight line represents the electron and the wavy lines represent photons

Fig. 18.2 The left diagram corresponds to absorption of the initial photon before emission of the final photon. The diagram on the right-hand side corresponds to emission of the final photon before absorption of the initial photon

The second term yields an expression for S (I ),2 which looks almost exactly like (18.185), except that the polarization vectors are swapped  α  (k  ) ↔  α (k), and ωn ,n − ck − i is replaced by ωn ,n + ck  − i in the denominator. After transformation into the length form, S (I ),0 and S (I ),2 yield the following expression, S (I ),0

(I ) Sn ,ζ  ;k  ,α  |n,ζ ;k,α

 μ0 ce2 = ωn ,n ωn ,n √ δ(ωn ,n + ωk  ,k ) 8π 2 ih¯ kk  n ,ζ     n , ζ | α  (k  ) · x|n , ζ  n , ζ  | α (k) · x|n, ζ  × ωn ,n − ck − i  n , ζ  | α (k) · x|n , ζ  n , ζ  | α  (k  ) · x|n, ζ  . + ωn ,n + ck  − i

The first term corresponds to absorption of the initial photon before emission of the final photon, whereas the second term corresponds to emission of the final photon before absorption of the initial photon, see Fig. 18.2. The total scattering matrix element in order e2 is  μ0 ce2 1 δn n δζ  ζ  α  (k  ) ·  α (k) Sn ,ζ  ;k  ,α  |n,ζ ;k,α = √ δ(ωn ,n + ωk  ,k ) 2  m 8π i kk     n , ζ | α  (k  ) · x|n , ζ  n , ζ  | α (k) · x|n, ζ  + ωn ,n ωn ,n h¯ ωn ,n − h¯ ck − i n ,ζ 

474

18 Quantization of the Maxwell Field: Photons

n , ζ  | α (k) · x|n , ζ  n , ζ  | α  (k  ) · x|n, ζ  + h¯ ωn ,n + hck ¯  − i

 .

We separate the energy conserving δ function for the calculation of the scattering cross section as in Eq. (13.119), Sn ,ζ  ;k  ,α  |n,ζ ;k,α = − iMn ,ζ  ;k  ,α  |n,ζ ;k,α δ(ωn ,n + ωk  ,k ).

(18.189)

The differential scattering rate per k space volume of incident photons is then dn,ζ ;k,α→n ,ζ  ;k  ,α  d 3k

3 

=d k =

 S

d 3k 2π

n ,ζ  ;k  ,α  |n,ζ ;k,α

2 

T  M

n ,ζ  ;k  ,α  |n,ζ ;k,α

2  δ(ωn ,n + ωk  ,k ), (18.190)

and the differential scattering cross section for polarized photons is with the incident ˆ photon current density per k space volume dj /d 3 k = ck/(2π )3 (18.151), dσn,ζ ;k,α→n ,ζ  ;k  ,α  = =

dn,ζ ;k,α→n ,ζ  ;k  ,α  dj (k) 2 4π 2  Mn ,ζ  ;k  ,α  |n,ζ ;k,α  δ(ωn ,n + ωk  ,k )d 3 k  . (18.191) c

This yields after integration over k  dσn,ζ ;k,α→n ,ζ  ;k  ,α  d

=

2  4π 2 2    M k .    n ,ζ ;k ,α |n,ζ ;k,α   k =k−(ωn ,n /c) c2

(18.192)

Substitution of our results for the scattering matrix element yields the result dσ = d

2    μ0 e2 k  1     δn n δζ ζ  α (k ) ·  α (k) + ωn ,n ωn ,n 4π k m n ,ζ     n , ζ | α  (k  ) · x|n , ζ  n , ζ  | α (k) · x|n, ζ  × hω ¯ − i ¯ n ,n − hck  2 n , ζ  | α (k) · x|n , ζ  n , ζ  | α  (k  ) · x|n, ζ   + (18.193)  hω ¯  − i ¯ n ,n + hck k  =k−(ω  /c)



n ,n

If there are non-vanishing transition matrix elements n , ζ  | α  (k  ) · x|n , ζ   and n , ζ  | α (k) · x|n, ζ  with the properties ωn ,n  ck and ωn ,n  − ck  , or if there are any non-vanishing matrix elements n , ζ  | α (k) · x|n , ζ   and n , ζ  | α  (k  ) · x|n, ζ  with the properties ωn ,n  − ck  and ωn ,n  ck, then the

18.9 Photon Scattering

475

n"

n’

ck n’

ck’ n ck’

n

ck

n"

Fig. 18.3 The left diagram corresponds to absorption of the initial photon with energy h¯ ck before emission of the final photon with energy h¯ ck  in resonantly enhanced scattering. The diagram on the right-hand side corresponds to emission of the final photon with energy hck ¯  before absorption of the initial photon with energy hck ¯

differential scattering cross section will be dominated by the resonantly enhanced contributions from those matrix elements, and we will have ωn ,n ωn ,n  − c2 kk  for the dominant terms. In these cases we can approximate our result (18.193) by the equation      n , ζ | α  (k  ) · x|n , ζ  n , ζ  | α (k) · x|n, ζ  dσ 2 2 3   αS c kk  d ωn ,n − ck − i n ,ζ   2 n , ζ  | α (k) · x|n , ζ  n , ζ  | α  (k  ) · x|n, ζ   + (18.194)  ωn ,n + ck  − i k  =k−(ω  /c) n ,n

This is an equation for photon scattering which was proposed already in 1924 by Kramers and Heisenberg based on the correspondence principle [102].7 However, note that the Kramers-Heisenberg formula (18.194) is only a suitable approximation to the actual cross section (18.193) if the near resonance conditions ωn ,n  ck and ωn ,n  − ck  , or ωn ,n  − ck  and ωn ,n  ck, can be fulfilled, and if there are allowed dipole transitions into the intermediate nearly resonant levels. Diagrams like Fig. 18.3 are sometimes used to illustrate the fact that nearly resonant scattering is only dominated by the contributions of the nearly resonant state to the virtual intermediate electron state, but that scattering does no actually proceed through the nearly resonant state. The virtual intermediate electron state corresponds to the central electron lines in the scattering diagrams (Fig. 18.2). The atom or molecule never really occupies the nearly resonant intermediate state |n , ζ  . Actual transition through the nearly resonant intermediate states would

Kramers-Heisenberg formula (18.194) also follows if a dipole approximation ex · E(t) is used for the photon-matter interaction [105]. Sakurai denotes (18.193) as the Kramers-Heisenberg formula [147].

7 The

476

18 Quantization of the Maxwell Field: Photons

not be described by the scattering formulas (18.193) or (18.194), but by the photon absorption and emission formulas from Sects. 18.7 and 18.6. Let us focus on the left scattering diagrams in Figs. 18.2 and 18.3, i.e. absorption of the incoming photon before emission of the outgoing photon. In an actual transition through a resonant intermediate state |n , ζ  , the initial photon would really be absorbed into that state, and then the state would spontaneously decay through emission of the second photon after an average lifetime n−1  →n . Such an event would be an example of fluorescence, but not of scattering. Nevertheless, the incoming photon is annihilated and the outgoing photon is created both in scattering and fluorescence. The difference is that in scattering, absorption and emission proceeds through a virtual state which has no lifetime of its own, whereas in fluorescence, absorption occurs into a real state which must satisfy the resonance condition for energy conservation, and emission of the final photon occurs with the characteristic time n−1  →n for decay of the intermediate real state into the final state.

Thomson Cross Section The contribution from the first term in (18.193) coincides with the classical Thomson cross section for elastic scattering of light which we will encounter again in Sect. 23.3 when we discuss photon scattering off free electrons. The first term yields for scattering of polarized photons  2 2   &2 dσT  μ0 e2 % μ0 e2   (k ) ·  α (k)  = = cos2 θαα  , α d α→α  4π m 4π m

(18.195)

The resulting cross section for unpolarized light involves a sum over final polarizations and an average over initial polarizations, 1  α  (k  ) ·  α (k) ⊗  α (k) ·  α  (k  ) 2  α,α

=

 1  α  (k  ) · 1 − kˆ ⊗ kˆ ·  α  (k  ) 2  α

=

 1 + cos2 θ 

1 

  tr 1 − kˆ ⊗ kˆ · 1 − kˆ ⊗ kˆ , = 2 2

(18.196)



where kˆ · kˆ = cos θ , i.e. θ is the scattering angle. This yields8

combination re ≡ μ0 e2 /4π me = αS h¯ /me c = αS λC /2π = 2.82 fm (cf. (1.56)) is also denoted as the classical radius of the electron.

8 The

18.9 Photon Scattering

477

dσT = d



μ0 e2 4π m

2

1 + cos2 θ , 2

(18.197)

and 8π σT = 3



μ0 e2 4π m

2 .

(18.198)

The first term in Eq. (18.193) would hypothetically dominate the cross section dσ/d if the photon energy is much larger than all the excitation energies of dipole allowed transitions, i.e. if ck  |ωn ,n | for all n , ζ  |x|n, ζ  = 0. However, there will always be allowed transitions into intermediate continuum states. Therefore the condition ck  |ωn ,n | for all dipole allowed transitions will not be fulfilled and the first term in (18.193) will never dominate light scattering by atoms or molecules.9 However, the Thomson cross section plays an important role in the scattering of light by free electrons, which will be discussed in Sect. 23.3.

Rayleigh Scattering Molecules in a gas or a liquid have many dense lying rotational and vibrational levels, and the condition of dipole allowed resonant excitation of intermediate levels will practically always be fulfilled. The Kramers-Heisenberg formula (18.194) will therefore always be an excellent approximation to (18.193) for molecules in a fluid phase. In particular, the cross section for elastic photon scattering |g; k, α → |g; k  , α   from a ground state |g or a state |g near the ground state will be    dσR  (αS ck 2 )2  d 



n,ζ,ωn,g ck

2  g| α  (k  ) · x|n, ζ n, ζ | α (k) · x|g   . ωn,g − ck − i 

(18.199)

A formula for resonance fluorescence which is equivalent to (18.199) was given for the first time by Viktor Weisskopf in his Ph.D. thesis [176].10 The reasoning with only one kind of resonantly enhanced terms is correct as long as the alternative resonance condition ωn,g  − ck cannot be fulfilled, i.e. as long as the energy Eg of the initial state |g is less than hck ¯ above the ground state energy. This applies e.g. to molecules at room temperature. These molecules will generically occupy states with energies less than 0.1 eV above their ground

9A

loophole in this argument concerns the remote possibility that all the matrix elements n , ζ  |x|n, ζ  with ωn ,n  ck are extremely small. 10 He used a dipole operator H = −ex · A(x, ˙ t) for atom-photon interactions throughout his calculations.

478

18 Quantization of the Maxwell Field: Photons

state energy. Scattering of optical photons by these molecules can be described by Eq. (18.199). We can connect (18.199) to the polarizability properties of the scattering centers by noting that the dynamical polarizability tensor (15.68) for ωmn  ω = ck has exactly the same form as the tensor multiplying the polarization vectors in (18.199). Therefore we can rewrite this equation also in the form 

μ 2

2 dσR  0 4    = ω (k ) · α ·  (k) , α α (g) d α→α  4π

(18.200)

where it is understood that the sum over intermediate levels in (15.68) is dominated by terms which are almost resonant with the frequency ω of the elastically scattered photons. Directional averaging over the orientation of the molecules will lead to an isotropic effective polarization tensor,  α  (k  ) · α (g) ·  α (k) = α(g)  α  (k  ) ·  α (k),

(18.201)

 2

μ dσR  0 α(g) ω4 cos2 θαα  , =  d α→α  4π

(18.202)

and averaging and summation over the polarizations of the incoming and scattered photons (18.196) yields the same angular dependence on the scattering angle as for Thomson scattering (18.197),

μ 2 1 + cos2 θ dσR 0 = α(g) ω4 d 4π 2

(18.203)

and σR =

2 8π μ0 α(g) ω4 . 3 4π

(18.204)

Equations (18.203), (18.204) are quantum mechanical versions of Lord Rayleigh’s ω4 law (Rayleigh 1871, 1899; see also Jackson [88] for a derivation of Rayleigh scattering in classical electrodynamics). It is sometimes stated (but neither in [88] nor in Weisskopf’s thesis) that Rayleigh scattering is a small frequency approximation in the sense that hω ¯ = h¯ ck should be small compared to the internal excitations of the scattering system. This is not true. The quantum mechanical derivation (as well as Jackson’s classical derivation) does not require this assumption. The only assumption that went into our derivation above was resonantly enhanced dipole scattering. Besides, energies of optical photons are not small compared to excitation energies for nitrogen or oxygen molecules. Indeed, the assumption of resonantly enhanced dipole scattering implies that the photon frequency ω = ck should be comparable to the transition frequencies of some dipole allowed transitions.

18.10 Problems

479

18.10 Problems 18.1 We consider a gauge invariant Lagrange density which contains matter fields (x) besides the electromagnetic fields Aμ (x), + L = Lm (, + , ∂ − i(q/h)A, ∂+ + i(q/h) A) − ¯ ¯

1 Fμν F μν . 4μ0

Equation (16.26) yields for the conserved charged current density from phase invariance δ(x) =

i qϕ(x), h¯

i δ+ (x) = − qϕ+ (x) h¯

(18.205)

after division by the irrelevant constant factor ϕ the expression jqμ = −

1 ∂L ∂L 1 δ · − δ+ · ϕ ∂(∂μ ) ϕ ∂(∂μ + )

∂L i ∂L i + q+ · . = − q · h¯ ∂(∂μ ) h¯ ∂(∂μ + )

(18.206)

On the other hand, the current density that appears in Maxwell’s equations ∂μ F μν = − μ0 j ν

(18.207) μ

is j μ = ∂L/∂Aμ . Why are those two current densities the same, jq = j μ ? 18.2 Prove that the vector field (18.23) satisfies ∇ · AJ (x, t) = 0. 18.3 In this problem  ≡ A0 /c is the electric potential. The Weyl gauge condition (or temporal gauge condition)  = 0 is dual to Coulomb gauge in the sense that here we consider  (instead of ∇ · A) as the gauge degree of freedom, and then Coulomb’s law determines ∇ · A (instead of ). ˙ = 18.3a Show that in Weyl gauge the conjugate momentum A = ∂L/∂ A ˙ yields the Hamiltonian density H (18.47) through the standard Lagrangian 0 A expression ˙ 2 − L. ˙ − L = 0 A H = A · A

(18.208)

Why was this clear from Eq. (18.46)? 18.3b Show that Maxwell’s equations in Weyl gauge take the form ∂ 1 ∇ · A(x, t) = − (x, t), ∂t 0

(18.209)

480

18 Quantization of the Maxwell Field: Photons



  t 1 ∂2 1 −  A(x, t) = μ j (x, t) + ∇ dt  (x, t  ). 0 0 c2 ∂t 2

(18.210)

18.3c Start from Eq. (18.210) and use charge conservation to show that the consistency condition    1 t    ∇ · A(x, t) + dt (x, t ) = 0 0

(18.211)

is fulfilled. Why is this condition important for the consistency of the system of Eqs. (18.209), (18.210)? 18.3d Use the decomposition A = ∇ϕ + ∇ × a ≡ A + A⊥ ,

E = −

∂ A , ∂t

E⊥ = −

∂ A⊥ , ∂t

(18.212)

to derive the Hamiltonian (18.100) in Weyl gauge. 18.3e Perform the gauge transformation for the electromagnetic potentials from Weyl gauge to Coulomb gauge explicitly. Hint for 3d and 3e: Calculate ϕ(x, t) as a functional of the charge density (x, t). For the gauge transformation use (18.12). 18.4 Consistency of canonical quantization in Coulomb gauge 18.4a Suppose that  F [A](t) =

d 3 x F(A(x, t), ∂μ A(x, t))

(18.213)

is a functional of the vector potential A(x, t) and its first order derivatives. Define the functional derivative with respect to the vector potential by δF [A](t) ∂F ∂F = (x, t) − ∂i (x, t). δA(x, t) ∂A ∂(∂i A)

(18.214)

Show that the commutation relations (18.51), (18.53) imply i δF [A](t) ˙ −∇⊗∇· 0 [F [A](t), A(x, t)] = − δA(x, t) h¯   δF [A](t) . ≡− δA(x, t) ⊥



d 3 x  G(x − x  )

δF [A](t) δA(x  , t) (18.215)

18.4b Show that Eq. (18.215) yields Eq. (18.17) from the Heisenberg equation

18.10 Problems

481

i ¨ ˙ A(x, t) = [H, A(x, t)] h¯

(18.216)

if we use the Hamiltonian (18.100), i.e. Coulomb gauge fully complies with canonical quantization if we use the commutation relations (18.51), (18.53). 18.5 Necessity of second quantization to describe photon emission 18.5a Show that Eq. (18.17) cannot generate radiation from excited hydrogen states in a semi-classical way. Hint: Show that A(x, t) = 0 solves Eq. (18.17) if an energy eigenstate ψn,,0 (x, t) is substituted into the current density on the right-hand side. Furthermore, substitution of a state ψn,,m =0 (x, t) yields a static potential A(x). 18.5b Substituting a superposition of hydrogen eigenstates C (x, t) =



Cn,,m ψn,,m (x, t)

(18.217)

n,,m

with contributions from at least two different energy levels into the right-hand side of Eq. (18.17) will yield a time-dependent vector potential A(x, t). Show that this will also not lead to emission of radiation from the hydrogen atom. Hint: How will A(x, t) scale with distance? How will the corresponding Poynting vector scale with distance? 18.6 The Hamiltonian density for an electron in the presence of a fixed charge Ze at X = 0 is in Coulomb gauge ∇ · A = 0 given by H=

   h¯ 2 αS hc e e ¯ ∇ψ + − i ψ + A · ∇ψ + i Aψ − Zψ + ψ 2m |x| h¯ h¯ 0 ˙ 2 1 + A + (18.218) (∇ × A)2 . 2 2μ0

This is the restriction of the corresponding Coulomb gauge Hamiltonian to the single-electron sector since we neglected the electron-electron Coulomb interaction. The corresponding energy current density j H satisfies ∂ H + ∇ · j H = 0. ∂t

(18.219)

Show that the energy current density is     h¯ 2 ∂ψ + e h¯ 2 e + ∂ψ + ∇ψ − i ψ A ∇ψ + i Aψ − jH = − 2m ∂t 2m ∂t h¯ h¯ −

∂A ∂ 1 ∂A × (∇ × A) − e G ◦ |ψ|2 , μ0 ∂t ∂t ∂t

(18.220)

482

18 Quantization of the Maxwell Field: Photons

where the convolution term is    ∂ e ∂ |ψ|2 (x  , t) eG ◦ |ψ|2 (x, t) = . d 3x ∂t 4π ∂t |x − x  |

(18.221)

18.7 When studying the hydrogen atom, one might wonder whether we should have included a vector potential in the Schrödinger equation for the hydrogen atom, and whether we should also take into account Ampère’s law (18.17). Why can we neglect the vector potential and Ampère’s law in the study of the hydrogen atom? Hint: How do the energy eigenstates ψn,,m (x, t) of the hydrogen atom scale in leading order with the charge e? How then would the vector potential scale in leading order with e? 18.8 We can solve the Coulomb equation (18.6) for the scalar potential  also without invoking any particular gauge. How does this generalize Eqs. (18.16) and (18.17)? Show that taking the divergence of the generalization of Eq. (18.17) yields a trivially fulfilled equation. 18.9 Conservation laws for electromagnetic fields from Lorentz invariance 18.9a The action (18.43) of electromagnetic fields is invariant under Lorentz transformations  μ = − δx μ = − ϕ μν xν ,

ϕ μν = − ϕ νμ ,

(18.222)

δAμ (x) = Aμ (x  ) − Aμ (x) = ϕμν Aν (x).

(18.223)

Use a procedure similar to the derivation of the energy-momentum tensor (18.46) to derive the densities and currents Mαβ μ =

& 1% xα Tβ μ − xβ Tα μ c

(18.224)

of the corresponding conserved charges  Mαβ =

d 3 x Mαβ 0 .

(18.225)

Hint: You have to add the improvement term ∂ν (xβ Aα F μν − xα Aβ F μν )/μ0 to j μ from Eq. (16.26) to get the gauge invariant expression (18.224) for the angular momentum densities and currents. 18.9b The angular momentum of the electromagnetic fields is M=

1 ei ij k Mj k = 2



 d 3x x × P =

d 3 x 0 x × (E × B),

(18.226)

18.10 Problems

483

but what is the meaning of the conserved quantities M 0i ? We define an energy weighted location of the electromagnetic fields, x =

1 E

 d 3 x xH,

(18.227)

where H is the energy density (18.47) of the electromagnetic fields. Show that the conservation of M 0i implies a conservation law for “center of energy” motion for the freely evolving electromagnetic fields, x(t) = x(0) +

c2 P t. E

(18.228)

18.10 The Coulomb gauge condition (18.13) eliminated the electric potential (x, t) as a dynamical degree of freedom through Coulomb’s law (18.16). However, in quantum optics this only means that (x, t) becomes the composite operator (18.97), where the matter fields satisfy canonical commutation or anti-commutation relations (recall that the products over Schrödinger fields in (18.97) include an implicit summation over spin variables, where applicable). Could this composite operator structure then induce commutation relations which could be interpreted as a commutator between (x, t) and a corresponding conjugate momentum operator? ˙  , t)]. Calculate the commutator [(x, t), (x 18.11 The quantum optics Hamiltonian (18.92) contains a particular interaction term   qa h¯ ↔ + HI = i A · a,σ ∇ a,σ = − A · j a. (18.229) 2ma a,σ a In the interaction picture, the field operators A(x, t) and a,σ (x, t) are free field operators with their corresponding Fourier expansions in k space. Express the interaction Hamiltonian  HI (t) = d 3 x HI (x, t) (18.230) as an integral in k space. You find two terms involving products of annihilation and creation operators. What do the two terms do? 18.12 A helium-neon laser produces a light wave with a central wavelength of 632.8 nm and a power of 5 mW. The aperture of the laser is 1 mm2 . Suppose the electric component is a sine oscillation |E(x, t)| ∝ sin(k · x − ckt) and is polarized in x direction. We also assume that the frequency profile is Gaussian with a relative width f/f = 3.16 × 10−6 . Which photon state describes this light wave? How many photons does the electromagnetic wave contain?

484

18 Quantization of the Maxwell Field: Photons

You need to use the Gaussian profile to calculate the number of photons, but you can use a monochromatic approximation to calculate the energy flux. 18.13 Show that the energy and momentum uncertainties for the coherent state (18.67) satisfy  E = c P = 2

2

2

d 3k

2

h¯ 2 c2 k 2 |ζα (k)|2 .

(18.231)

α=1

18.14 Show that the state 

d 3 x n,,m (x)ψσ+ (x)aα+ (k)|0

(18.232)

H0 |n, , m , σ ; k, α = (En, + hck)|n, , m , σ ; k, α, ¯

(18.233)

|n, , m , σ ; k, α = satisfies

where 



h¯ 2 H0 = d x ∇ψσ+ · ∇ψσ + ψσ+ V ψσ 2m σ σ    0 ˙ 2 (∇ × A)2 + d 3x A + . 2 2μ0



3

(18.234)

You have to use that the atomic orbital satisfies −

h¯ 2 n,,m (x) + V (x)n,,m (x) = En, n,,m (x). 2m

(18.235)

It is also useful to keep the x representation for the electronic part of H0 , but to use the k representation for the photon contributions in H0 . 18.15 Calculate the emission rate for unpolarized photons from the 2p state to the ground state of hydrogen in first order and dipole approximation. Which estimate do you get from this for the lifetime of 2p states? Which estimate do you get from this for the radiated power from decay of 2p states? 18.16 In a photon counting detector with sensitivity proportional to spectral widths |dω| = c|dk| = 2π c|dλ|/λ2 , the absorption cross section (18.154) (or rather (18.156) with a natural line width γ = τ −1 ) describes the absorption spectrum measured by the detector in the region of the n ,  → n,  transition, if this transition is sufficiently well separated from other transitions. On the other hand, a detector with sensitivity proportional to |dλ| would measure the line shape with an

18.10 Problems

485

extra factor 2π/k 2 . A detector with logarithmic dependence of the sensitivity would not depend on frequency or wavelength representation of the absorption spectrum. Calculate the integrated photon absorption cross section 

∞

G1s→2p = 0

dk σ1,0→2,1 (k) k

(18.236)

due to the transition from 1s to 2p states in hydrogen. You can use the monochromatic approximation (18.154). 18.17 Semi-classical approximations for single-photon wave functions, length gauge, and their limitations 18.17a Show that the first order scattering matrix elements (18.116) and (18.145) for emission and absorption can also be gotten in a semi-classical approximation from a perturbation operator V (t) = − qx · E(x, t)

(18.237)

with E(x, t) corresponding to a single photon electric field ¯ 0 c3 k ˙ α(+) (x, t) = − i hμ E α(+) (x, t) = − A  α (k) exp[− i(k · x − ckt)] 16π 3 for emission, and to ¯ 0 c3 k ˙ α(−) (x, t) = i hμ E α(−) (x, t) = − A  α (k) exp[i(k · x − ckt)] 16π 3 for absorption. This provides another example for the necessity of photon quantization: We cannot describe photons through the same classical electromagnetic potentials for both emission and absorption, Aα(+) (x, t) = Aα(−) (x, t) (and we cannot generate them from the classical Maxwell equations anyway, as demonstrated in Problem 18.5). 18.17b If we would use the same substitution of semi-classical perturbation operators V (t) from (18.117) to (18.237) for the calculation of scattering in dipole approximation exp(± ik · x)  1, we would find the Kramers-Heisenberg formula (18.194) from (18.237), while (18.117) yields the correct result (18.193). Why does the substitution (18.117) → (18.237) not work beyond first order perturbation theory, except in the case of resonances? Hint: The justification for the transition from the velocity form to the length form of matrix elements is based on

486

18 Quantization of the Maxwell Field: Photons

i p p = [H, x] ⇒ f | |i = iωf i f |x|i. m m h¯

(18.238)

18.17c On the other hand, we could use a gauge function ϕ(x, t) = − x · A(t) if dipole approximation A(x, t)  A(t) is applicable, which is assumed in (18.193). Substitution into Eqs. (15.14) and (15.16) yields   (x, t) = exp[− iqx · A(t)/h](x, t), ¯ A (x, t) = 0,

 (x, t) = (x, t) + x ·

(18.239) d A(t). dt

The Schrödinger equation for the new wave function then has the form   h¯ 2 ∂  ih¯  (x, t) = −  + q(x, t) − qx · E ⊥ (t)   (x, t), ∂t 2m

(18.240)

i.e. now we seem to have accomplished the substitution (18.117) → (18.237) through a gauge transformation! However, quantum optics is gauge independent, and therefore one might expect that (18.194) should be equivalent to (18.193) beyond the approximations described in Sect. 18.9 for the Kramers-Heisenberg relation. Why is this expectation incorrect? Hint: If you pay close attention, you may recognize that we actually did not perform a complete gauge transformation. We only employed the field redefinition (18.239) of the Schrödinger field to the dressed Schrödinger field   (x, t), but we expressed the gauge transformed potentials in term of the original electromagnetic potentials. Furthermore, and more important, the field redefinition included arbitrary high powers of the coupling constant q, i.e. what is second order perturbation theory in the original Schrödinger equation, which yields (18.193), is not the same as apparent second order perturbation theory for (18.240), which yields (18.194). Remark The transition to the Schrödinger equation (18.240) through the field redefinition (18.239) is commonly denoted as length gauge. The original Schrödinger equation with the interaction operator (18.117) is then denoted as the equation in velocity gauge. 18.18 Show that the transition rate (18.129) can formally be derived by incorrectly assuming a Golden Rule for transition between the discrete states |n, , m  → |n ,  , m  in a semi-classical approximation (18.237) for the monochromatic perturbation V (t), if we use the density of final states 3 (E)dE = d 3 k = dk 2 dk = dE 2 dE/(hc) ¯ .

(18.241)

This works because (18.241) is the density of continuous final states of the emitted photon in the infinite volume limit, but we would have missed that important piece of information if we would just have naively insisted on using the Golden Rule

18.10 Problems

487

for calculating the transition rate between states |n, , m  → |n ,  , m  due to the monochromatic perturbation V (t). Instead, we would have tried to make sense of the energy preserving δ function by invoking a final electron density of states (En ), e.g. by using some finite energy width of the final electron state. Any such guess would certainly not have produced the correct factor E 2 , and we would also have missed the factor d because the final electron state |n ,  , m  uses angular momentum quantum numbers instead of angles. 18.19 Ultraviolet photons with an energy Eγ = 10.15 eV are nearly resonant with the n = 1 → n = 2 transition in hydrogen. Use both the result (18.193) and the Kramers-Heisenberg formula (18.194) to estimate the differential scattering cross section for a photon scattering angle of π/2 if the incident photons are polarized in z direction and move in x direction. Assume that the scattered photons move in y direction with polarization ez cos α + ex sin α. 18.20 Express the photon absorption cross sections from Sect. 18.7 using the velocity form (instead of the length form) for the matrix elements. 18.21 Electron capture revisited Section 18.6 discussed photon emission due to transitions between discrete electron states. However, when a free electron is captured into an atomic or molecular state, we are dealing with a continuous initial electron state and a discrete final electron state, as in our first discussion of capture cross sections in Sect. 13.5. Here we will revisit that problem, but in the meantime we have learned three more things: We know the relevant perturbation operator HI (18.109), we know that the final state will also contain a photon,11 and we know that we have to apply second quantization to take into account the photon emission during the electron capture. Derive the electron capture cross section using these three additional pieces of information. Solution The pertinent scattering matrix element for emission of a photon with momentum hq ¯ and polarization  α (q) from the incoming Mott-Gordon state (11.90) is  i ∞ eh¯ Sf i = dt exp [i(ωnk + cq)t] i 2m h¯ −∞    ↔ 3 + ×n, , m ; q, α| d x A(x) · ψ (x) ∇ ψ(x) |kMG . (18.242) Here we already omitted the spin labels since the photon perturbation operator HI preserves spin projections. We used m for the electron mass instead of the reduced mass μ. The scattering matrix element after evaluation of the field operators,

11 We

assume that the capture event under investigation primarily appears due to photon emission, but not due to interactions with spectator particles.

488

18 Quantization of the Maxwell Field: Photons

Sf i = 2π δ(ωnk + cq)

e imh¯

hμ ¯ 0c 16π 3 q

×n, , m | α (q) · p exp(− iq · x)|kMG ,

(18.243)

is a first quantized scattering matrix element with perturbation (18.117) (charge − e) and vector potential (18.118) with photon momentum q. Using again dipole approximation q · x % 1 and the transformation (18.126) into length form, we find  Sf i = δ(ωnk + cq)

αS ωnk n, , m | α (q) · x|kMG . q

(18.244)

The differential emission rate for polarized photons from electron capture is then d (α) (q)k→n,,m = d 3 q = d 3q

d 3 k |Sf i |2 8π 3 T

d 3k αS c δ(ωkn − cq) ωkn |n, , m | α (q) · x|kMG |2 , 3 2π 8π

(18.245)

and integration over the photon momentum yields (α)

dk→n,,m d

=

d 3 k αS 3 ω |n, , m | α (q) · x|kMG |2 . 8π 3 2π c2 kn

(18.246)

There is a volume factor d 3 k/8π 3 for the incoming electron because the incoming asymptotic state was normalized without the factor (2π )−3/2 , as usual in the calculation of cross sections. This will cancel in the usual way when we normalize with the correctly normalized incoming current density dj (k)/d 3 k = h¯ k/8π 3 m, to calculate the capture cross section, (d 3 k/8π 3 )/dj (k) = m/hk. ¯ The incoming state is independent of ϕ, and therefore the final state must have m ∈ {0, ±1}. The reasoning between equations (18.129) and (18.139) for integration over the solid angle for photon emission and summation over photon polarizations proceeds in the same way. This leads to an unpolarized differential photon emission rate per momentum volume of the captured electron, dk→n,,m =

d 3 k 4αS 3 ω |n, , m |x|kMG |2 . 8π 3 3c2 kn

(18.247)

This is also the differential electron capture rate, and normalization with the incoming current density yields the capture cross section due to spontaneous photon emission, σk→n,,m =

4mαS 3 dk→n,,m ωkn |n, , m |x|kMG |2 . = dj (k) 3hc ¯ 2k

(18.248)

18.10 Problems

489

If we compare this with the result (13.110), the major difference is that integration over the photon momentum eliminated the δ-function. We could not do this in (13.110) because we did not know back then how to quantize photons, and therefore we could not associate the monochromatic perturbation W exp(iωt) with spontaneous photon emission and a corresponding momentum integration. Equation (18.247) describes the differential photon emission rate per k-space volume d 3 k for a small volume d 3 k at the location k = kez in k-space. If we would want to integrate this over a finite k-space volume, we would first need to generalize the Mott-Gordon state |kMG to arbitrary incoming electron momenta using the state |k+ (11.91) (which comes with k-space measure d 3 k). However, for the calculation of the capture cross section, we do not integrate over d 3 k, but divide dk→n,,m /d 3 k by the differential current density dj (k)/d 3 k. The result for fixed magnetic quantum number m of the final state still depends on the direction of the incoming electron, but the average over m would be independent of direction. Calculations of radiative capture cross sections for electron-proton recombination into arbitrary hydrogen shells were performed in parabolic coordinates by Oppenheimer [127] and by Bethe and Salpeter [12]. Calculations in polar coordinates had been performed by Wessel, Stückelberg and Morse, and Stobbe [165, 167, 178]. All these authors had noticed that the electron capture cross sections for ions from radiative recombination were much too small to explain the experimental values, and it was eventually recognized that collisional relaxation due to interactions with spectator particles dominated the observed recombination rates. Therefore modern calculations of electron-ion recombination rates focus on collisional relaxation, which means that the relevant perturbation operators V are not determined by photon emission but by Coulomb interactions in a plasma, and the spectator particles also have to be taken into account in the initial and final states. Electron-ion recombination rates are particularly important for plasma physics and astrophysics. 18.22 Repeat the discussion of the difference between fluorescence and scattering after Fig. 18.3 in the case that corresponds to the right scattering diagrams in Figs. 18.2 and 18.3, i.e. emission of the final photon before absorption of the initial photon. 18.23 Free electrons cannot emit a single photon due to energy-momentum conservation. However, they can emit photons when simultaneously interacting with another charged particle, e.g. through the Coulomb interaction or through virtual photon exchange. The emitted radiation helps electrons to lose energy when interacting with matter, thus slowing them down. This radiation is therefore denoted as bremsstrahlung12 (German for “braking radiation”). Show that the scattering matrix element for bremsstrahlung from low energy electrons moving through the Coulomb potential of a nucleus of charge Ze located in the point X is

12 Please

see [77] for a review of the early work on bremsstrahlung.

490

18 Quantization of the Maxwell Field: Photons

Sf i =

h¯ c exp[i(k − k  − q) · X] 3/2  Zα δ(c|q| + ω(k ) − ω(k)) 2π 2 im S (k − k  − q)2 |q|1/2   k k × α (q) · + , (18.249) ω(k) − ω(k  + q) ω(k  ) − ω(k − q)

where  α (q) is the polarization of the emitted photon of momentum h¯ q. The  incoming and outgoing electron momenta are hk ¯ and h¯ k , respectively, and ω(k) = 2 hk ¯ /2m. Use the result (18.249) to calculate the differential scattering cross section for bremsstrahlung emission from low energy electrons.

Chapter 19

Epistemic and Ontic Quantum States

Equipped with the tools of time-dependent perturbation theory and second quantization for the Schrödinger and Maxwell fields, we are now at a stage where we can start to discuss interpretations of quantum mechanics. Interpretations of quantum mechanics are primarily concerned with two related questions: - What happens during an observation of a quantum system? - What (if anything) does quantum mechanics tell us about the world outside of our minds? The philosophical study of the world as an objective existence beyond and independent from our minds is the field of ontology. The second question can therefore also be addressed as the question for a quantum ontology. Quantum mechanics textbooks are sometimes criticized for not addressing these questions, but there are good and defendable reasons for this: Textbooks should generically focus on established and widely accepted knowledge, while eventually adding some refinement here or there, instead of dwelling on unsettled questions. This chapter, on the other hand, deals with unsettled questions. The debate about the interpretation of quantum mechanics has remained an active and controversial field of scientific inquiry on the intersection of philosophy and physics ever since the 1920s, and will likely remain so for many more years to come. Another (likely legitimate) reason is that quantum mechanics is one of the most successful theories ever conceived, and fortunately it can be used perfectly well without ever worrying about quantum ontology. What merit could there possibly be in distracting and maybe even side-tracking our students with the notoriously difficult interpretational questions of quantum mechanics, when all the successful and proven applications of the theory happily exist without any generally agreed upon interpretation? Finally, since the community of philosophers and physicists actively studying these problems has not settled on one widely accepted solution, a thorough © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 R. Dick, Advanced Quantum Mechanics, Graduate Texts in Physics, https://doi.org/10.1007/978-3-030-57870-1_19

491

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19 Epistemic and Ontic Quantum States

discussion of the field requires full research monographs, whereas a single chapter has to be limited to basic discussions and eventually a presentation of the author’s preferred views on the topic. Therefore I should also caution the reader that many assertions in this chapter are debatable and, depending on their own views, could provoke negative responses from other researchers of quantum foundations. The reader can rest assured that all the other chapters in this book, just like the chapters in many other standard quantum mechanics textbooks, do not depend on any particular interpretation of the formalism. Readers who would like to study the subject in depth, and would like to get a balanced view, are well advised to read many of the pertinent research monographs to gain insight from many different perspectives. A limited number of textbook discussion of quantum ontology from the philosophy perspective include [106, 116, 146]. On the physics side, Ballentine presents the view of quantum mechanics as a many-particle theory without single-particle interpretation [6], while Holland discusses and favors pilot-wave theories [82]. Griffiths discusses a decoherent histories approach to quantum mechanics through wave packet motion [67]. Schlosshauer provides an introduction to the emergence of preferred bases of quantum states and suppression of quantum interferences (i.e. decoherence) through entanglement with the environment [148]. Decoherence affords a better understanding of the quantum-to-classical transition [90, 181, 184, 185]. Like traditional Copenhagen interpretations (to be defined below), the decoherence approach usually presumes a realist or ontic interpretation of quantum states as records of complete, objective information about a system.1 On the other hand, many of the more problematic aspects of ontological interpretations of quantum states can be avoided or resolved in epistemic interpretations of quantum states as records of collective or individual knowledge of observers performing an experiment (see e.g. [26, 58, 164]). The most widely, but not universally accepted interpretation of quantum mechanics is based on Born’s rule for probabilities (which is universally accepted as a basic property of quantum mechanics), combined with the assertion that the state of a quantum system after observation of the observable A is in an eigenstate of A. Specifically, if A|ψi  = ai |ψi , measurement of A for a system in a state |φ =  c |ψ  will produce the value ai with probability |ci |2 (this is Born’s rule), j j j every individual measurement will yield some eigenvalue ai , and every subsequent observation on the same system will yield the same eigenvalue ai if [A, H ] = 0 (with appropriate modifications if [A, H ] = 0, e.g. if we are observing a particle

1 If

a quantum state would only be a subjective statement of an observer’s knowledge about a system, entanglement with the environment should not impact the outcome of quantum measurements, except in cases where entanglement with the environment is deliberate and controlled. However, as a tool to achieve classical behavior, environmental decoherence can just as well be implemented in epistemic interpretations of quantum mechanics, because “classicality” and awareness of the ubiquity of environmental effects on a classical object is part of the knowledge of an observer of a classical system.

19 Epistemic and Ontic Quantum States

493

trajectory A = x). The last part is the projection postulate of the Copenhagen interpretation [172], which asserts that observation of A reduces the quantum state to an eigenstate of A. Usually, this is read as implying ontological collapse of an ontic quantum state in the Copenhagen interpretation, although Heisenberg’s formulations in Z. Phys. 43, 172 (1927) are ambiguous in that regard. Other interpretations include what I would like to call 1. Epistemic collapse: The quantum state changes not as a property of the observed system per se, but as an information update for the observers who performed the measurement. It is still a collapse in the sense that the observers would always observe the eigenvalue ai in repetitions of the measurement, if they have observed it the first time. Or 2. Epistemic reduction: The quantum state changes not as a property of the observed system per se, but as an information update for the observers who performed the measurement. Repetition of the measurement will reproduce the eigenvalue ai with high probability, but not with absolute certainty. Indeed, the central role of observation in the Copenhagen interpretation would hint at an epistemic interpretation of quantum states rather than an ontological interpretation. The puzzling aspects of the measurement process can be summarized as follows: Q1. If reduction of the quantum state during the  measurement process is indeed an ontological collapse of the state from |φ = j cj |ψj  into an eigenstate |ψi , how is that collapse accomplished?  Q2. On the other hand, if the collapse |φ = j cj |ψj  → |ψi  is a subjective epistemic collapse for the observers performing the measurement, why should repeated observation by the same observers always return ai without the system presumably “knowing” that the same observers are looking? And why should the system reproduce ai in a measurement by a completely different set of observers who did not communicate with the first team, and therefore did not experience the same subjective collapse of the system’s quantum state? The latter property would apparently require some objectivity of the subjective collapse, which would seem to point back to collapse of the quantum state as an ontological property of the system that is described by the quantum state. Q3. Finally, if the reduction to |ψi  is only an updated best guess for the state of the system (epistemic reduction), why does every individual observation yield  2 a with an eigenvalue ai , instead of the expectation value φ|A|φ = |c | i i i   uncertainty A2 = i |ci |2 ai2 −( i |ci |2 ai )2 ? Stated differently: Why would a projection property for outcomes of observations apply? An in-depth discussion of modern developments of quantum foundations and quantum ontology is beyond the scope of this book. Rather, I will point out how second quantization resolved one of the early foundational puzzles of quantum

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19 Epistemic and Ontic Quantum States

mechanics, and then discuss what archetypal quantum observations may tell us about the quest for a quantum ontology. To set the stage, we will start with a brief review of spin projection measurements in Sect. 19.1, followed by a discussion of the Einstein-Podolsky-Rosen objection to the completeness of quantum mechanics in Sect. 19.2. We will then discuss in Sect. 19.3 how the second quantized scattering matrix solves two basic problems in quantum mechanics: reconciliation of continuous evolution of quantum states on the one hand with quantum jumps on the other hand, and the problem of spontaneous emission. Sections 19.4 and 19.5 will finally lead us to a proposal that there are ontic states of quantum systems which evolve continuously between spontaneous quantum jumps. The epistemic states which we calculate from the integration of quantum mechanical wave equations in the second quantized formalism allow to both describe the continuous evolution of free quantum states and the rates of quantum jumps due to interactions. Quantum jumps deteriorate the agreement between the state of a quantum system and the continuously evolving epistemic state which we use to describe evolution of the system as best as we can from integration of the wave equations. Observation allows us to realign our epistemic state with the state of the system. We will therefore find both an ontic and an epistemic layer of quantum states in Fock space.

19.1 Stern-Gerlach Experiments Suppose we consider spin-1/2 particles moving in y direction. In the standard Pauli basis (8.13) of spin matrices, spin-polarized states in ±z direction are | ↑ =

  1 , 0

| ↓ =

  0 . 1

(19.1)

The states polarized in ±x direction are √ |± = (| ↑ ± | ↓) / 2,

(19.2)

and the states with forward or backward polarization (±y direction) are √ | → = (| ↑ + i | ↓) / 2,

√ | ← = (| ↑ − i | ↓) / 2.

(19.3)

Stern and Gerlach had observed that a beam of silver atoms traversing an inhomogeneous magnetic field B(x) with primary orientation in z direction, splits into two components corresponding to spin projections sz = ±1/2. This can be understood as a consequence of the force term (cf. Eq. (8.4)) & % F (x) = ∇ μs · B(x) ,

μs = − g s

e s, 2m

(19.4)

19.1 Stern-Gerlach Experiments

495

and the fact that silver atoms have total spin s = 1/2. However, the observation seems to imply that every silver atom responds to the magnetic field in a way that corresponds to the silver atom either being in a state | ↑ or | ↓. Furthermore, if one of the two beams is then sent through another inhomogeneous magnetic field B 2 (x) with primary orientation in x direction, the beam splits again equally into two beams corresponding to polarization states |+ an |−. This is in agreement with the expectation that the beam entering the second field should √ √ only contain polarization states | ↑ = (|+ + |−)/ 2 or | ↓ = (|+ − |−)/ 2, which according to the Born rule implies 50% probability each for the silver atoms in the beam to be right or left polarized. To observe the splitting of the beams after the first magnetic field or after the second magnetic field, the silver atoms can be caught on a glass pane put into the beams. However, the Schrödinger equation only tells us that the two orbital wave function components ↑ (x, t) and ↓ (x, t) of a silver atom entering the first magnetic field should leave that magnetic field in different directions, and the residual wave function components e.g. in the ↑ (x, t) beam, √ ↑,+ (x, t) and ↑,− (x, t), which satisfy ↑,+ (x, t) = ↑,− (x, t) = ↑ (x, t)/ 2 before entering the second magnetic field, should also leave it in different directions. Nowhere does the Schrödinger equation tell us that the silver atom at any time should be forced into a choice between up or down polarization, or between right and left polarization. The projection postulate is needed to explain this behavior, and therefore it is an additional important piece of information in quantum mechanics, in addition to the wave equations and second quantization. We can try to reduce the collapse of the silver atom’s spin projection to the atom’s collapse in location, because the Stern-Gerlach experiment only tells us that projection of the atom after the first magnetic field forces it to make a choice between upper or lower part of the glass pane, and projection of the residual beams after the second magnetic field forces the atom to make a choice between right and left part of the glass pane, or to escape undetected in the complementary beam emanating from the first field. It is just another confirmation of the Born interpretation of |(x, t)|2 = |↑ (x, t)|2 + |↓ (x, t)|2 after the first polarization experiment, or |(x, t)|2 = |↑,+ (x, t)|2 + |↑,− (x, t)|2 + |↓ (x, t)|2 after the second experiment, as a probability density for particle location. The electron biprism experiments discussed in Sect. 1.7 told us that we cannot assign a location to a particle inside of its wave function, until it is located through interaction with a glass pane or a scintillation screen or another particle detector. The Schrödinger equation only tells us how the multi-component wave function (x, t) of a particle with spin evolves, but it does not imply that a particle is an extended object, nor does it imply that a particle must at all times have specific values for its observables, neither for momentum nor location nor spin orientation. It is only after observation that we can assign location and spin polarization to a particle, until those pieces of information get wiped out again if the Hamiltonian of the particle does not commute with x or with the particular spin operator that was measured. The added feature in the spin projection experiments is particle spin, and after polarization experiments with sufficient separation of upper and lower (or left and

496

19 Epistemic and Ontic Quantum States

right) wave function components, particle location is correlated with spin projection. Therefore observation of particle location also implies observation of spin projection in the Stern-Gerlach experiments. Note that although we can perform the experiment in ultra-high vacuum conditions and with electromagnetic shielding, there are necessarily interactions preceding the observation, viz. the interactions with the magnetic fields. Those interactions are described by the minimally coupled Schrödinger equation and spatially separate the different spin components of the wave function. However, they do not enforce projection on spin eigenstates parallel or anti-parallel to the magnetic fields. The particle is still described by the full two-component wave function, no matter how far apart the two components are, and we still cannot assign it to any particular location within any of the two components until the localization on the glass pane or scintillation screen has been performed. Localization projects onto a particular spin component only because of the correlation between location and spin projection, if the two wave function components are spatially separated in magnetic fields. We could therefore try to argue that spin projection is only a consequence of location projection in the Stern-Gerlach experiments, and if we could abandon the projection postulate for location measurements by resolving wave function collapse into a location dynamically, we could hope to thereby also eliminate the projection postulate for spin measurements and for any other measurements of particle properties which require observations of positions. However, the Schrödinger equation per se does not imply dynamical wave function collapse, and the program of extending the Schrödinger equation in “dynamical collapse theories” was not met with success so far. Another feature that is easily understood with the spin experiments is the preparation of states. While quantum mechanics tells us that we cannot think of a particle as having values assigned to its observables at all times, we can experimentally select states which assign only one particular value to a particle observable, e.g. spin up states by separating particles in a spin polarizer. We cannot assign a location to a particle within its own wave function, and we cannot generate sharply localized wave functions with a finite amount of energy. However, we can in principle assign a sharp value to a spin projection of a particle, e.g. sz = h/2, at the ¯ expense of maximal uncertainty in the orthogonal projections, sx = sy = h¯ /2. Indeed, the finite dimension of the space of spin states implies that, if it would not be for position or momentum, we could assign some sharp spin projection at any time t to every particle. This holds because because an arbitrary spin state2 |ˆs+ (t) = cos[ϑ(t)/2]| ↑ + exp[iϕ(t)] sin[ϑ(t)/2]| ↓ % & = cos[ϑ(t)/2]| ↑ + i sin[ϑ(t)/2] sin[ϕ(t)]σ1 − cos[ϕ(t)]σ2 | ↑

2 The

rotation in Eq. (19.5) corresponds to a rotation by the angle ϑ(t) around the axis (sin ϕ(t), − cos ϕ(t), 0) in the passive interpretation, or around the axis (− sin ϕ(t), cos ϕ(t), 0) in the active interpretation of transformations.

19.2 Non-locality from Entanglement?

497



 % & i = exp ϑ(t) sin[ϕ(t)]σ1 − cos[ϕ(t)]σ2 | ↑ 2

(19.5)

corresponds to sharp spin projection h¯ /2 onto the spin vector % & sˆ (t) = sin ϑ(t) · cos ϕ(t), sin ϑ(t) · sin ϕ(t), cos ϑ(t) , sˆ (t) · σ |ˆs+ (t) = |ˆs+ (t).

(19.6) (19.7)

However, the inevitable presence of a variable for particle position (or momentum) implies that we cannot assign some sharp spin polarization to every particle, because the general wave packet x|(t) = x|↑ (t)| ↑ + x|↓ (t)| ↓

(19.8)

implies that generically we can only think of the particle also as a wave packet in spin space, with local polarization vectors sˆ (x, t) attached to the wave packet. However, contrary to the impossibility of infinitely sharp localization of particle position, we can have particles with sharp spin projection values corresponding to wave packets % & x|(t) = x|(t) cos[ϑ(t)/2]| ↑ + exp[iϕ(t)] sin[ϑ(t)/2]| ↓ .

(19.9)

These special wave packets have the same polarization vector sˆ (t) (19.6) attached to every position at time t, sˆ (t) · σ · x|(t) = x|(t).

(19.10)

19.2 Non-locality from Entanglement? A puzzling property of quantum mechanics which was first discussed by Einstein, Podolsky and Rosen (EPR) on the basis of location and momentum of particles [48],3 and then rephrased by Bohm [15] using spin states, is the apparent non-locality of quantum state reduction during observations of components of entangled twoparticle systems. Suppose a spin-0 state decays into two spin-1/2 fermions A and B. The resulting 2-particle state must be a singlet state, 1 |AB (t) = √ 2(1 + δAB ) 3 Reid



 d 3x

d 3 x  AB (x, x  , t)

et al. [143] provide a discussion of EPR experiments.

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19 Epistemic and Ontic Quantum States

  + + + + × ψA↑ (x)ψB↓ (x  ) − ψA↓ (x)ψB↑ (x  ) |0.

(19.11)

The factor 1 + δAB takes into account the additional normalization requirement if the two particles are identical. In that case the singlet state is with (x, x  , t) = (x  , x, t),   3 |(t) = d x d 3 x  (x, x  , t)ψ↑+ (x)ψ↓+ (x  )|0 1 = 2 =

1 2



 3

d x 

 d 3x

d 3 x  [(x, x  , t) + (x  , x, t)]ψ↑+ (x)ψ↓+ (x  )|0 d 3 x  (x, x  , t)

  × ψ↑+ (x)ψ↓+ (x  ) − ψ↓+ (x)ψ↑+ (x  ) |0.

(19.12)

Please note that the state (19.11) does not imply that particle A and particle B have opposite polarizations along a fixed quantization axis like the z axis, but it states that particle A and particle B have opposite polarizations along any quantization axis. This follows from the identities4 | ↑| ↓ − | ↓| ↑ = |−|+ − |+|− = i (| →| ← − | ←| →),

(19.13)

which also hold for the corresponding products of creation operators. This implies that we can write the state (19.11) also in terms of polarizations along the x axis,   1 3 |AB (t) = √ d x d 3 x  AB (x, x  , t) 2(1 + δAB ) " ! + + + + (x)ψB+ (x  ) − ψA+ (x)ψB− (x  ) |0, × ψA−

(19.14)

or along any other quantization axis. The singlet state (19.11) tells us that the two particles have opposite polarizations with respect to any quantization axis, although none of the two particles has an individual polarization with respect to any quantization axis. This is one of the fascinating features of quantum mechanics: Entangled particles can have correlations of their properties, although none of the particles has a specific value for that property until we observe it. Suppose now that the particles are moving far away from each other, and we also assume that the particles’ spin entanglement is not perturbed through interactions with the environment until we perform a measurement on one or both of the particles. Every simultaneous observation of the spins s A and s B of particles A and B along the same polarization direction rˆ will then yield opposite results,

4 See

Problem 19.4 for a more general explanation of rotational invariance of the singlet state.

19.2 Non-locality from Entanglement?

(ˆr · s A )(ˆr · s B ) = − h¯ 2 /4,

499

(19.15)

see also Problems 19.6 and 19.7. The prediction (19.15) has been tested e.g. in the two-proton experiment of Lamehi-Rachti and Mittig [104]. Their results were in agreement with quantum mechanics, and this indicates instantaneous collapse of the entangled two-particle quantum state over macroscopic separations. Corresponding photon experiments had been performed by Aspect et al. [3, 4]. Although we are usually dealing with entanglement of the properties of two particles in these systems, we are still dealing with a single quantum state encompassing those two particles, and the theoretical prediction and experimental confirmation of instantaneous quantum state collapse over macroscopic distances remains one of the most intriguing aspects of quantum mechanics. Indeed, collapse of quantum states over macroscopic scales is also observed in Tonomura and Stern-Gerlach type particle location experiments, see Sects. 1.7 and 19.1. We know, and can infer from the build up of the interference patterns, that the single particle wave functions are distributed over the area of the interference pattern, and yet the individual particles appear in locations with atom scale resolution. In a sense, already the observations of single particle locations are coincidence measurements by observing coincidence of appearance of the particle in one location with non-appearance in all the other locations which would be compatible with the particle’s wave function. From this perspective, the outcomes of the EPR type experiments should be expected. The only difference is that now we have two positions associated with the collapse instead of a single position, but it is still one macroscopic quantum state which collapses. However, we cannot localize any physical property like energy, momentum, spin projection, or location within a normalizable quantum state, neither in the x-representation for a single-particle state or (x, x  )-representation for a 2-particle state, nor in the corresponding momentum representations. We can calculate corresponding Hamiltonian or momentum or spin or location densities, but they do not allow us to say “this particle has that much energy with that momentum and spin projection in this point at time t”. Why is this relevant for understanding the EPR experiments? Let us revisit the Tonomura experiment again: The electron appears with a certain amount of energy in a certain point on a scintillation screen, although its wave function after exiting the biprism is laterally spread over several centimeters. We cannot argue that any spread out energy or mass or momentum or spin of the electron was instantaneously transported from all the other parts of the macroscopic wave function into the point where the electron appeared. However, then we also cannot infer that any energy or momentum or spin was transferred between any parts of a 2-particle wave function (x, x  , t) when the particles appeared in two separate scintillation screens or photomultipliers. There is no energy or momentum or spin transport within a quantum state, and therefore there is also no signal transport. Instantaneous collapse of entangled quantum states involves rigid correlations in coincidence measurements, but it is not associated with energy or signal transmission.

500

19 Epistemic and Ontic Quantum States

It is apparently interaction, e.g. with the atoms in a scintillation screen or a photographic plate, or the molecules in a cloud chamber, or the electrons in a photocathode or CCD camera, that triggers the collapse of the quantum states in various versions of Tonomura, Stern-Gerlach, or EPR type experiments. What all these interactions have in common is a change in (quasi-)particle composition of the experimental system. We need to generate an electron or a photon or a phonon or some other suitable quantum excitation to either provide the basic primary signal of the experiment, or otherwise to release energy from the observed system if we wish to stop e.g. silver atoms on a glass pane. It is then also the interactions, which at a microscopic level require jumps between different sectors in Fock space, which are responsible for the energy distribution between different separable (i.e. not entangled) factors of the final quantum state. We will return to this observation after discussing the dichotomy between quantum jumps, as observed in atomic or molecular transitions, and the continuous evolution of quantum states, as described e.g. by the Schrödinger and Maxwell equations.

19.3 Quantum Jumps and the Continuous Evolution of Quantum States Continuously evolving complex wave functions ψ(x, t) can be used to describe some aspects of quantum mechanics, but by no means all of it. Quantum mechanics is incomplete without second quantization. Therefore discussion of its ontological implications cannot only be based on the evolution of wave functions, but needs to take into account the evolution of quantum states in Fock space. A simple illustration for the shortcomings of continuously evolving wave functions is provided by the Schrödinger-Maxwell system. The Schrödinger equation (15.10) for the quasiparticle of charge q = − e and mass m = me mp /(me + mp ),

(19.16)

which describes relative motion in a hydrogen atom, coupled to Maxwell’s equations with charge and current densities q , j q (16.61) and (16.62) is in Coulomb gauge, ih¯ 

 2 h¯ 2 e ∂ψ(x, t) αS hc ¯ + ∇ + i A(x, t) ψ(x, t) + ψ(x, t) = 0, ∂t 2m |x| h¯

(19.17)

 1 ∂2 eh¯

∇ψ + (x, t) · ψ(x, t) − ψ + (x, t)∇ψ(x, t) −  A(x, t) = μ 0 2im c2 ∂t 2

19.3 Quantum Jumps and the Continuous Evolution of Quantum States

 e − 2i ψ + (x, t)A(x, t)ψ(x, t) h¯  x − x μ0 e ∂ − |ψ(x  , t)|2 . d 3x 4π ∂t |x − x  |3

501

(19.18)

This system would predict stability of excited atomic energy eigenstates unless we promote the wave function A(x, t) to an operator through field quantization. This can easily be inferred from the fact that a Poynting vector for radiation must drop with distance r like r −2 , whereas any Poynting vector formed from solutions of (19.18) with an atomic energy eigenstate on the right-hand side would be suppressed at large distance with a factor r −4 . This happens because the solutions of Eq. (19.18) are time-independent for stationary source terms, whence both E = −∇ and B = ∇ × A drop like r −2 . This implies that the coupling terms of the atom-photon system cannot explain spontaneous emission from excited atomic eigenfunctions as a consequence of continuous time evolution according to the wave equations (19.17) and (19.18). The continuously evolving atomic energy eigenfunctions ψn,,m (x, t) with n > 1 would not spontaneously decay through emission of electromagnetic radiation at the level of wave equations. Even with the coupling terms included, the Schrödinger equation and Ampère’s law predict that excited atomic eigenstates would be stable within the “first quantized” semi-classical formalism, and no classical radiation could be emitted under any circumstances. Absence of radiation from excited states on the level of the non-quantized wave equations (19.17) and (19.18) can also be seen from the fact that ψ(x, t) = ψn,,0 (x, t) and A(x, t) = 0 solve these equations. Furthermore, the system would also not account for exchange interactions, and therefore completely ignore magnetism in materials, if we would not also promote the wave function ψ(x, t) to an operator. Indeed, second quantization and the scattering matrix are key concepts to understand the coexistence of continuous, causal evolution of quantum states according to |ψ(t) = U (t, t  )|ψ(t  )

(19.19)

with discrete, random quantum jumps like the Lyman α emission or absorption processes |2, 1, m1  ↔ |1, 0, 0; k, α, where  α (k) is the polarization of the emitted or absorbed photon of momentum hk = E2 − E1 . The ¯ and energy hck ¯ corresponding scattering matrix elements were formulated and evaluated in leading order in Sects. 18.6 and 18.7. The point is that every quantum jump from an initial state |ψi  to a final state |ψf  during the time window t  → t corresponds to a continuously and causally evolving scattering matrix element Sf i (t, t  ) = ψf (t)|ψi (t) = ψf (t)|U (t, t  )|ψi (t  ) = ψf (t)|U + (t  , t)|ψi (t  ) = ψf (t  )|ψi (t  ).

(19.20)

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19 Epistemic and Ontic Quantum States

Usually only the first line of (19.20) is written down, and is interpreted as continuous evolution of the state |ψi  from time t  to time t, when it is suddenly converted into the state |ψf , involving e.g. the spontaneous absorption or emission of a photon. However, the second line of Eq. (19.20) tells us that the spontaneous transition could just as well have happened at time t  , and then the final state |ψf  evolves to time t. Furthermore, the composition law U (t, t  ) = U + (t0 , t)U (t0 , t  ) tells us that we can just as well think of the quantum jump as having happened at any other time t0 ! This is exactly how the scattering matrix ties together continuous evolution of quantum states with quantum jumps, and resolves the apparent dichotomy of quantum dynamics. Scattering matrix elements provide the continuously evolving transition 2  probability densities Sf i (t, t  ) which yield continuously evolving densities of transition rates. These are continuously evolving densities of rates of quantum jumps. The Schrödinger equation, Maxwell’s equations, the relativistic generalizations of the Schrödinger equation, and non-abelian generalizations of Maxwell’s equations (Yang-Mills equations) describe both the evolution of states between quantum jumps and the rate of quantum jumps through time evolution operators. Quantum mechanics appears to be complete. There is no theoretical need nor experimental evidence for any further dynamical resolution of quantum jumps.5 Quantum states therefore provide a best possible description of everything that we can know about the evolution of a physical system. However, the quantum states which we use to describe the system evolve continuously according to Eq. (19.19) until we update our knowledge about the system through observation, while the physical system evolves continually, including spontaneous quantum jumps through particle emission or absorption. Quantum mechanics tells us two things about physical systems: - Quantum mechanics tells us what can happen during the evolution of a physical system; and - it tells us the time-dependent probabilities for events to happen during the evolution of the system. However, quantum mechanics does not tell us whether anything will happen with certainty, nor does it allow us to pinpoint the moment of an event to happen. Furthermore, while our records for quantum states evolve continuously between observations according to Eq. (19.19), whereas the physical system can go through discontinuous quantum jumps, we also introduce subjective discontinuities in the description of quantum systems. We do this by replacing the previously used quantum state with another quantum state after performing a measurement on the system, because (judging on the basis of what we have learned during the measurement) the new state provides a higher fidelity description of the quantum

5 This

is the majority view in physics, and also my own opinion. However, I would be remiss to not point out that the interpretational difficulties of quantum mechanics have led to ongoing discussions whether quantum mechanics really is a complete theory of dynamics at the single-particle level.

19.3 Quantum Jumps and the Continuous Evolution of Quantum States

503

system. We will denote these subjective discontinuities in the quantum states as quantum leaps to separate them from the quantum jumps. This indicates two aspects to quantum states. There are the complete records of quantum systems, and there are our best guesses for those records. This seems to be trivial. After all, in classical mechanics we have the coordinates x(t) of a particle and our best estimates for those coordinates from observation. What makes the distinction non-trivial in quantum physics is that integration |ψ(t) = U (t, t  )|ψ(t  ) of the differential equations of quantum field theory provide us with the continuous evolution of our epistemic states as best guesses for the state of the system, under the assumption that at time t  the system was in the state |ψ(t  ). These best guesses automatically include estimates for the occurrence of quantum jumps, when our best guesses necessarily start to deviate from the actual state of the system, because the jump has occurred or not, and we won’t know until we update our knowledge of the system through another observation. We think of the evolution equations of classical mechanics as objective evolution equations for the classical physical system, but quantum mechanics tells us a more complicated story: Only the linear parts of evolution equations describe continuous evolution of ontic quantum states, whereas the nonlinear interaction parts describe spontaneous quantum jumps in the ontic states. At what times these jumps occur cannot be predicted. Integration of the evolution equations only provides us with estimates how often the jumps should occur. The quantum jumps introduce inherent deviations between the quantum state of the system and the epistemic quantum state which we use, until we reset our epistemic record through observation. If we do not distinguish the state of the system and our records, and always think of |ψ(t) = U (t, t  )|ψ(t  ) as an ontic state, the occurrence of quantum jumps in U (t, t  ) might indeed tempt us into a many-worlds interpretation. Fortunately, the story is not that complicated. Schrödinger’s time-dependent continuous evolution equation did not do away with Bohr’s quantum jumps. It only provided us with a tool to estimate the rates of those jumps. However, every jump deteriorates the agreement between the system’s state and our best guess for the state, and contributes to the principal limitation of predictability of quantum evolution. In a nutshell, this means that there are ontic quantum states, but integration of the evolution equations of quantum mechanics only provides us with best guesses for those states. This observation would also be compatible with a purely epistemic interpretation of quantum states as records of evolution of our knowledge between observations. However, I am reluctant to go that far. Combining a purely epistemic interpretation of quantum states with the assertion of completeness of quantum mechanics would seem to imply a lack of any ontology of quantum systems, or an assumption that we cannot know anything about the ontology of quantum systems. And yet quantum systems certainly exist. The assumption that our epistemic quantum states provide complete records of quantum systems up to quantum jumps, and in that sense describe an ontology of quantum systems, appears to me like the best possible interpretation of quantum mechanics. We can illustrate these concepts with further examples in the following sections.

504

19 Epistemic and Ontic Quantum States

19.4 Photon Emission Revisited Creation or annihilation of particles constitute the most basic examples of quantum processes, and we can hope to learn something about the interpretation of quantum mechanics from its application to these processes. Suppose we start out with a hydrogen atom in a 2p state, e.g. |2, 1, 0. We know that this hydrogen atom will sooner or later relax into a 1s state through spontaneous emission of a Lyman α photon. This process constitutes an example where the quasiparticle representing relative motion between the proton and the electron in the 2p state is annihilated and replaced with a quasiparticle in the 1s state and the Lyman α photon. We can calculate the corresponding emission rate  = 6.3 × 108 s−1 from Eq. (18.136). This implies a lifetime estimate τ =  −1 = 1.6 ns for the 2p state. The principles of time-dependent perturbation theory then imply that in leading order of the atom-photon interaction term, the initial state evolves into a superposition of an atomic state with atom+photon states (cf. Eq. (18.127)). For the 2p state as initial state this yields |ψ(t)  |2, 1, 0 exp(− iω2 t) −



 d 3 k δ(ω2,1 − ck)c αS k

α

×|1, 0, 0; k, α1, 0, 0| α (k) · x|2, 1, 0 exp[− i(ω1 + ck)t]   √ 2 5 h¯ = |2, 1, 0 exp(− iω2 t) − 4 2 √ exp(− iω1 t) 3 m αS  √ × d 3 k kδ(ω2,1 − ck)αz (k)|1, 0, 0; k, α exp(− ickt).

(19.21)

α

The atom+photon state in the second term contains a Lyman α photon which is delocalized in direction with preferential polarization along the z-axis.6 The primary question with respect to interpretations of quantum mechanics is now: What, if anything, does the state |ψ(t) tell us about the ontological properties of the system which initially consisted of a hydrogen atom in a 2p state? And by extension: What, if anything, do quantum states tell us about the world beyond probabilistic predictions for observations?

f i ≡ Sf i (∞, −∞) instead of the finite time scattering matrix Sf i (t, 0) (19.20). Otherwise the δ function would be resolved as

6 Equation (19.21) is simplified by using the standard scattering matrix S

δ(ω2,1 − ck) →

sin[(ω2,1 − ck)t/2] exp[i(ω2,1 − ck)t/2]. π(ω2,1 − ck)

(19.22)

The basic feature that quantum evolution turns the initial atomic state into a superposition of the initial atomic state with another atomic state tensored with a photon state holds either way.

19.5 Particle Location

505

Unless we are willing to entertain simultaneous existence and non-existence of a photon as a possibility, nor a corresponding many-worlds interpretation, we conclude that the state |ψ(t) is epistemic. It provides us with probability measures to find the initial excited hydrogen atom after time t, or the de-excited hydrogen atom with an accompanying Lyman α photon. According to this reasoning, every quantum state which describes a change in particle composition of a system as a consequence of particle annihilation or creation, is epistemic. On the other hand, the state |2, 1, 0 of the hydrogen atom before de-excitation and the atom+photon state after de-excitation are ontic. The point is that ontic states are replaced with new ontic states during every particle creation or annihilation event, and the transition between these ontic states is described by epistemic superpositions of ontic states [37], like the state |ψ(t). The epistemic interpretation of the superposition |ψ(t) in Eq. (19.21) is also in agreement with the fact that no known method of observation can collapse or reduce the atom+photon system into the particular state (19.21) with simultaneous presence and absence of a photon. We start out with the knowledge of the ontic 2p state of the system, and integration of this ontic state with the evolution equations of quantum theory provides us with the epistemic state (19.21) as an estimate for when the system will decay from the ontic 2p state into the ontic 1s+γ state. Eventually, observation of the Lyman α photon updates our knowledge about the system. We then abandon the state (19.21) and replace it with another epistemic state which arises from integration of the evolution equations using the ontic 1s as initial state (tensored with a corresponding photon state if the photon has not been destroyed in the observation). Quantum theory tells us which ontic states a system can take, and it also provides us with epistemic states as best guesses for how ontic states will evolve until we update our knowledge of the system through observation. The states |ψ(t) that we calculate for ontic initial states |ψ(0) from integration of the time evolution equations are epistemic, because we cannot predict quantum jumps. We can only calculate how often quantum jumps should happen on average. Stated differently: Schrödinger’s cat is always in one of two ontic states: dead or alive. However, the state that we calculate for the evolution of the Schrödinger cat + lethal quantum device system through integration of the evolution equation of the quantum device is epistemic, because we cannot calculate when exactly the lethal quantum device will trigger (e.g. through a quantum decay). Looking at the cat will update our knowledge and we will reset our epistemic state accordingly. Once the cat has died, our principally epistemic calculated state will actually coincide with the ontic state of dead cat plus triggered quantum device, because no unpredictable quantum jump is involved anymore in the future evolution of the system.

19.5 Particle Location Observation of particle location can involve two principally different techniques. On the one hand, we can observe consequences of particle flyby on atoms or molecules or electrons. Observation of particle trajectories through bremsstrahlung emission

506

19 Epistemic and Ontic Quantum States

in the Coulomb field of nearby atomic nuclei, through Coulomb excitation and fluorescence of nearby atoms, through condensation in a cloud chamber, or through electrical signals in a wire chamber are examples for this technique. On the other hand, we can locate the particle on a substrate or in a magnetic trap, and then observe it. Molecular beam-epitaxy and scanning tunneling microscopy provide examples for particle depositions and observations on substrates. We have seen that observation of particle location is the key part of Stern-Gerlach experiments to observe spin polarization, because traversal of an inhomogeneous magnetic field correlates spin projection with location. Momentum and energy measurements through time-of-flight methods or through observation of the Larmor radius of a particle in a magnetic field are also ultimately observations of particle location with subsequent inferences on other particle properties. Observation of particle location therefore plays a key role in our further discussion of interpretations of quantum mechanics. We can always assign locations to macroscopic objects, but whether or not we can assign a location to an elementary particle depends on the situation. For example, the expectation value for the distance vector between the proton and the electron in a hydrogen atom in an energy eigenstate |n, , m  is r = n, , m |r|n, , m  = 0,

(19.23)

and yet, within the size a (the Bohr radius) of the atom, we would not claim that the electron is in any way located at the same point as the proton. However, in the coordinate system x of a laboratory experiment, where the atom may be located centimeters or meters from the origin of the coordinate system, we think of the location of the atom and its constituents as being identical, since r =  r 2  % |x|, although we are well aware that zooming into the atom without perturbing it would not provide us with a location of the electron within the atom. On the other hand, searching for the electron in the atom with spatial resolution |x| ≤ 0.1a would provide us with an electron location, but at the expense of seriously perturbing the atom by introducing large uncertainties pi ≥ h/2|x| ≥ 10 keV/c ¯

(19.24)

in electron momentum, which would usually ionize the atom. We could not even claim to have observed the electron inside the atom. A fairer assertion would be that we destroyed the atom to localize the electron with subatomic resolution. In general, the value of an observable O does not exist at a level below the corresponding uncertainty O, although we can always define the expectation value O. An electron in an atom does not have a location within the atom. The notion of “electron location within an atom” is vacuous. Indeed, based on double slit experiments, Feynman had masterfully argued that we also cannot assign a location to a free particle, as long as we do not perform an observation of the location [53, 54]. The combination of single-particle wave function interference with final single-particle projection in electron biprism

19.5 Particle Location

507

experiments (see Sect. 1.7) implies that we cannot think of the particle to exist anywhere within its own wave function as long as we do not observe that location, while at the same time the observation of particle location on the photographic film with higher resolution than the width of |ψ(x, t)| implies that we also cannot think of the particle as an extended object (like a drop of a fluid) with an extension given by the width of its wave function. Contrary to classical beliefs, a particle does not have position and momentum, neither in a pointlike sense nor in a center of mass sense, but it constitutes a “statistical package” of position and momentum values with minimal uncertainty products xi pj ≥ δij h¯ /2. We cannot picture this object in our minds, just like we cannot visualize four-dimensional spacetime.7 We also cannot assign a trajectory to the particle as long as we do not observe a trajectory. However, once the particle has been located through interaction with the screen, the wave function of the particle would be reset to include the improved position information [75]. This constitutes an example of collapse of the wave function of the particle, although Heisenberg used the less spectacular term “reduction of the wave function”. In the light of our discussion in the opening paragraphs to this chapter, this begs the question whether localization of the particle corresponds to collapse of the particle’s ontic quantum state (ontological collapse)? Or is this an epistemic collapse that is not tied to an ontic quantum state, but nevertheless would always lead to the same location if the particle is not moving? Or is this just an epistemic reduction of our best guess for the particle’s quantum state, with only a high probability to find the same location in repetitions of the measurement? Each of these interpretations has puzzling implications, as indicated by the questions Q1–Q3 above, and yet none of them is easily dismissed. Quantum state collapse as a consequence of an interaction cannot be dismissed, since we know from spectroscopy and the interpretation of the scattering matrix that interactions can be related to spontaneous, discrete quantum jumps between quantum states. Wave function collapse then just appears as another spontaneous, discrete quantum transition during an interaction, and indeed, we use the scattering matrix formalism all the time to estimate rates for wave function collapse in momentum space when we calculate scattering cross sections, just like we calculate rates for Lyman α transitions in hydrogen. It is therefore not surprising that Niels Bohr, the father of the theory of spontaneous transitions in atoms, was also one of the most assertive proponents of ontological wave function collapse. Whichever of the three basic collapse or reduction interpretations applies, it is clear that the quantum state cannot reduce to a sharp position eigenstate, which would require a diverging amount of energy. The position uncertainty of the measurement must apparently determine the widths xi of the quantum state after observation. There is no doubt that silver atoms were localized on the glass pane of Stern and Gerlach after traversing the magnetic field, although the wave functions for

7 Sadly,

biological evolution had no need to equip us with that level of proficiency.

508

19 Epistemic and Ontic Quantum States

unpolarized silver atoms entering the magnet were spread over two macroscopically separated regions when hitting the glass pane. There are also no doubts that the electrons traversing an electron biprism are finally localized on the screen cutting off their paths, although again their wave functions are spread over macroscopic regions after leaving the biprism. The same applies to depositions of atoms or molecules on surfaces through molecular beam epitaxy, and localized particles on surfaces can be seen through scanning tunneling microscopy. One might hope that these localization effects are due to energy loss mechanisms, like e.g. bremsstrahlung emission, during the observation, thus avoiding the need for a genuine wave function collapse. However, the lessons from the observations of the discrete spin degrees of freedom show that quantum state collapse is inevitable in quantum mechanics, and e.g. a free wave packet with low energy is generically wider than a free wave packet with high energy.8 Therefore, energy loss in a flyby observation without noticeable impact of an attractive potential should make a wave packet wider rather than narrower. Localization is therefore also a manifestation of quantum state collapse. Bohr and von Neumann suggested that collapse should be a consequence of observation in a way that would require presence of a conscious observer. However, I would rather consider collapse as a consequence of quantum jumps between different sectors in Fock space and note that observations necessarily involve quantum jumps. The outcomes of quantum jumps therefore set the initial condition for the next phase of continuous evolution of the system. This removes the need of the presence of a conscious observer during quantum state collapse. The ensuing picture of quantum mechanics therefore includes ontic particle states, which can go through spontaneous quantum jumps through particle absorption or emission, as described by the scattering matrix. On top of the ontic states, we also have the epistemic states which we use to approximate the evolution of quantum systems between observations. The epistemic states evolve continuously and allow us to calculate rates of quantum jumps. Every quantum jump deteriorates the agreement between ontic and epistemic states, until we update our epistemic states through observation. Observation involves quantum jumps through bremsstrahlung emission and collisions, but it is the quantum jumps, not the presence of observers, which triggers collapse of quantum states. In terms of the terminology introduced in the opening paragraphs to this chapter, we have ontic quantum state reduction in the sense of “particle localization” (or rather reduction of the width of wave packets in x space) through quantum jumps during bremsstrahlung emission and collisions. In the sense of the calculation of jump rates, these collapse mechanisms are captured by the Schrödinger equation and its relativistic generalizations. We also have an epistemic reduction (or rather, realignment) in our epistemic approximations to the ontic quantum states, when we perform an observation. In a nutshell this means that quantum mechanics appears

8 This

is simply a consequence of the fact that the kinetic energy of a non-relativistic particle is proportional to the curvature of the wave function.

19.5 Particle Location

509

to be complete in the following sense, which I will cautiously formulate as seven hypotheses: H1. The evolution equations of quantum theory describe the possible set of states of quantum systems. H2. These sets of states are a complete basis for the possible ontic states of quantum systems, and also a complete basis for the epistemic states which we calculate to approximate the time evolution of quantum systems. H3. A quantum system is always in an ontic state. The ontic state evolves continuously within a single sector of Fock space, until it goes through a spontaneous quantum jump into another sector in Fock space. The continuous evolution follows the projection of the evolution equations of quantum theory into the Fock space sector of the system, until a jump occurs. H4. If we have a good guess |ψ(t0 ) for the ontic state of a quantum system at time t0 (e.g. from observation or preparation of the state), we integrate the evolution equations to calculate a continuously evolving epistemic state |ψ(t) = U (t, t0 )|ψ(t0 ).

(19.25)

This epistemic state evolves continuously as a superposition of different sectors of Fock space (indeed, the state superimposes all sectors of Fock space if we include the interactions to arbitrary high order). It describes quantum jumps in a probabilistic sense by allowing us to calculate jump rates between the different sectors of Fock space, but we cannot predict with certainty which jump will occur, nor when it will occur. H5. Updating our knowledge of the system through observation of the ontic state |ψ(t1 ) allows us to update our epistemic state |ψ(t) for t ≥ t1 as |ψ(t) = U (t, t1 )|ψ(t1 ).

(19.26)

H6. Quantum jumps imply collapse of quantum states. H7. Physical quantities or signals are not transferred within continuously evolving ontic quantum states. Transferral of physical quantities or signals occurs only as a consequence of quantum jumps. This means e.g. that we cannot say that the two electrons within a singlet state each share 50% of the energy of the state, nor can we say that energy is transferred from one electron to the other when one electron is deflected down in an inhomogeneous magnetic field while the other electron goes up. However, energy from the state is transmitted into bremsstrahlung and fluorescence photons when the electrons are observed on a scintillation screen. The continuous evolution equations of Maxwell, Schrödinger, Klein and Gordon, Dirac, and Yang and Mills did not excise Bohr’s quantum jumps from quantum theory. They only allow us to construct scattering matrices and calculate rates of quantum jumps (which is no small feat!). The success of quantum theory and the absence of any contradicting experimental evidence suggests that the

510

19 Epistemic and Ontic Quantum States

resulting epistemic-ontic interface describes a hard boundary for the mathematical description of nature. No further resolution of quantum jumps appears to be needed for the description of quantum systems, nor may any further resolution be possible. If we do not find any experimental evidence for violations of quantum mechanics, then we have to assume that the unresolvable spontaneous occurrence of quantum jumps is a basic property of nature. Calculating expected rates of occurrence of the quantum jumps would then be the limit for how far differential evolution equations can take us at the microscopic level.

19.6 Problems 19.1 Show that the spin state |ˆs− (t) = sin[ϑ(t)/2]| ↑ − exp[iϕ(t)] cos[ϑ(t)/2]| ↓   & ϑ(t) − π % = exp i sin[ϕ(t)]σ1 − cos[ϕ(t)]σ2 | ↑ 2

(19.27)

is orthogonal to the state (19.5) and corresponds to sharp spin projection − h/2 ¯ onto the spin vector sˆ (t) (19.6), sˆ (t) · σ |ˆs− (t) = − |ˆs− (t).

(19.28)

19.2 Show that the eigenstates |ˆs+  and |ˆs−  (19.5) and (19.27) of the operator sˆ · σ satisfy ˆs± |σ |ˆs±  = ± sˆ .

(19.29)

This implies in particular that the expectation value for any spin projection aˆ · σ is ˆs± |aˆ · σ |ˆs±  = ± aˆ · sˆ .

(19.30)

This equation has the following meaning: If we select a fermion beam which is polarized in direction ± sˆ and send this beam through a Stern-Gerlach experiment ˆ then we will see again (up to) two signals with spin projection with direction a, ˆ However, the ensemble averaged value of the values ± h/2 ¯ onto the direction of a. spin projection (i.e. weighted by intensity of the two signals) will be ± h¯ aˆ · sˆ /2, depending on which of the two sˆ -polarizations we selected, although we will never observe the individual spin projection value ± h¯ aˆ · sˆ /2 (unless aˆ · sˆ = ± 1). This illustrates again the projection property of quantum mechanics: Observed values correspond to eigenvalues of the observable, irrespective of whether the initial state was prepared in a superposition of eigenstates of the observable.

19.6 Problems

511

19.3 Show that with respect to √ the x axis and the y axis, the particles in the |1, 0 triplet state (| ↑| ↓ + | ↓| ↑)/ 2 have parallel spins, | ↑| ↓ + | ↓| ↑ = |+|+ − |−|− = i (| ←| ← − | →| →).

(19.31)

19.4 Equations (19.13) are special expressions of the fact that the anti-symmetric spin combination ab = δa1 δb2 − δa2 δb1 is the scalar component 0 in the reduction 1/2 ⊗ 1/2 = 1 + 0 of representations of SU (2), [exp(iϕ · σ /2)]ac [exp(iϕ · σ /2)]bd cd = ab .

(19.32)

Verify this equation. For agreement between this more general statement and Eqs. (19.13), observe that a rotation around the y-axis by π/2 transforms | ↑ → |−,

| ↓ → |+,

(19.33)

and a rotation around the x-axis by π/2 transforms | ↑ → | →,

| ↓ → i | ←.

(19.34)

If you like to verify these equations graphically, recall that we use a passive interpretation of coordinate transformations, i.e. we rotate the coordinate systems but not the spins. 19.5 The Pauli matrices in the second quantized formalism are given by the operators (17.132). 19.5a Show that the local spin up and down states | ↑, x = ψ↑+ (x)|0,

| ↓, x = ψ↓+ (x)|0,

(19.35)

are eigenstates of σ 3 with eigenvalues ±1. 19.5b Demonstrate the transformation laws (19.2) and (19.3) for the corresponding creation operators and local spin states, e.g.

 √ + ψ± (x) = ψ↑+ (x) ± ψ↓+ (x) / 2, + |±, x = ψ± (x)|0,

σ 1 |±, x = ± |±, x.

(19.36) (19.37)

19.6 Any actual observation of spins in the singlet state (19.11) will require separation of the spin components of the particles through Stern-Gerlach experiments, and one might worry whether this complicates things. Show that Stern-Gerlach spin polarizers transform the state (19.11) into a superposition of a singlet state and a triplet state.

512

19 Epistemic and Ontic Quantum States

Solution Let us assume that we send our particles through inhomogeneous magnetic fields, although we do not observe them yet, i.e. we do not project the particles on a glass pane or scintillation detector. The inhomogenous magnetic fields will deflect the spin up and spin down components of the particles differently and map the initial singlet state (19.11) into a state of the form   1 + + (x)ψB↓ (x  )|0 d 3 x d 3 x  A↑,B↓ (x, x  , t)ψA↑ 2(1 + δAB )    + + 3 3    − d x d x A↓,B↑ (x, x , t)ψA↓ (x)ψB↑ (x )|0 . (19.38)

|AB (t) = √

Generically, this new state generated by the magnetic fields is a superposition of a singlet state and a triplet state, (0)

(1)

|AB (t) = |AB (t) + |AB (t),

(19.39)

with the singlet state (0) |AB (t)

  ! " 1 3 = √ d x d 3 x  A↑,B↓ (x, x  , t) + A↓,B↑ (x, x  , t) 8(1 + δAB )   + + + + (x)ψB↓ (x  ) − ψA↓ (x)ψB↑ (x  ) |0, (19.40) × ψA↑

and the triplet state   ! " 1 (1) |AB (t) = √ d 3 x d 3 x  A↑,B↓ (x, x  , t) − A↓,B↑ (x, x  , t) 8(1 + δAB )   + + + + (x)ψB↓ (x  ) + ψA↓ (x)ψB↑ (x  ) |0. (19.41) × ψA↑ This equation holds irrespective of whether we send both particles through inhomogeneous magnetic fields, or only one of the particles (e.g. A, when the labels B ↓ and B ↑ become redundant). Note that we will have A↑,B↓ (x, x  , t) = A↓,B↑ (x, x  , t),

(19.42)

because inhomogeneous magnetic fields deflect opposite spins into different locations. We should even have negligible overlap between the different spin components of each particle in an ideal spin measurement,   + A↑,B↓ (x, x , t) · A↓,B↑ (x, x , t) = 0,

(19.43)

to avoid cross contamination of the beams with the opposite spin polarization. This implies equal strength and orthogonality of the singlet and triplet contributions if A

19.6 Problems

513

and B are different kinds of particles, (0)

(0)

(1)

(1)

AB (t)|AB (t) = AB (t)|AB (t) = 1/2, (0)

(1)

AB (t)|AB (t) = 0.

(19.44) (19.45)

The result (19.44) implies that admixture of a triplet state through preparation for a spin observation is inevitable if A and B are different kinds of particles. On the other hand, if A and B are indistinguishable because they are the same kind of particle, + + we have ψA,s (x) = ψB,s (x) ≡ ψs+ (x), and the triplet component (19.41) would vanish for ↑↓ (x, x  , t) = ↑↓ (x  , x, t),

↓↑ (x, x  , t) = ↓↑ (x  , x, t).

(19.46)

We will actually have ↑↓ (x, x  , t) = ↓↑ (x  , x, t), because ↑↓ (x, x  , t) is the probability amplitude to find a particle with spin up in x at time t and a particle of the same kind with spin down in x  . However, the conditions (19.46) cannot hold after the particles went through a Stern-Gerlach spin polarizer, because the spins of opposite direction were split while traveling through the polarizer, and we cannot have e.g. the same probability amplitude to find particles with spin up and particles with spin down for all locations x. 19.7 Spin-spin correlations 19.7a Assume that A and B are different kinds of spin-1/2 particles. Show that the j spin-spin correlations AB (t)|σAi σB |AB (t) for the state (19.38) are j

AB (t)|σAi σB |AB (t) = − δ i3 δ j 3 .

(19.47)

j

Here σAi and σB are the Pauli operators (17.132) acting on particle A and particle B, respectively. Assume normalization of the orbital wave functions, 

 d 3x

2  d 3 x  As,Bs  (x, x  , t) = 1,

(19.48)

and also assume perfect separation (19.43) of the spin components. 19.7b Show that the spin-spin correlations (t)|σ i σ j |(t) for the state (19.38) with identical particles, |(t) ≡ |A=B (t), are also given by (t)|σ i σ j |(t) = − δ i3 δ j 3 . Use the normal ordered spin correlation operator

(19.49)

514

19 Epistemic and Ontic Quantum States

σ iσ j =

  1 j i d 3 x d 3 x  ψs+ (x)ψu+ (x  )σa(s),a(s  ) σa(u),a(u ) 2   s,s ,u,u

×ψu (x  )ψs  (x),

(19.50)

which eliminates spin-spin correlations for single particle states, and assume perfect separation of the spin components,   + ↑↓ (x, x , t) · ↓↑ (x, x , t) = 0.

(19.51)

Remark The Eqs. (19.47) and (19.49) show that preparation of the singlet state (19.11) for observation through spin polarizers does not reduce the correlation (19.15) along the quantization axis which is probed. However, the correlation with respect to the orthogonal axes is lost (we only get − δ i3 δ j 3 , but not − δ ij ), because the splitting of spins in z-direction destroyed the singlet state. Preparation of the singlet state for observation of the characteristic correlation of spins along quantization axes only preserved the correlation along the particular quantization axis that we wish to probe, at the expense of destroying the correlations along the orthogonal axes. 19.7c Repeat the calculation of the spin-spin correlation for a pair of identical particles in the state  |(t) =

 3

d x

d 3 x  ↑↓ (x, x  , t)ψ↑+ (x)ψ↓+ (x  )|0,

(19.52)

but instead of the condition (19.51) for perfect separation of spins after a SternGerlach polarizer, assume that the singlet condition of symmetric orbital wave function holds, ↑↓ (x, x  , t) = ↑↓ (x  , x, t). Show that the spin-spin correlation in this case yields the singlet property (19.15), (t)|σ i σ j |(t) = − δ ij .

(19.53)

This equation says that if we observe the spin projections of the two particles in ˆ respectively, we will get the result the singlet state along quantization axes aˆ and b, 2 ˆ − h¯ aˆ · b/4. It does not say that we can afterwards repeat the measurement on the same pair of particles with different quantization axes cˆ and dˆ and then expect ˆ the result − h¯ 2 cˆ · d/4, because after the first measurement the pair of particles is not in a singlet state anymore. Indeed, Problems 19.6 and 19.7b demonstrated that preparation for observation of the spin-spin correlation already destroys the singlet state and reduces Eq. (19.53) to (19.49).

Chapter 20

Quantum Aspects of Materials II

We have already seen in Chap. 10 that basic properties of electron states in materials are determined by quantum effects. This impacts all properties of materials, including their mechanical properties, electrical and thermal conductivities, and optical properties. Examples of the inherently quantum mechanical nature of electromagnetic properties of materials are provided by the role of virtual intermediate states in the polarizability tensor in Sect. 15.3, and the importance of exchange interactions for magnetism in materials, as discussed in Sect. 17.7. We will now continue to illustrate quantum effects in materials with a focus on effects that require the use of second quantization or Lagrangian field theory, or at least the knowledge of exchange interactions for a proper treatment. We will start at the molecular level and then discuss the second quantization of basic excitations in condensed materials. The inception of the Schrödinger equation was accompanied by a large number of immediate successes, including atomic theory, the quantum theory of photon-atom interactions, and quantum tunneling. Another one of these important successes was the development of the theory of covalent chemical bonding, which was initiated by Burrau [20, 21], Heitler and London [78], and others. This is an extremely important and well studied subject in chemistry and molecular physics, and yet it never seemed to reach the level of popularity and recognition that other areas of applied quantum mechanics enjoy. One reason for this lack of popularity might be the lack of simple, beautiful model systems which can be solved analytically. Solvable model systems are of great instructive and illustrative value, and often provide a level of insight that is very hard to attain with systems which can only be analyzed by approximation methods. However, the existence and stability of covalent bonds is clearly an important property of molecules and of materials in general, and a basic quantitative understanding of the covalent bond should be part of the toolbox of every chemist, physicist and materials scientist. Indeed, there is a model system which can be analyzed to some extent by analytic methods. If only basic qualitative features are required, the analytic formulation can then be used for numerical evaluations © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 R. Dick, Advanced Quantum Mechanics, Graduate Texts in Physics, https://doi.org/10.1007/978-3-030-57870-1_20

515

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20 Quantum Aspects of Materials II

which do not require a huge amount of effort. This model system is the hydrogen molecule ion H+ 2 , which is also known as the dihydrogen cation. The analysis of electron states for fixed locations of the two protons in this simplest molecular system have been investigated already in the early years of quantum mechanics [85, 89, 168, 180], and have been a subject of research ever since, both in terms of the semi-analytic analysis in prolate spheroidal coordinates (see e.g. [5, 84, 155]) used in Sect. 20.2, and in terms of high precision variational calculations [25, 66, 107]. Before specializing to H+ 2 we will discuss the interplay of nuclear and electronic coordinates and the role of the Born-Oppenheimer approximation in molecular physics.

20.1 The Born-Oppenheimer Approximation The interactions of non-relativistic electrons and atomic nuclei are dominated by the Coulomb interaction.1 Molecules can therefore be described by first quantized Hamiltonians of the form H =

p2 P2 ZI ZJ e2 e2 i I + + + 2me 2MI 4π 0 |R I − R J | 4π 0 |r i − r j | i

−

I

i,J

ZJ e2 4π 0 |r i − R J |

I 0, and the energy levels of the free states are E(k  , k⊥ ) =

 h¯ 2 2 2 k  + k⊥ . 2m

(21.32)

The energy-dependent Green’s function x  , z|G(E)|x  , z  ≡ z|G(x  − x  , E)|z  ≡ −

h¯ 2 z|G(x  − x  , E)|z  2m

of this system must satisfy   2m  + 2 [E + Wδ(z − z0 )] z|G(x  , E)|z  = − δ(x  )δ(z − z ). h¯

(21.33)

We would not have to solve this equation explicitly, since we know the complete set of energy eigenstates of the system. However, there is a neat way to solve these kinds of problems which also works for interfaces in which particles move with different effective mass [32].4

4 See

also the previous references.

572

21 Dimensional Effects in Low-Dimensional Systems

We can solve Eq. (21.33) in a mixed representation using k  , k⊥ |G(E)|k  , z  = √



1 2π

5

 d 2x

d 2 x 



dz x  , z|G(E)|x  , z 



 × exp i k  · x  − k  · x  − k⊥ z 

= k⊥ |G(k  , E)|z δ k  − k  ,

(21.34)

where 1 k⊥ |G(k  , E)|z  = √ 2π



 d 2x

dz z|G(x  , E)|z 

&" ! % × exp − i k  · x  + k⊥ z .

(21.35)

Substitution into Eq. (21.33) yields with κ = mW/h¯ 2 &" ! %   exp ik⊥ z0 − z 2mE 2 exp(ik⊥ z0 ) k⊥ |G(k  , E)|z  − 2 = k 2 + k⊥ √ h¯ 2π  κ (21.36) − dq⊥ exp(iq⊥ z0 ) q⊥ |G(k  , E)|z . π This result implies that the retarded Green’s function k⊥ |G(k  , E)|z  must have the form   ! % &" exp ik⊥ z0 − z   exp(ik⊥ z0 ) k⊥ |G(k  , E)|z  = + f (k  , E, z ) √ 2π ×

1 2 k⊥

+ k 2

− (2mE/h¯ 2 ) − i

,

(21.37)

with the yet to be determined function f (k  , E, z ) satisfying f (k  , E, z ) −

κ π

 dk⊥

! % &" √  exp ik⊥ z0 − z / 2π + f (k  , E, z ) 2 + k 2 − (2mE/h2 ) − i k⊥ ¯ 

= 0,

(21.38) which follows from substituting (21.37) back into (21.36). The integral is readily evaluated with the residue theorem, 

(h¯ 2 k 2 − 2mE) exp(ik⊥ z) dk⊥  = 2 + k 2 − (2mE/h2 ) − i π k⊥ ¯  k 2 − (2mE/h¯ 2 )

21.2 Inter-Dimensional Effects in Interfaces and Thin Layers

 × exp − k 2 −

2mE h¯ 2

573



  (2mE − h¯ 2 k 2 ) 2mE |z| + i  exp i − k 2 |z| . 2 2 2 h ¯ (2mE/h¯ ) − k 

This yields the condition for f (k  , E, z ) in the form ⎡

⎛

⎞⎤ (h¯ 2 k 2 − 2mE) (2mE − h¯ 2 k 2 ) ⎣1 − hκ ⎠⎦ f (k  , E, z ) +i  ¯ ⎝  2 2 2 2 2mE − h¯ k  h¯ k  − 2mE ⎡    2 2 hκ |z − z0 | ¯ ⎣ (h¯ k  − 2mE) 2 2  exp − h¯ k  − 2mE =√ h¯ 2π h¯ 2 k 2 − 2mE ⎤   − z | (2mE − h¯ 2 k 2 ) |z 0 ⎦, +i  exp i 2mE − h¯ 2 k 2 (21.39) h 2 2 ¯ 2mE − h¯ k  and therefore we find with the proper treatment of poles for retarded Green’s functions the result  1 1 exp(− ik⊥ z ) k⊥ |G(k  , E)|z  = √ 2 + k 2 − (2mE/h2 ) − i 2π k⊥ ¯    2 2  hκ( h¯ k  − 2mE) ¯ |z − z0 | exp − ik⊥ z0 − h¯ 2 k 2 − 2mE + h¯ h¯ 2 k 2 − 2mE − h¯ κ − i     ihκ(2mE − h¯ 2 k 2 ) ¯ 2 |z − z0 | 2 exp − ik⊥ z0 + i 2mE − h¯ k  . (21.40) + h¯ 2mE − h¯ 2 k 2 − ihκ ¯ Fourier transformation of Eq. (21.40) with respect to k⊥ yields finally     2 h( ¯ h¯ 2 k  − 2mE) |z − z |  exp − h¯ 2 k 2 − 2mE h¯ 2 h¯ 2 k 2 − 2mE    hκ |z − z0 | + |z − z0 | ¯ exp − h¯ 2 k 2 − 2mE + h¯ h¯ 2 k 2 − 2mE − h¯ κ − i z|G(k  , E)|z  =

    h(2mE − h¯ 2 k 2 ) ¯ 2 |z − z | 2 exp i 2mE − h¯ k  +i  h¯ 2 2mE − h¯ 2 k 2    ihκ ¯ 2 |z − z0 | + |z − z0 | 2 . (21.41) exp i 2mE − h¯ k  + h¯ 2mE − h¯ 2 k 2 − ihκ ¯

574

21 Dimensional Effects in Low-Dimensional Systems

The limit κ → 0 in Eqs. (21.40) and (21.41), as well as in Eq. (21.55) below reproduces the corresponding representations of the free retarded Green’s function in three dimensions. Our results describe the Green’s function for a particle in the presence of the thin quantum well, but for arbitrary energy and both near and far from the quantum well. Therefore we cannot easily identify any two-dimensional limit from the Green’s function. To explore this question further, we will look at the density of electron states in the presence of the quantum well. The quantum well at z0 breaks translational invariance in z direction, and we have with Eq. (21.21) (E, z) =

4m

m x  , z|G(E)|x  , z =  2 π h¯ π 3 h¯ 2

 d 2 k  z|G(k  , E)|z,

where a factor g = 2 was taken into account for spin 1/2 states. If there is any quasi two-dimensional behavior in this system, we would expect it in the quantum well region. Therefore we use the result (21.41) to calculate the density of states (E, z0 ) in the quantum well. Substitution yields (E, z0 ) =

m

  2

d 2 k  z0 |G(k  , E)|z0  π 3 h¯    ∞ m dk k δ = h¯ 2 k 2 − 2mE − h¯ κ π h¯ 0   √2mE/h¯ k 2mE − h¯ 2 k 2 m dk , + 2 (E) π h¯ 2mE − h¯ 2 k 2 + h¯ 2 κ 2 0

(21.42)

and after evaluation of the integrals, (E, z0 ) = (2mE + h¯ 2 κ 2 )κ + (E)

m π 2 h¯ 3



m π h¯ 2

√ 2mE − hκ ¯ arctan

√  2mE . hκ ¯

(21.43)

We can also express this in terms of the free two-dimensional and three-dimensional densities of electron states (cf. (21.22)),

 (E, z0 ) = κ d=2 E + (h¯ 2 κ 2 /2m)  √  2mE hκ ¯ + d=3 (E) 1 − √ arctan . h¯ κ 2mE

(21.44)

21.3 Problems

575

We note that the states which are exponentially suppressed perpendicular to the quantum well indeed contribute a term proportional to the two-dimensional density of states d=2 (E  ) with the kinetic energy E  of motion of particles along the quantum well, but with a dimensional proportionality constant κ which is the inverse penetration depth of those states. Such a dimensional factor has to be there, because densities of states in three dimensions enumerate states per energy and per volume, while d=2 (E  ) counts states per energy and per area. Furthermore, the unbound states yield a contribution which approaches the free three-dimensional density of states d=3 (E) in the limit κ → 0. The result can also be derived directly from the energy eigenstates (21.28)–(21.31) and the definition (12.42) of the local density of states, see Problem 21.8. However, the derivation from the Green’s function, while more lengthy for the pure quantum well, has the advantage to also work in the case of an interface in which the electrons move with different effective mass. The density of states in the quantum well region is displayed for binding energy B = h¯ 2 κ 2 /2m = 1 eV, mass m = me = 511 keV/c2 , and different energy ranges in Figs. 21.3 and 21.4.

21.3 Problems 21.1 Derive the d-dimensional version of Eq. (11.38) for scattering off spherically symmetric potentials. 21.2 Calculate the differential scattering cross sections for the potentials 21.2a V (r) = V0 (R − r), 21.2b V (r) = V0 exp(− r/R), 21.2c V (r) = V0 exp(− r 2 /R 2 ), in d dimensions in Born approximation. Which results do you find in particular for d = 2? 21.3 Derive an analog of the optical theorem in one dimension. Hint: The asymptotic wave function in one dimension is ψk (x) = [1 + (x)fk (0)] exp(ikx) + (− x)fk (π ) exp(− ikx),

(21.45)

and the total scattering cross section is σk = |fk (0)|2 + |fk (π )|2 . 21.4 The solution (3.48) can also be considered as the bound state in a onedimensional pointlike quantum dot V (x) = − Wδ(x), ψd=1 (x) = κ exp(− κ|x|), with binding energy

κ=

m h¯ 2

W,

(21.46)

576

21 Dimensional Effects in Low-Dimensional Systems

Fig. 21.3 The density of states in the quantum well location z = z0 for binding energy B = 1 eV, mass m = me = 511 keV/c2 , and energies −B ≤ E ≤ 3 eV. The red curve is the contribution from states bound inside the quantum well, the blue curve is the pure three-dimensional density of states in absence of a quantum well, and the black curve is the density of states according to Eq. (21.43)

B = −E =

m h¯ 2 κ 2 = 2 W 2. 2m 2h¯

(21.47)

No such states exist for higher-dimensional pointlike quantum dots V (x) = − Wδ(x), unless we also let the depth W go to zero in a judicious way. Show that the following wave functions describe bound states in two-dimensional and three-dimensional pointlike quantum dots if we let W go to zero, κ ψd=2 (r) = √ K0 (κr), π

 ψd=3 (r) =

κ exp(− κr) . 2π r

(21.48)

The following equations also hold for the binding energies in two and three dimensions, B = −E =

h¯ 2 κ 2 . 2m

(21.49)

21.3 Problems

577

Fig. 21.4 The density of states (21.43) in the quantum well location z = z0 for higher energies 0 ≤ E ≤ 100 eV. The binding energy, mass and color coding are √ the same as in Fig. 21.3. The full density of states (21.43) approximates the three-dimensional E behavior for energies E  B, but there remains a finite offset compared to d=3 due to the presence of the quantum well

Hint: Show that the bound states must be proportional to the energy-dependent retarded Green’s functions. Note that we cannot extend this construction to four or more dimensions because the corresponding Green’s functions are not square integrable any more. 21.5 Suppose we consider a proton and an electron in d ≥ 3 spatial dimensions. The electromagnetic interaction potential of these particles is Vd (r) = −

e2 Gd (r). 0

(21.50)

Suppose also that there are normalizable bound energy eigenstates in this system. Which relation between the expectation values K and V  of kinetic and potential energy would then be implied by the virial theorem (4.101)? Can atoms exist in d ≥ 4 dimensions? 21.6 According to Eq. (21.6), the Yukawa potential with screening length R in d spatial dimensions is

578

21 Dimensional Effects in Low-Dimensional Systems

r  W , V (r) = √ d √ d−2 K d−2 2 R 2π rR

(21.51)

where W is a coupling constant with dimensions of energy × lengthd−2 , e.g. W ∝ qQ if two particles interact through the Yukawa potential with charges q and Q. 21.6a Calculate the scattering amplitude for scattering in the potential (21.51) in Born approximation. Answer   d−3  W m k R2 d +1 f (k) = exp − iπ . 2π h¯ 2 4 2π 1 + (kR)2

(21.52)

21.6b Show that the potential (21.51) reduces to the d-dimensional Coulomb interaction potential V (r) = W Gd (r) in the limit R → ∞. The result (21.52) shows that the forward singularity for scattering angle θ → 0 in dσ/d = |f (k)|2 ∝ sin−4 (θ/2) holds in this form for Coulomb scattering in every number of dimensions. Nevertheless, it becomes integrable to a finite total cross section σ if d is sufficiently large. Use Eq. (J.43) to determine the minimal value of d for which dσ/d is integrable. Calculate the total Coulomb cross sections σ in Born approximation for the values of d where the result is finite. 21.7 Show that substitution of the Fourier transform  k  , k⊥ |G(E)|k  , k⊥ =



 d 2x

d 2 x 



 dz

dz

x  , z|G(E)|x  , z 

(2π )3 

   × exp i k  · x  + k⊥ z − k  · x  − k⊥ z 

 (21.53) δ k  − k  , = k⊥ |G(k  , E)|k⊥

with  k⊥ |G(k  , E)|k⊥ =

1 2π



 d 2x

dz z|G(x  , E)|z 

! % &"   × exp − i k  · x  + k⊥ z − k⊥ z

(21.54)

in Eq. (21.33) yields with the same technique that we used to solve (21.36) the result  k⊥ |G(E, k  )|k⊥ 

=

1 2 + k 2 − (2mE/h2 ) − i k⊥ ¯ 



 ) δ(k⊥ − k⊥

21.3 Problems

κ + π

579

  − k )z ] exp[i(k⊥ ⊥ 0 2 + k 2 − (2mE/h2 ) − i k⊥ ¯ 



h¯ 2 k 2 − 2mE (h¯ 2 k 2 − 2mE)  h¯ 2 k 2 − 2mE − hκ ¯ − i

 2mE − h¯ 2 k 2 (2mE − h¯ 2 k 2 )   . + 2mE − h¯ 2 k 2 − ihκ ¯

(21.55)

Show also that Fourier transformation yields again the result (21.40). 21.8 Derive the result (21.43) directly from the energy eigenstates (21.28), (21.30) and (21.31) for particles in the presence of the quantum well. Solution The decomposition of unity in terms of the eigenstates is  d 2 k  |k  , κk  , κ| +



 d 2k

±

∞ 0

dk⊥ |k  , k⊥ , ±k  , k⊥ , ±| = 1.

For the application of the definition (12.42) we have to take into account that d 2 k  = k dk dϕ →

m h¯ 2

dEdϕ

holds both for the two dimensional integration measure d 2 k  from E = h¯ 2 (k 2 − κ 2 )/2m, and also in the three-dimensional integration measure d 2 k  ∧ dk⊥ , where 2 )/2m. This yields with a factor of 4π from g = 2 for electrons and E = h¯ 2 (k 2 + k⊥ from integration over ϕ the result (E, z0 ) =

4π m



κ (2mE + h¯ 2 κ 2 ) 2 + (E) 4π

h¯ 2

√



2mE/h¯

2 dk k⊥ ⊥



2) 4π 3 (κ 2 + k⊥

0

.

Evaluation of the integral yields again the result (21.43). 21.9 Use the techniques from the previous problem to show that the local density of states on and off the interface can be written in the form (with z0 = 0 and g = 2 for spin), (E, z) = (2mE + h¯ 2 κ 2 ) √

 × 0

κm π h2



2mE/h¯

dk⊥

¯

exp(−2κ|z|) + (E)

m π 2 h¯ 2

 [k⊥ cos(k⊥ z) − κ sin(k⊥ |z|)]2 sin (k⊥ z) + . 2 κ 2 + k⊥ 2

21.10 Generalize the derivation of the relation (21.21) to the relativistic case.

580

21 Dimensional Effects in Low-Dimensional Systems

Solution The relativistic scalar Green’s function is G=

h¯ 2 , p2 + m2 c2 − i

k|G|k   =

δ(k − k  ) , k 2 + (mc/h) ¯ 2 − i

(21.56)

 see also (J.124)–(J.128). We can write this with H = c p2 + (mc)2 in the form h¯ 2 c2 h¯ 2 c2 G=− 2 = − 2E E − H 2 + i



 1 1 + . E − H + i E + H − i

(21.57)

Here E = cp0 is still an operator, but we can make the transition to the energydependent Green’s operator G(E) with classical variable E = hck ¯ 0 through |k = 0 |k ⊗ |k  and k 0 |G|k 0  = G(E)δ(k 0 − k 0 ).

(21.58)

Use of the Sokhotsky-Plemelj relation (2.26) yields π h¯ 2 c2 [δ(E − H ) − δ(E + H )] 2E  π h¯ 2 c2 = [δ(E − En ) − δ(E + En )] |n, νn, ν|, 2E n,ν

G(E) =

(21.59)

and therefore [183] x|G(E)|x =

" π h¯ 2 c2 ! (E) − (E) . 2E

(21.60)

Here (E) and (E) denote the densities of states of particles of energy E, and of anti-particles (or holes) of energy E = − E, respectively. We can test our result in the free (anti-)particle case where the density of states per helicity state is (E) ˆ = (E) + (E) =

 d−2 2(E 2 − m2 c4 ) E 2 − m2 c 4 , |E| √ d (2 π hc) ¯ (d/2)

(21.61)

see Eq. (12.60). The x-representation x|G(E)|x   = G(x − x  , ω) of the energy-dependent free Green’s function has been calculated in Appendix J, Eq. (J.91). The modified Bessel function Kν (z) with real argument is real, and the imaginary part of the Hankel function iHν(1) (z) for real z satisfies [1] lim Hν(1) (z) =

z→0

(z/2)ν . (ν + 1)

(21.62)

21.3 Problems

581

Substitution into (J.91) for r = |x − x  | → 0 yields x|G(E)|x =

d−2 π h¯ 2 c2 (E 2 − m2 c4 )  2 E − m2 c 4 , √ d (d/2) (2 π hc) ¯

in agreement with Eqs. (21.60) and (21.61).

(21.63)

Chapter 22

Relativistic Quantum Fields

The quantized Maxwell field provided us already with an example of a relativistic quantum field theory. On the other hand, the description of relativistic charged particles requires Klein-Gordon fields for scalar particles and Dirac fields for fermions. Relativistic fields are apparently relevant for high energy physics. However, relativistic effects are also important in photon-matter interactions, spectroscopy, spin dynamics, and for the generation of brilliant photon beams from ultra-relativistic electrons in synchrotrons and free electron lasers. Quasirelativistic effects from linear dispersion relations E ∝ p in materials like Graphene and in Dirac semimetals have also reinvigorated the need to reconsider the role of Dirac and Weyl equations in materials science. In applications to materials with quasirelativistic dispersion relations, c and m are replaced with effective velocity and mass parameters to describe cones or hyperboloids in regions of (E, k) space. We start our discussion of relativistic matter fields with the simpler Klein-Gordon equation and then move on to the more widely applicable Dirac equation. We will also discuss covariant quantization of photons, since this is more convenient for the calculation of basic scattering events than quantization in Coulomb gauge.

22.1 The Klein-Gordon Equation A limitation of the Schrödinger equation in the framework of ordinary quantum mechanics is its lack of covariance under Lorentz transformations.1 On the other

1 However,

we will see that in the second quantized formalism in the Heisenberg and Dirac pictures, the time evolution of the field operators is given by Heisenberg equations of motion, and the corresponding time evolution of states in the Schrödinger and Dirac pictures is given by corresponding Schrödinger equations with relativistic Hamiltonians.

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 R. Dick, Advanced Quantum Mechanics, Graduate Texts in Physics, https://doi.org/10.1007/978-3-030-57870-1_22

583

584

22 Relativistic Quantum Fields

hand, we have encountered an example of a relativistic wave equation in Chap. 18, viz. the inhomogeneous Maxwell equation % & ∂μ ∂ μ Aν − ∂ ν Aμ = − μ0 j ν .

(22.1)

This equation is manifestly covariant (or rather, form invariant) under Lorentz transformations because it is composed of quantities with simple tensorial transformation behavior under Lorentz transformations, and it relates a 4-vector ∂μ F μν to a 4vector j ν , such that the equation holds in this form in every inertial reference frame. Another, simple reasoning to come up with a relativistic wave equation goes as follows. We know that the standard Schrödinger equation for a free massive particle arises from the non-relativistic energy-momentum dispersion relation E = −cp0 = p2 /2m upon substitution of the classical energy-momentum vector through differential operators, pμ → −ih∂ ¯ μ . Following the same procedure in the relativistic dispersion relation −

E2 + p 2 + m2 c 2 = p 2 + m2 c 2 = 0 c2

(22.2)

yields the free Klein-Gordon equation [63, 96, 151]     1 ∂2 m2 c 2 m2 c 2 2 ∂ − 2 φ(x) =  − 2 2 − 2 φ(x) = 0. c ∂t h¯ h¯

(22.3)

Furthermore, the gauge principle or minimal coupling prescription ∂μ → Dμ = ∂μ − i(q/h)A ¯ μ yields the coupling of the charged Klein-Gordon field to electromagnetic potentials,    2 2 q q m2 c 2 ∂ − i A(x) − 2 φ(x) = ∇ − i A(x) h¯ h¯ h¯   2 q 1 ∂ m2 c 2 + i (x) − 2 φ(x) = 0. − 2 h¯ c ∂t h¯

(22.4)

Complex conjugation of Eq. (22.4) leads to the Klein-Gordon equation for a scalar field with charge − q. Therefore the charge conjugate Klein-Gordon field is simply gotten by complex conjugation, φ c (x) = φ ∗ (x).

(22.5)

The Klein-Gordon field is relevant in particle physics. E.g. π -mesons are described by Klein-Gordon fields as soon as their kinetic energy becomes comparable to their mass mc2  140 MeV, when relativistic effects have to be taken into account.

22.1 The Klein-Gordon Equation

585

Another important application of the Klein-Gordon field is the Higgs field for electroweak symmetry breaking in the Standard Model of particle physics. The Klein-Gordon field also provides a simple introduction into the relativistic quantum mechanics of charged particles. Therefore it is also useful as a preparation for the study of the Dirac field. We will focus in particular on the canonical quantization of freely evolving Klein-Gordon fields, since this describes Klein-Gordon operators in the practically relevant interaction picture representation. Conservation laws for full scalar quantum electrodynamics are discussed in Problems 22.6 and 22.7.

Mode Expansion and Quantization of the Klein-Gordon Field Fourier transformation of Eq. (22.3) yields the general solution of the free KleinGordon equation in k = (ω/c, k) space, φ(k) = φ(k, ω) = k|φ(ω)  " π ! a(k)δ(ω − ωk ) + b+ (− k)δ(ω + ωk ) , = ωk

(22.6)

where ωk is just the k space expression for the relativistic dispersion relation,  ωk = c k 2 + (m2 c2 /h¯ 2 ).

(22.7)

Frequency-time Fourier transformation (5.35) yields k|φ(t) = √

" 1 ! a(k) exp(− iωk t) + b+ (− k) exp(iωk t) 2ωk

(22.8)

and the general free Klein-Gordon wave function in x = (ct, x) space is 

d 3k

a(k) exp[i(k · x − ωk t)] √ 2ωk  + b+ (k) exp[− i(k · x − ωk t)] .

1 φ(x) = x|φ(t) = √ 3 2π

(22.9)

For the inversion of the Fourier transformation in the sense of solving for a(k) and b(k) we need Eq. (22.9) and 



ωk

− a(k) exp[i(k · x − ωk t)] 2  + b+ (k) exp[− i(k · x − ωk t)] .

i ˙ φ(x, t) = √ 3 2π

3

d k

(22.10)

586

22 Relativistic Quantum Fields

Inversion of both equations yields 

1

a(k) = √ 3 2π 1 = √ 3 2π 1

b(k) = √ 3 2π 1 = √ 3 2π

  

& d 3x % ˙ ωk φ(x, t) + iφ(x, t) exp[− i(k · x − ωk t)] √ 2ωk ↔ d 3x exp[− i(k · x − ωk t)] i ∂t φ(x, t), √ 2ωk

(22.11)

& d 3x % ωk φ + (x, t) + iφ˙ + (x, t) exp[− i(k · x − ωk t)] √ 2ωk ↔ d 3x exp[− i(k · x − ωk t)] i ∂t φ + (x, t). √ 2ωk

(22.12)

Here the alternating derivative is defined as ↔

f ∂t g = f

∂f ∂g − g. ∂t ∂t

(22.13)

Substituting (22.11) and (22.12) back into (22.9) and formal exchange of integrations yields  φ(x, t) =

↔

d 3 x  K(x − x  , t − t  ) ∂t  φ(x  , t  )

(22.14)

with the time evolution kernel for free scalar fields, 1 K(x, t) = (2π )3



d 3k exp(ik · x) sin(ωk t). ωk

(22.15)

This distribution satisfies the initial value problem2  ∂2 −

m2 c 2 h¯ 2

 K(x, t) = 0,

K(x, 0) = 0,

  ∂ K(x, t) = δ(x). ∂t t=0

(22.17)

2 At

first sight, Eq. (22.14) may appear puzzling because the left-hand side does not depend on the initial time t  whereas the right-hand side seems to depend on it. However, the fact that both K(x − x  , t − t  ) and φ(x  , t  ) satisfy the Klein-Gordon equation implies  ↔ (22.16) ∂t  d 3 x  K(x − x  , t − t  ) ∂t  φ(x  , t  ) = 0. The properties (22.17) also imply that ±(±t)K(x, t) yields retarded or advanced Green’s functions for propagation of driving terms of the Klein-Gordon equation, see Eqs. (J.95) and (J.98).

22.1 The Klein-Gordon Equation

587

For canonical quantization we need the Lagrange density for the complex KleinGordon field L = h¯ φ˙ + · φ˙ − h¯ c2 ∇φ + · ∇φ − 2 + = − hc ¯ ∂φ · ∂φ −

m2 c 4 + φ ·φ h¯

m2 c 4 + φ · φ, h¯

(22.18)

or the real Klein-Gordon field L=

hc h¯ c2 h¯ m2 c 4 2 m2 c 4 2 ¯ 2 φ˙ · φ˙ − ∇φ · ∇φ − φ =− (∂φ)2 − φ . 2 2 2h¯ 2 2h¯

(22.19)

In the following we will continue with the discussion of the complex Klein-Gordon field. Canonical quantization proceeds from (22.18) without any problems. The conjugate momenta "φ =

∂L = h¯ φ˙ + , ∂ φ˙

"φ + =

∂L ˙ = h¯ φ, ∂ φ˙ +

(22.20)

yield the canonical commutation relations in x space, [φ(x, t), φ˙ + (x  , t)] = iδ(x − x  ),

˙  , t)] = iδ(x − x  ), [φ + (x, t), φ(x

˙  , t)] = 0, [φ(x, t), φ(x

[φ(x, t), φ(x  , t)] = 0,

˙ ˙  , t)] = 0, [φ(x, t), φ(x

˙ [φ(x, t), φ˙ + (x  , t)] = 0,

[φ(x, t), φ + (x  , t)] = 0, (22.21)

and in k space, [a(k), a + (k  )] = δ(k − k  ), [b(k), b(k  )] = 0,

[a(k), a(k  )] = 0,

[a(k), b(k  )] = 0,

[b(k), b+ (k  )] = δ(k − k  ),

[a(k), b+ (k  )] = 0.

(22.22)

The Lagrangian for interacting Klein-Gordon and Maxwell fields is & % μ & m2 c 4 + c2 % + + μ φ ·φ h∂ ¯ μ φ + iqφ · Aμ · h¯ ∂ φ − iqA · φ − h¯ h¯ 1 Fμν F μν . (22.23) − 4μ0

L=−

588

22 Relativistic Quantum Fields

The Charge Operator of the Klein-Gordon Field The Klein-Gordon Lagrangian (22.18) is invariant under phase transformations 

 q φ(x) → φ (x) = exp i α φ(x), h¯ 

q δφ(x) = i αφ(x). h¯

(22.24)

According to Sect. 16.2 this implies a local conservation law (16.26) for a conserved charge Q. After cancelling the superfluous factor α, the charge following from (16.27) is (after normal ordering of the integrand in k space, see the remarks following Eqs. (18.55) and (18.56)) Q = −i

q h¯

= − iq  =q



 d 3x 

∂L ∂L · φ − φ+ · ∂ φ˙ ∂ φ˙ +



% & ˙ t) d 3 x φ˙ + (x, t) · φ(x, t) − φ + (x, t) · φ(x,

% & d 3 k a + (k)a(k) − b+ (k)b(k) .

(22.25)

↔

The charge density iqφ + ∂t φ is not positive definite, and therefore division of the charge density by q does not yield a probability density for the location of a particle, contrary to the Schrödinger field. Lack of a single-particle interpretation is a generic property of relativistic fields which we had also encountered for the Maxwell field.

Hamiltonian and Momentum Operators for the Klein-Gordon Field The invariance of the Klein-Gordon Lagrangian (22.18) under constant translations x μ → x μ = x μ + δx μ

(22.26)

implies a local conservation law (16.30) with corresponding conserved Hamilton and momentum operators (16.32). This yields the following expressions for energy and momentum of Klein-Gordon fields, m2 c 4 + φ · φ, h¯   & % + + H = d 3 x H = d 3 k hω ¯ k a (k)a(k) + b (k)b(k) , 2 + H = − 0 0 = h¯ φ˙ + · φ˙ + hc ¯ ∇φ · ∇φ +

(22.27) (22.28)

22.1 The Klein-Gordon Equation

P=

589

1 ∂L ∂L e i i 0 = − · ∇φ − ∇φ + · c ∂ φ˙ ∂ φ˙ +

+ ˙ = − h¯ φ˙ + · ∇φ − h∇φ · φ, ¯   % & + + P = d 3 x P = d 3 k hk ¯ a (k)a(k) + b (k)b(k) .

(22.29) (22.30)

The commutation relations and the charge operator (22.25), the Hamilton operator (22.28), and the momentum operators (22.30) imply that the operator a + (k) creates a particle of momentum hk, ¯ energy h¯ ωk and charge q, while b+ (k) creates a particle of momentum hk, ¯ energy hω ¯ k and charge − q. The operators (22.9) and a(k, t) = a(k) exp(− iωk t),

a + (k, t) = a + (k) exp(iωk t),

(22.31)

are the field operators in the Dirac picture, or the free field operators in the Heisenberg picture. They satisfy the Heisenberg evolution equations i ∂ a(k, t) = [H, a(k, t)], ∂t h¯

i ∂ φ(x, t) = [H, φ(x, t)], ∂t h¯

(22.32)

with the free Hamiltonian (22.28). The corresponding integrals follow in the standard way,  a(k, t) = exp  φ(x, t) = exp

   i i H t a(k) exp − H t , h¯ h¯

(22.33)

   i i H t φ(x) exp − H t , h¯ h¯

(22.34)

etc. In the Schrödinger picture theory, this amounts to operators a(k), φ(x), and time evolution of the states ih¯

d |(t) = H |(t) dt

(22.35)

with the free Hamiltonian (22.28) for free states or a corresponding minimally coupled Hamiltonian which follows from (22.23) for interacting states, see Problem 22.6. This is the statement that we have Heisenberg and Schrödinger type evolution equations also in relativistic quantum field theory. The Klein-Gordon equation also follows from the iterated Heisenberg equation, ∂2 1 φ(x, t) = − 2 [H, [H, φ(x, t)]], 2 ∂t h¯ cf. (18.64) for photons.

(22.36)

590

22 Relativistic Quantum Fields

Non-relativistic Limit of the Klein-Gordon Field We have in the non-relativistic limit ωk = c k 2 +

m2 c 2 h¯ 2



2 hk mc2 ¯ , + 2m h¯

(22.37)

 √ 2. and therefore in leading order also 1/ 2ωk  h/2mc ¯ + Suppose that the k-space amplitudes a(k) and b (k) are negligibly small unless h|k| % mc. In this case we can approximate Eq. (22.9) by ¯ 



   h¯ k 2 mc2 φ(x, t)  √ 3 t d k a(k) exp ik · x − i t exp − i h¯ 2m 2π    2 mc2 hk ¯ + t exp i + b (k) exp − ik · x + i t . (22.38) 2m h¯ 1



h¯ 2mc2 



3

However, this expression automatically contains two fields 1



ψ(x, t) = √ 3 2π



2 hk ¯ t d k a(k) exp i k · x − 2m



3

(22.39)

and 1

ϕ(x, t) = √ 3 2π





2 hk ¯ t d k b(k) exp i k · x − 2m



3

,

(22.40)

which satisfy the free Schrödinger equation, i.e. the complex Klein-Gordon field will reduce to a Schrödinger field ψ(x, t) if the k-space amplitudes also satisfy |a(k)|  |b+ (k)|. Substitution of the remaining approximation  φ(x, t) 

  mc2 h¯ t ψ(x, t) exp − i h¯ 2mc2

(22.41)

into the charge, current, energy and momentum densities of the Klein-Gordon field yields the corresponding expressions for the Schrödinger field for the electric charges and currents, % & = − iq φ˙ + · φ − φ + · φ˙  qψ + ψ, % & & h¯ % + ψ · ∇ψ − ψ · ∇ψ + , j = iqc2 ∇φ + · φ − φ + · ∇φ  q 2im

(22.42) (22.43)

22.1 The Klein-Gordon Equation

591

and the energy-momentum densities,3 m2 c 4 + h¯ 2 φ ·φ  ∇ψ + · ∇ψ + mc2 ψ + · ψ, h¯ 2m & m h¯ % + + ˙ P = − h¯ φ˙ + · ∇φ − h∇φ ψ · ∇ψ − ψ · ∇ψ + = j . (22.45) ·φ  ¯ 2i q 2 + H = h¯ φ˙ + · φ˙ + hc ¯ ∇φ · ∇φ +

Furthermore, the free Klein-Gordon equation (22.3) becomes with 1 ∂2 φ(x, t)  c2 ∂t 2



   m2 c 2 mc2 2m ∂ h¯ t − ψ(x, t) exp − i ψ(x, t) − i h¯ h¯ ∂t 2mc2 h¯ 2

the free Schrödinger equation ih¯

∂ h¯ 2 ψ(x, t) = − ψ(x, t), ∂t 2m

(22.46)

as it should, because we have already observed in the derivation of (22.41) that ψ(x, t) satisfies the free Schrödinger equation. For the non-relativistic limit of the real Klein-Gordon field we find     h¯ mc2 t φ(x, t)  ψ(x, t) exp − i h¯ 2mc2   mc2 + + ψ (x, t) exp i t , (22.47) h¯ but we have to include subleading terms from h¯ φ˙ 2 /2 and m2 c4 φ 2 /2h¯ in the calculation of the non-relativistic limit of H to find that remnant fast oscillation terms proportional to exp(±2imc2 t/h) ¯ cancel, i.e. we cannot simply use (22.47) and apply the same shortcut as for the complex Klein-Gordon field, but we actually have to use (22.44) in the approximation for the field φ to calculate the non-relativistic limit of H . The best way in this case is to study the non-relativistic limit of the

3 We

take a common shortcut here for the Hamiltonian density H by considering only the leading terms from each of the three contributions to H. This is strictly speaking not correct, because the contribution from h¯ c2 |∇φ|2 is subleading compared to the leading order mc2 contributions from the other two terms, which contain subleading terms of the same order as the leading term ˙ 2 and m2 c4 |φ|2 /h¯ cancel each other. from h¯ c2 |∇φ|2 . However, those subleading terms from h¯ |φ| Calculating those subleading terms properly requires to use    1 h¯ h¯ 2 k 2 1 − (22.44)  √ 2mc2 4m2 c2 2ωk in the approximation (22.39) and (22.41).

592

22 Relativistic Quantum Fields

momentum space representations for energy and momentum, and see that this limit yields exactly the same results as for the Schrödinger field (22.39).

22.2 Klein’s Paradox The commutation relations for the field operators a(k) and b+ (k) imply that the operator φ(x, t) (22.9) describes both particles and anti-particles simultaneously, and therefore the Klein-Gordon equation cannot support a single-particle interpretation. This is also obvious from the charge operator (22.25) and the corresponding lack of a conserved probability density for Klein-Gordon particles. Klein’s paradox provides a particularly neat illustration of the failure of single particle interpretations of relativistic wave equations. Klein observed that using relativistic quantum fields to describe a relativistic particle running against a potential step yields results for the transmission and reflection probabilities which are incompatible with a single-particle interpretation.4 This observation can be explained by pair creation in strong fields and the fact that relativistic fields describe both particles and anti-particles simultaneously. We will explain Klein’s paradox for the Klein-Gordon field. In the following we can neglect the y and z coordinates and deal only with the x and t coordinates. We are interested in a scalar particle of charge q scattered off a potential step of height V > 0. The step is located at x = 0, and can be implemented through an electrostatic potential (x), V (x) = q(x) = − cqA0 (x) = V (x).

(22.48)

Minimal coupling then yields the free Klein-Gordon equation for x < 0, and5 2 2 2 2 2 4 (h∂ ¯ t + iV ) φ − h¯ c ∂x φ + m c φ = 0

(22.49)

for x > 0. A monochromatic solution without any apparent left moving component for x > 0 is (after omission of an irrelevant constant prefactor) ! φ(x, t) =

4 Klein

" exp(ikx) + β exp(− ikx) exp(− iωt), x < 0 θ exp[i(κx − ωt)], x > 0.

(22.50)

[97] actually discussed reflection and transmission of relativistic spin 1/2 fermions which are described by the Dirac equation (22.71). 5 We cannot try to discuss motion of particles of mass m in the presence of a potential by simply & % including a scalar potential term in the form h¯ 2 ∂t2 − h¯ 2 c2 ∂x2 + m2 c4 φ = (x)V 2 φ in the Klein Gordon equation. This would correspond to a local mass M(x)c2 = m2 c4 − (x)V 2 rather than to a local potential, and yield tachyons in x > 0 for V 2 > m2 c4 .

22.2 Klein’s Paradox

593

The frequency follows from the solution of the Klein-Gordon equation in the two domains, ω = c k2 +

m2 c 2 h2 ¯

=

V m2 c 2 ± c κ2 + 2 . h¯ h¯

(22.51)

It has to be the same in both regions for continuity of the wave function at x = 0. The sign in the last equation of (22.51) depends on the sign of hω ¯ − V . We apparently have to use the minus sign if and only if hω ¯ − V < 0. Note that in our solution we always have h¯ ω ≥ mc2 . Solving Eq. (22.51) for κ yields (hω ¯ − V )2 − m2 c2 ∈ R, c2

2 2 4 (hω ¯ −V) > m c ,

(22.52)

i (hω ¯ − V )2 m2 c 2 − ∈ iR+ , h¯ c2

2 2 4 (hω ¯ −V) < m c .

(22.53)

κ=± κ=

1 h¯

However, we have to be careful with the sign in (22.52). The group velocity in x > 0 for h¯ ω + mc2 < V (i.e. for the negative sign in (22.51)) is hκ dω ¯ = −c , 2 2 dκ h¯ κ + m2 c2

(22.54)

i.e. we have to take the negative root for κ for V > hω ¯ + mc2 to ensure positive group velocity in the region x > 0. We can collect the results for κ in the equations V < h¯ ω − mc2 :

κ=

h¯ ω − mc2 < V < hω ¯ + mc2 : κ = V > h¯ ω + mc2 :

κ=



1 (h¯ ω−V )2 − m2 c2 ∈ R+ , h¯  c2 )2 i m2 c2 − (h¯ ω−V ∈ iR+ , h¯  c2 1 (h¯ ω−V )2 2 − h¯ − m c2 ∈ R− . c2

(22.55)

The current density j = iqc2 (∂x φ + · φ − φ + · ∂x φ) is & % j = 2qc2 k 1 − |β|2 , x < 0, x > 0, κ ∈ R, j = 2qc2 κ|θ |2 , j = 0, x > 0, κ ∈ iR.

(22.56)

Note that in x > 0 we have j/q < 0 if V > hω ¯ +mc2 , in spite of the fact of positive group velocity in the region. Since charges q cannot move to the left in x > 0, this means that the negative value of j/q in x > 0 for V > hω ¯ + mc2 must correspond to right moving charges −q. We will see that this arises as a consequence of the generation of anti-particles near the potential step for V > hω ¯ + mc2 .

594

22 Relativistic Quantum Fields

Table 22.1 Reflection and transmission for different relations between height V of the potential step and energy h¯ ω of the incident particle −∞ < V ≤ 0 0 ≤ V ≤ hω ¯ − mc2 2 hω ¯ − mc < V < h¯ ω + mc2 hω ¯ + mc2 ≤ V ≤ 2h¯ ω 2h¯ ω ≤ V < ∞

∞>κ≥k k≥κ≥0 κ ∈ iR+ 0 ≥ κ ≥ −k −k ≥ κ > −∞

1>R≥0 0≤R≤1 R=1 1≤R≤∞ ∞≥R>1

0 1 for V > hω ¯ + mc2 ≥ 2mc2 , 2 recall that the solution for V > h¯ ω + mc in x > 0 has κ < 0. If we write the solution as φ(x, t) = θ exp[− i(−κx + ωt)],

x > 0,

(22.61)

and compare with the anti-particle contribution to the free solution (22.9), we recognize the solution in the region x > 0 as an anti-particle solution with momentum hκ ¯ > 0 and energy ¯  = −hκ E p = − hω ¯ < 0,

mc2 − V ≤ E p ≤ −mc2 .

(22.62)

This is acceptable, because the anti-particle has charge −q and therefore experiences a potential U = −V in the region x > 0. Further support for this energy assignment for the anti-particles comes from the equality for the kinetic+rest energy of the anti-particles,

22.3 The Dirac Equation

595

 Kp = c h¯ 2 κ 2 + m2 c2 = V − h¯ ω,

mc2 ≤ Kp ≤ V − mc2 .

(22.63)

We expect Ep = Kp − V at least in the non-relativistic limit for the anti-particles. The anti-particles move to the right, d(−ω)/d(−κ) > 0, and yield a negative particle current density j/q ∝ −qκ  /q = −κ  < 0 due to the opposite charge. We therefore get R > 1 and T < 0 for V − hω ¯ > mc2 due to pair creation. The generated particles move to the left because they are repelled by the potential V > mc2 + hω. ¯ They add to the reflected particle in x < 0 to generate a formal reflection coefficient R > 1. The anti-particles move to the right because they can only move in the attractive potential −V in x > 0. The movement of charges −q to the right generates a negative apparent transmission coefficient T = jx>0 /jin < 0. Please note that the last two lines in Table 22.1 do not state that extremely large potentials V  2mc2 are less efficient for pair creation. They only state that a potential V > 2mc2 is particularly efficient for generation of particle–anti-particle pairs with energies Ep = Kp = −Ep = hω ¯ = V /2. The conclusion in a nutshell is that if we wish to calculate scattering in the potential V > 2mc2 for incident particles with energies in the pair creation region mc2 ≤ hω ¯ ≤ V − mc2 , then the ongoing pair creation will yield the seemingly paradoxical results R > 1 and T = 1 − R < 0, see Fig. 22.1. A more satisfactory discussion of energetics of the problem would also have to take into account the dynamics of the electromagnetic field  = V /q, and then use the Hamiltonian density (22.248) of quantum electrodynamics with scalar matter. This would also imply an additional energy cost for separating the oppositely charged particles and anti-particles. The potential V would therefore decay due to pair creation until it satisfies the condition V ≤ 2mc2 , when pair creation would seize and the standard single-particle results 0 ≤ T = 1 − R ≤ 1 apply for incident particles with any energy, or the potential would have to be maintained through an external energy source.

22.3 The Dirac Equation We have seen in Eq. (22.25) that the conserved charge of the complex Klein-Gordon field does not yield a conserved probability, and therefore has no single-particle interpretation. This had motivated Paul Dirac in 1928 to propose a relativistic wave equation which is linear in the derivatives,6 μ ihγ ¯ ∂μ (x) − mc(x) = 0.

6 Dirac’s

(22.64)

[40] relativistic wave equation was a great success, but like every relativistic wave equation, it also does not yield a single-particle interpretation. It immediately proved itself by explaining the anomalous magnetic moment of the electron and the fine structure of spectral lines, and by predicting positrons.

596

22 Relativistic Quantum Fields

V

V−mc 2 V−hf hf Particle energy

Energy region for particles from pair creation

mc 2

x

−mc 2 Energy region for anti−particles from pair creation −hf

mc 2 −V

Anti−particle kinetic+rest energy V−hf

−V Fig. 22.1 Particles of charge q experience the potential V for x > 0, while anti-particles with charge −q experience the potential −V . If the potential satisfies V > 2mc2 , it can produce particles with energy Ep , mc2 ≤ Ep = hf ≤ V − mc2 , in the region x < 0 and anti-particles with energy Ep = −hf , mc2 − V ≤ Ep ≤ −mc2 , in the region x > 0. This corresponds to a kinetic+rest energy Kp = V − hf , mc2 ≤ Kp ≤ V − mc2 , see Eq. (22.63). Pair creation is most efficient for hf = Ep = Kp = −Ep = V /2

Since the relativistic dispersion relation p2 + m2 c2 = 0 implies that the field  should still satisfy the Klein-Gordon equation, Eq. (22.64) should imply the KleinGordon equation. Applying the operator ihγ ¯ μ ∂μ + mc yields − h¯ 2 γ μ γ ν ∂μ ∂ν (x) − m2 c2 (x) = 0.

(22.65)

This is the Klein-Gordon equation if the coefficients γ μ can be chosen to satisfy {γ μ , γ ν } = − 2ημν .

(22.66)

22.3 The Dirac Equation

597

In four dimensions, Eq. (22.66) has an up to equivalence transformations unique solution in terms of (4 × 4)-matrices (see Appendix G for the relevant proofs and for the construction of γ matrices in d spacetime dimensions). The Dirac basis for γ matrices is  γ0 =

 −1 0 , 0 1

 γi =

 0 σi , −σ i 0

(22.67)

where the (4×4)-matrices are expressed in terms of (2×2)-matrices. Another often used basis is the Weyl basis:  γ0 =

 0 1 , 1 0

 γi =

 0 σi . −σ i 0

(22.68)

The two bases are related by the orthogonal transformation μ γW

1 = 2

μ γD

1 = 2





1 1 −1 1 1 −1 1 1



 1 −1 , 1 1

(22.69)

 1 1 . −1 1

(22.70)

 μ · γD

·



 μ · γW

·

The Dirac equation with minimal photon coupling, γ μ (ih∂ ¯ μ + qAμ )(x) − mc(x) = 0,

(22.71)

follows from the Lagrange density of quantum electrodynamics, ! " 1 L = c γ μ (ih∂ Fμν F μν , ¯ μ + qAμ ) − mc  − 4μ0

 =  +γ 0.

(22.72)

The conserved current density for the phase invariance   q  = exp i α  h¯ 

(22.73)

is j μ = cqγ μ ,

= j 0 /c = q + ,

j = cqγ .

(22.74)

Variation of (22.72) with respect to the vector potential shows that j μ appears as the source term in Maxwell’s equations, ∂μ F μν = − μ0 j ν .

(22.75)

598

22 Relativistic Quantum Fields

Solutions of the Free Dirac Equation We temporarily set h¯ = 1 and c = 1 for the construction of the general solution of the free Dirac equation. Substitution of the Fourier ansatz  (x) =

d 4p (p) exp(ip · x) (2π )2

(22.76)

into (22.64) yields the equation (γ μ pμ + m)(p) = 0.

(22.77)

We can use any representation of the γ matrices to find det(γ μ pμ + m) = (m2 + p2 )2 = (m2 + p 2 − E 2 )2 = (E 2 (p) − E 2 )2 ,

(22.78)

i.e. the solutions of (22.77) must have the form  (p) = with E(p) =

π u(p)δ(E − E(p)) + E(p)





π v(− p)δ(E + E(p)) E(p)

(22.79)

p2 + m2 and   γ · p − γ 0 E(p) + m · u(p) = 0,

(22.80)

  γ · p + γ 0 E(p) + m · v(− p) = 0.

(22.81)

The normalization factors in (22.79) are included for later convenience when we quantize the Dirac field. To find the eigenspinors u(p), v(− p), we observe (γ μ pμ + m)(m − γ μ pμ ) = m2 + p2 ,

(22.82)

i.e. the columns ζi+ (p) of the matrix (m − γ μ pμ )E=E(p) solve Eq. (22.80) while the columns ζi− (p) of the matrix (m − γ μ pμ )E=−E(p) solve Eq. (22.81). However, only two columns of each of the two matrices ζi± (p) are linearly independent. We initially use a Dirac basis (22.67) for the γ matrices. A suitable basis for the general solution of the free Dirac equation is then given by the spin basis in the Dirac representation, u(p, 12 ) = u↑ (p) = √

1 ζ1+ (p) E(p) + m

22.3 The Dirac Equation

599

⎞ E(p) + m ⎟ ⎜ 1 0 ⎟, ⎜ = √ ⎠ ⎝ p3 E(p) + m p+ ⎛

1 ζ2+ (p) E(p) + m ⎞ ⎛ 0 ⎜ E(p) + m ⎟ 1 ⎟, ⎜ = √ p− ⎠ E(p) + m ⎝

(22.83)

u(p, − 12 ) = u↓ (p) = √

(22.84)

− p3

1 ζ3− (p) E(p) + m ⎞ ⎛ − p3 ⎜ − p+ ⎟ 1 ⎟, ⎜ = √ E(p) + m ⎝ E(p) + m ⎠

v(− p, − 12 ) = v↓ (− p) = √

(22.85)

0

1 ζ4− (p) E(p) + m ⎞ ⎛ − p− ⎟ ⎜ 1 p3 ⎟, ⎜ = √ ⎠ 0 E(p) + m ⎝

v(− p, 12 ) = v↑ (− p) = √

(22.86)

E(p) + m

where p± = p1 ± ip2 was used. The spin labels indicate that u(p, ± 12 ) describes spin up or down particles, while v(p, ± 12 ) describes spin up or down anti-particles. It is also convenient to express the 4-spinors (22.83)–(22.86) in terms of the 2spinors χ↑ =

  1 , 0

χ↓ =

  0 , 1

(22.87)

in the form 1 u↑ (p) = √ E(p) + m



 (E(p) + m)χ↑ , (p · σ ) · χ↑

(22.88)

600

22 Relativistic Quantum Fields



 (E(p) + m)χ↓ , (p · σ ) · χ↓   1 (p · σ ) · χ↑ , v↓ (p) = √ E(p) + m (E(p) + m)χ↑   1 (p · σ ) · χ↓ . v↑ (p) = √ E(p) + m (E(p) + m)χ↓

1 u↓ (p) = √ E(p) + m

(22.89) (22.90) (22.91)

The general solution of the free Dirac equation then has the form 

! d 3p bs (p)u(p, s) exp(ip · x) 2E(p) s∈{↓,↑} " + ds+ (p)v(p, s) exp(− ip · x) , 1

(x) = √ 3 2π

√

(22.92)

where p0 = E(p) is understood: p · x = p · x − E(p)t. Calculations involving 4-spinors are often conveniently carried out with h¯ = 1 and c = 1, and restoration of the constants is usually only done in the final results from the requirement of correct units. For completeness I would also like to give the general solution of the free Dirac equation with the constants h¯ and c restored. We can choose the basic spinors (22.83)–(22.86) to have units of square roots of energy, e.g. ⎛

⎞ E(k) + mc2 ⎜ ⎟ 1 0 ⎜ ⎟, u↑ (k) =  ⎝ ⎠ h¯ ck3 E(k) + mc2 hck ¯ +

(22.93)

and the solution (22.92) is 

! d 3k bs (k)u(k, s) exp(ik · x) 2E(k) s∈{↓,↑} " + ds+ (k)v(k, s) exp(− ik · x) 1

(x) = √ 3 2π

√

(22.94)

with k · x ≡ k · x − ω(k)t. In these conventions the Dirac field has the same dimensions length−3/2 as the Schrödinger field. The free field (x) also describes the freely evolving field operator D (x) in the interaction picture. Some useful algebraic properties of the spinors (22.83)–(22.86) are frequently used in the calculations of cross sections and other observables, u+ (k, s) · u(k, s  ) = 2E(k)δss  ,

v + (k, s) · v(k, s  ) = 2E(k)δss  ,

(22.95)

22.3 The Dirac Equation

601

u+ (k, s) · v(− k, s  ) = 0, u(k, s) · u(k, s  ) = 2mc2 δss  , u(k, +) · v(− k, −) = − 2cp3 , u(k, −) · v(− k, −) = − 2cp+ ,

u(k, s) · v(k, s  ) = 0,

(22.96)

v(k, s) · v(k, s  ) = − 2mc2 δss  ,

(22.97)

u(k, +) · v(− k, +) = − 2cp− ,

(22.98)

u(k, −) · v(− k, +) = 2cp3 .

(22.99)

The following equations contain 4 × 4 unit matrices 1 on the right-hand sides,

u(k, s)u+ (k, s) +



s

v(− k, s)v + (− k, s) = 2E(k)1,

(22.100)

s



  u(k, s)u(k, s) = mc2 1 − cγ μ pμ 

s



cp0 =E(k)

  u(− k, s)u(− k, s) = mc2 1 + cγ μ pμ 

s



  v(k, s)v(k, s) = − mc2 1 − cγ μ pμ 

s



(22.101)

,

cp0 =−E(k)

cp0 =E(k)

  v(− k, s)v(− k, s) = − mc2 1 + cγ μ pμ 

s

(22.102)

,

(22.103)

,

cp0 =−E(k)

.

(22.104)

It is actually clumsy to write down unit matrices when their presence is clear from the context, and the action e.g. of the scalar mc2 on a 4-spinor  has the same effect as the matrix mc2 1. Therefore we will usually adopt the practice of not writing down 4 × 4 unit matrices explicitly. Equations (22.95) and (22.96) are used e.g. in the inversion of the Fourier representation (22.94), 1 bs (k) = √ 3 2π 1 ds (k) = √ 3 2π

 

d 3x exp(− ik · x)u+ (k, s) · (x), √ 2E(k)

(22.105)

d 3x exp(− ik · x) + (x) · v(k, s). √ 2E(k)

(22.106)

Substituting these equations back into (22.94) yields  (x, t) =

d 3 x  W(x − x  , t − t  ) · (x  , t  )

with the time evolution kernel

(22.107)

602

22 Relativistic Quantum Fields

W(x, t) =

1 (2π )3



d 3k exp(ik · x) [u(k, s)u+ (k, s) exp(− iω(k)t) 2E(k) s

+ v(− k, s)v + (− k, s) exp(iω(k)t)]  3 1 d k exp(ik · x)[E(k) cos(ω(k)t) = (2π )3 E(k) 0 + ic(hγ ¯ · k − mc)γ sin(ω(k)t)].

(22.108)

This satisfies the initial value problem μ (ihγ ¯ ∂μ − mc)W(x, t) = 0,

W(x, 0) = δ(x).

(22.109)

It is related to the time evolution kernel (22.15) of the Klein-Gordon field through   mc K(x, t). iW(x, t)γ = c iγ · ∂ + h¯ 0

(22.110)

It is sometimes useful to express equation (22.92) and the corresponding equation in k space in bra-ket notation, similar to Eqs. (18.37) and (18.38) for the Maxwell field. With the definitions b+,s (k) = bs (k),

b−,s (k) = ds+ (− k),

(22.111)

u+,s (k) = us (k),

u−,s (k) = vs (− k),

(22.112)

we can write the free Dirac field in the forms k, σ, s|(t) = bσ,s (k) exp[− iσ ω(k)t]

(22.113)

and  x, a|(t) =

d 3k √ 3 2π



bσ,s (k)uaσ,s (k) exp[i(k · x − σ ω(k)t)], √ 2hω(k) ¯ σ ∈{+,−} s∈{↑,↓}

where a ∈ {1, . . . 4} is a Dirac spinor index, σ ∈ {+, −} labels particles (+) or anti-particles (−), and s is the spin label. Equations (22.95) and the first equation in (22.96) are u+ ¯ σ,σ  δs,s  . σ,s (k) · uσ  ,s  (k) = 2hω(k)δ

(22.114)

uσ,s (k) · uσ,s  (k) = 2mc2 σ δss  ,

(22.115)

Equation (22.97) is

22.3 The Dirac Equation

603

and Eq. (22.100) is

uσ,s (k)u+ ¯ σ,s (k) = 2hω(k)1.

(22.116)

σ,s

The x representations of the spinor momentum eigenstates are x, a|k, σ, s =

exp(ik · x) a uσ,s (k), √ 4π π h¯ ω(k)

(22.117)

and using Eqs. (22.114) and (22.116) we can easily verify the relations k, σ, s|k  , σ  , s   = δσ,σ  δs,s  δ(k − k  ),

(22.118)

x, a|x  , a   = δa,a  δ(x − x  ).

(22.119)

Charge Operators and Quantization of the Dirac Field We can apply the results from Sect. 16.2 to calculate the energy and momentum operator for the Dirac field. The free Dirac Lagrangian % μ & L = c ihγ ¯ ∂μ − mc 

(22.120)

yields the positive definite normal ordered Hamiltonian  H =

d 3 x c(x, t) (mc − ihγ ¯ · ∇) (x, t) 

=

d 3 k h¯ ω(k)

! " bs+ (k)bs (k) + ds+ (k)ds (k) ,

(22.121)

s∈{↓,↑}

but only if we assume anti-commutation properties of the ds and ds+ operators. The normal ordered momentum operator is then 

h¯ d 3 x  + (x, t) ∇(x, t) i  ! " = d 3 k hk bs+ (k)bs (k) + ds+ (k)ds (k) . ¯

P =

s∈{↓,↑}

The electromagnetic current density (22.74) yields the charge operator  Q=q

d 3 x  + (x, t)(x, t)

(22.122)

604

22 Relativistic Quantum Fields

 =q

d 3k

! " bs+ (k)bs (k) − ds+ (k)ds (k) .

(22.123)

s∈{↓,↑}

The normalization in Eq. (22.94) has been chosen such that the quantization condition {α (x, t), β + (x  , t)} = δαβ δ(x − x  )

(22.124)

for the components of (x) yields {b(k, s), b+ (k  , s  )} = δss  δ(k − k  ),

{d(k, s), d + (k  , s  )} = δss  δ(k − k  ),

with the other anti-commutators vanishing. Equations (22.121)–(22.123) then imply that the operator b+ (k, s) creates a fermion of mass m, momentum h¯ k and charge q, while d + (k, s) creates a particle with the same mass and momentum, but opposite charge −q. For an explanation of the spin labels of the spinors u(k, ± 12 ), we notice that the spin operators corresponding to the rotation generators Mi = − iLi =

1 ij k Mj k 2

(22.125)

are both in the Dirac and in the Weyl representation given by h¯ h¯ ih¯ Si = ij k Sj k = ij k γj γk = 2 4 2



 σi 0 , 0 σi

(22.126)

see Appendix H for an explanation of generators of Lorentz boosts and rotations for Dirac spinors. Equation (22.126) implies that the rest frame spinors u(0, ± 12 ) transform under rotations around the z axis as spinors with z-component of spin h¯ s = ±h/2. ¯ For an explanation of the spin labels of the spinors v(p, ± 12 ), we have to look at charge conjugation. Both in the Dirac and the Weyl representation of γ matrices we have γμ∗ = γ2 γμ γ2 .

(22.127)

Therefore complex conjugation of the Dirac equation [iγ μ ∂μ + qγ μ Aμ (x) − m](x) = 0,

(22.128)

followed by multiplication with iγ2 from the left yields [iγ μ ∂μ − qγ μ Aμ (x) − m] c (x) = 0

(22.129)

22.4 The Energy-Momentum Tensor for Quantum Electrodynamics

605

with the charge conjugate field  c (x) = iγ2  ∗ (x).

(22.130)

v c (k, 12 ) = iγ2 v ∗ (k, 12 ) = u(k, 12 )

(22.131)

In particular, we have

and v c (k, − 12 ) = iγ2 v ∗ (k, − 12 ) = − u(k, − 12 ),

(22.132)

i.e. the negative energy spinors for charge q, momentum h¯ k and spin projection hs ¯ correspond to positive energy spinors for charge −q, momentum h¯ k and spin projection h¯ s. Please note that Eq. (22.130) uses the property (22.127), which holds in the Dirac and Weyl representations, and in all representations which are related to the Dirac and Weyl representations through real orthogonal transformations, γμ = Rγμ R T , RR T = 1 (recall that the Dirac and Weyl bases are related through an orthogonal transformation (22.70)). If we switch to any other representation of γ matrices with a unitary transformation γμ = U · γμ · U + , then Eq. (22.130) generalizes to  c (x) = − iC ∗ ·  ∗ (x),

(22.133)

with C = U ∗ γ2 U + ,

C −1 = C ∗ = − U γ2 U T ,

(22.134)

see Problem 22.21.

22.4 The Energy-Momentum Tensor for Quantum Electrodynamics We use the symmetrized form of the QED Lagrangian (22.72),     1 ¯ ↔ μ ih ∂μ + qAμ − mc  − Fμν F μν . L = c γ 2 4μ0 This yields according to (16.31) a conserved energy-momentum tensor μ ν = ημ ν L − ∂μ 

∂L ∂(∂ν )

− ∂μ 

∂L ∂L − ∂μ Aλ ∂(∂ν ) ∂(∂ν Aλ )

(22.135)

606

22 Relativistic Quantum Fields

= ημ −

ν

      1 ¯ ↔ λ ih κλ ∂λ + qAλ − mc  − Fκλ F c γ 2 4μ0

↔ 1 ih¯ cγ ν ∂μ  + ∂μ Aλ F νλ . 2 μ0

(22.136)

According to the results of Sect. 16.2, this yields on-shell conserved charges, i.e. we can use the equations of motion to simplify this expression. The Dirac equation then implies μ ν = −

↔ ih¯ 1 1 cγ ν ∂μ  + ∂μ Aλ F νλ − ημ ν Fκλ F κλ . 2 μ0 4μ0

(22.137)

We can also add the identically conserved improvement term −

% & 1 1 1 ∂λ Aμ F νλ = − ∂λ Aμ F νλ − Aμ ∂λ F νλ μ0 μ0 μ0 =−

1 ∂λ Aμ F νλ − qcAμ γ ν , μ0

(22.138)

where Maxwell’s equations ∂μ F μν = − μ0 qcγ ν  have been used. This yields the gauge invariant tensor tμ ν = μ ν − +

↔ % & 1 ih¯ ∂λ Aμ F νλ = − cγ ν ∂μ  − qcγ ν Aμ  μ0 2

1 1 Fμλ F νλ − ημ ν Fκλ F κλ . μ0 4μ0

(22.139)

However, we can go one step further and replace tμ ν with a symmetric energymomentum tensor. The divergence of the spinor term in tμ ν is  ∂ν

↔ ih¯ γ ν ∂μ  + qγ ν Aμ  2

 = − qFμν γ ν ,

(22.140)

where again the Dirac equation was used. The symmetrization of tμ ν also involves the commutators of γ matrices, Sμν =

i + [γμ , γν ] = γ0 · Sμν · γ0 . 4

(22.141)

Since we can write a product always as a sum of an anti-commutator and a commutator, we have γμ · γν = − ημν − 2iSμν ,

(22.142)

22.4 The Energy-Momentum Tensor for Quantum Electrodynamics

607

and the commutators also satisfy7 ημα γβ − ημβ γα + i[Sαβ , γμ ] = 0.

(22.143)

Equations (22.141)–(22.143) together with % & μ ν h¯ 2 ∂ 2  = ihγ ¯ ∂μ mc − qγ Aν  ! % μ & " ! " μν ν = ihq ¯ ∂μ A  + 2iS ∂μ (Aν ) + mc mc − qγ Aν  (22.144) imply also  ∂ν

↔ ih¯ γμ ∂ ν  + qγμ Aν  2

 = − qFμν γ ν .

(22.145)

Therefore the local conservation law ∂ν Tμ ν = 0 also holds for the symmetrized energy-momentum tensor Tμ

ν

  ↔ c ih¯ ν ↔ ih¯ ν ν ν γ ∂μ + γμ ∂ + qγ Aμ + qγμ A  =−  2 2 2 +

1 1 Fμλ F νλ − ημ ν Fκλ F κλ . μ0 4μ0

(22.146)

This yields in particular the Hamiltonian density  ih¯ ↔ H = − T0 = c ∂0 + qA0  + 2   ih¯ ↔ = c mc − γ · ∇ − qγ · A  + 2 0

+



0 2 1 2 E + B 2 2μ0 0 2 1 2 E + B , 2 2μ0

(22.147)

and the momentum density with components Pi = Ti 0 /c,     1 ih¯ ↔ 1 + h¯ ↔ ∇ − qA  + γ ∂0 + qA0  + 0 E × B. P=  2 2i 2 2

(22.148)

Elimination of the time derivatives using the Dirac equation yields P = +

7 The



 1 h¯ ↔ ∇ − qA  + 0 E × B + ∇ × ( + · S · ). 2i 2

(22.149)

commutators Sμν provide the spinor representation of the generators of Lorentz transformations. Furthermore, Eq. (22.143) is the invariance of the γ matrices under Lorentz transformations, see Appendix H.

608

22 Relativistic Quantum Fields

The spin contribution P S = ∇×( + ·S ·)/2 with the vector of 4×4 spin matrices S = ihγ ¯ × γ /4 (22.126) appears here as an additional contribution compared to the orbital momentum density P O = P − P S that follows directly from the tensor (22.139). The spin term in the momentum density (22.149) generates the spin contribution in the total angular momentum density J = x × P = M + S from S = x × P S →  + · S ·  if the symmetric energy-momentum tensor is used in the calculation of angular momentum. This is explained in Problem 22.17c, see in particular Eqs. (22.283)–(22.287).

Energy and Momentum in QED in Coulomb Gauge In materials science it is convenient to explicitly disentangle the contributions from Coulomb and photon terms in Coulomb gauge ∇ · A = 0. We split the electric field components in Coulomb gauge according to E  = − ∇,

E⊥ = −

∂A . ∂t

(22.150)

The equation for the electrostatic potential decouples from the vector potential in Coulomb gauge,  = −

q +  , 0

(22.151)

and is solved by (x, t) =

q 4π 0



d 3x

1  + (x  , t)(x  , t). |x − x  |

(22.152)

Furthermore, the two components (22.150) of the electric field are orthogonal in Coulomb gauge, 

 d 3 x E  (x, t) · E ⊥ (x, t) =

d 3 k E  (k, t) · E ⊥ (− k, t) 

=−

d 3 x (x, t)

∂ ∇ · A(x, t) = 0, ∂t

and the contribution from E  to the Hamiltonian is   0 0 d 3 x E 2 (x, t) = − d 3 x (x, t)(x, t) 2 2  1 = d 3 x (x, t) (x, t) 2

HC =

(22.153)

22.4 The Energy-Momentum Tensor for Quantum Electrodynamics

= q2



 d 3x

d 3x

ss 

609

s+ (x, t)s+ (x  , t)s  (x  , t)s (x, t) , 8π 0 |x − x  |

(22.154)

where the summation is over 4-spinor indices. The presentation of the ordering of the field operators was conventionally chosen as the correct ordering in the nonrelativistic limit, cf. (18.99), but (22.154) must actually be normal ordered such that the particle and anti-particle creation operators bs+ (k) and ds+ (k) appear leftmost in the Coulomb term in the forms b+ d + db, d + d + dd, etc. Substituting the mode expansions  ∼ b + d + and normal ordering therefore leads to the attractive Coulomb terms between particles and their anti-particles. The resulting Hamiltonian in Coulomb gauge therefore has the form  H =

d 3 x c(x, t) [mc − γ · (ih∇ ¯ + qA(x, t))] (x, t)

 1 2 0 2 E ⊥ (x, t) + B (x, t) 2 2μ0   s+ (x, t)s+ (x  , t)s  (x  , t)s (x, t) d 3x d 3x + q2 . 8π 0 |x − x  | 

+

(22.155)

ss

This Hamiltonian yields the corresponding Dirac equation in the Heisenberg form ih¯

∂ (x, t) = [(x, t), H ] ∂t

(22.156)

if canonical anti-commutation relations are used for the spinor field. The Coulomb gauge wave equation (18.17) with the relativistic current density j (22.74) follows in the form ih¯

∂ A(x, t) = [A(x, t), H ], ∂t

∂2 1 A(x, t) = 2 [H, [A(x, t), H ]]. ∂t 2 h¯

(22.157)

if the commutation relations (18.50) and (18.53) are used. This confirms the canonical relations between Heisenberg, Schrödinger and Dirac pictures, and the consistency of Coulomb gauge quantization with the transverse δ function (18.41) also in the fully relativistic theory. It also implies appearance of the Dirac picture time evolution operator in the scattering matrix in the now familiar form. The momentum operator in Coulomb gauge follows from (22.149) and 

 d 3 x 0 E  × B = −

as

 d 3 x 0 A =

 d 3 x A = q

d 3 x  + A

610

22 Relativistic Quantum Fields

 P =

  ¯ +h ∇ + 0 E ⊥ × B , d x  i 3

(22.158)

where boundary terms at infinity were discarded.

22.5 The Non-relativistic Limit of the Dirac Equation The Dirac basis (22.67) for the γ -matrices is convenient for the non-relativistic limit. Splitting off the time dependence due to the rest mass term       mc2 mc2 ψ(x, t) t = t exp − i (x, t) = ϒ(x, t) exp − i φ(x, t) h¯ h¯

(22.159)

in the Dirac equation (22.71) yields the equations (ih∂ ¯ t − q)ψ + cσ · (ih∇ ¯ + qA)φ = 0,

(22.160)

2 (ih∂ ¯ + qA)ψ = 0. ¯ t − q + 2mc )φ + cσ · (ih∇

(22.161)

This yields in the non-relativistic regime φ−

1 σ · (ih∇ ¯ + qA)ψ 2mc

(22.162)

and substitution into the equation for ψ yields Pauli’s equation8 ih∂ ¯ tψ = −

1 q h¯ 2 (h∇ σ · Bψ + qψ. ¯ − iqA) ψ − 2m 2m

(22.163)

The spin matrices for spin-1/2 Schrödinger fields are the upper block matrices in the spin matrices (22.126) for the full Dirac fields, S = hσ ¯ /2, see also Sect. 8.1 and in particular Eq. (8.17). If the external magnetic field B is approximately constant over the extension of the wave function ψ(x, t) we can use A(x, t) =

8 Pauli

1 B(t) × x. 2

(22.164)

[131] actually only studied the time-independent Schrödinger equation with the Pauli term in the Hamiltonian, and although he mentions Schrödinger in the beginning, he seems to be more comfortable with Heisenberg’s matrix mechanics in the paper.

22.5 The Non-relativistic Limit of the Dirac Equation

611

Substitution of the vector potential in Eq. (22.163) then yields the following linear terms in B in the Hamiltonian on the right-hand side, i

% & q q q h¯ (B × x) · ∇ − B · S = − B · M + 2S 2m m 2m & % q μB =− B · M + 2S . e h¯

(22.165)

Here μB = eh/2m is the Bohr magneton, and we used the shorthand notation ¯ −ihx × ∇ → M for the x representation of the angular momentum operator. Recall ¯ that this operator is actually given by  M = x × p = − ih¯

d 3 x |xx × ∇x|.

(22.166)

Equation (22.165) shows that the Dirac equation explains the double strength magnetic coupling of spin as compared to orbital angular momentum (often denoted as the magneto-mechanical anomaly of the electron or the anomalous magnetic moment of the electron). The corresponding electromagnetic currents in the nonrelativistic regime are = qψ + ψ, % & j = cq ψ + σ φ + φ + σ ψ & q % + + =− ψ σ ⊗ σ · (ih∇ − qψ + A) · σ ⊗ σ ψ , ¯ + qA)ψ − (ih∇ψ ¯ 2m where σ ⊗ σ is the three-dimensional tensor with the (2 × 2)-matrix entries σ i · σ j (we can think of it as a (3 × 3)-matrix containing (2 × 2)-matrices as entries). Substitution of σ ⊗ σ = 1 + iei ⊗ ej εij k σ k

(22.167)

yields j=

& q % + + ψ · h∇ψ − h∇ψ · ψ − 2iqψ + Aψ + j s , ¯ ¯ 2im

(22.168)

with a spin term js =

% & q h¯ ∇ × ψ +σ ψ . 2m

(22.169)

However, this term does not accumulate or diminish charges in any volume, ∇·j s = 0, and can therefore be neglected in the calculation of electric currents.

612

22 Relativistic Quantum Fields

The non-relativistic approximations for the Lagrange density L, the energy density H and the momentum density P are L=

  ∂ ∂ q h¯ + ih¯ ψ + · ψ − ψ + · ψ − qψ + ψ + ψ σ · Bψ 2 ∂t ∂t 2m +

1 1 + − qψ + A) · (ih∇ψ + qAψ) − Fμν F μν , (ih∇ψ ¯ ¯ 2m 4μ0

1 q h¯ + + (h∇ψ ψ σ · Bψ + iqψ + A) · (h∇ψ − iqAψ) − ¯ ¯ 2m 2m 1 2 0 B , + E2 + 2 2μ0 & h¯ % + ψ · ∇ψ − ∇ψ + · ψ − qψ + Aψ + 0 E × B. P= 2i

(22.170)

H=

(22.171) (22.172)

The Hamiltonian and momentum operators in Coulomb gauge are  H =

 1 + 2 d x − ψ (x, t)[h∇ ¯ − iqA(x, t)] ψ(x, t) 2m 3

 1 2 q h¯ + 0 2 B (x, t) − E ⊥ (x, t) + ψ (x, t)σ · B(x, t)ψ(x, t) 2 2μ0 2m  2  ψs+ (x, t)ψs+ (x  , t)ψs  (x  , t)ψs (x, t) d 3x d 3x . (22.173) + q2 8π 0 |x − x  |  +

ss =1

and (cf. Eq. (22.158))  P =

  ¯ +h ∇ψ + 0 E ⊥ × B . d x ψ i 3

(22.174)

It is interesting to note that if we write the current density (22.168) as j =J +

% & % & q h¯ q ∇ × ψ + σ ψ = J + μB ∇ × ψ + σ ψ 2m e

(22.175)

we can write Ampère’s law with Maxwell’s correction term as

 ∂ q ∇ × B − μ0 μB ψ + σ ψ = ∇ × B class = μ0 J + μ0 0 E, e ∂t

(22.176)

i.e. the “spin density" S(x, t) =

h¯ + ψ (x, t)σ ψ(x, t) 2

(22.177)

22.5 The Non-relativistic Limit of the Dirac Equation

613

adds a spin magnetic field to the magnetic field B class which is generated by orbital currents J and time-dependent electric fields E, B(x, t) = B class (x, t) +

q 2q μ0 μB S(x, t) = B class (x, t) + μ0 S(x, t). eh¯ m

Higher Order Terms and Spin-Orbit Coupling We will discuss higher order terms in the framework of relativistic quantum mechanics, i.e. our basic quantum operators are x and p etc., but not quantum fields. This also entails a semi-classical approximation for the electromagnetic fields and potentials. For the discussion of higher order terms, we write the Dirac equation in Schrödinger form, ih¯

d |ϒ(t) = H (t)|ϒ(t), dt

(22.178)

with the Hamilton operator H (t) = (γ 0 − 1)mc2 + q(x, t) + c α · [p − qA(x, t)].

(22.179)

The operator α is α = γ 0γ ,

α i ab = a|α i |b = γ 0 ac γ i cb ,

(22.180)

and x, a|ϒ(t) = ϒa (x, t) is the a-th component of the 4-spinor ϒ (22.159) in x representation. We continue to use the Dirac basis (22.67) of γ matrices in this section, such that as a matrix valued vector α is given by  α=

 0 σ . σ 0

(22.181)

The part of the Hamiltonian (22.179) which mixes the upper and lower components of the 4-spinor ϒ is K(t) = c α · [p − qA(x, t)].

(22.182)

Operators which mix upper and lower 2-spinors in 4-spinors are also denoted as odd terms in the Hamiltonian. We can remove the odd contribution K(t) by using the anti-hermitian operator

614

22 Relativistic Quantum Fields

γ0 1 γ · [p − qA(x, t)], K(t) = 2 2mc 2mc

T (t) =

(22.183)

[T (t), γ 0 mc2 ] = −K(t),

(22.184)

which implies subtraction of K(t) from the new transformed Hamiltonian exp[T (t)]H (t) exp[−T (t)]. However, we also have to take into account that the transformed state |ϒT (t) = exp[T (t)]|ϒ(t) satisfies the equation ih¯

d |ϒT (t) = exp[T (t)]H (t) exp[−T (t)]|ϒ(t) dt d exp[T (t)] exp[−T (t)]|ϒ(t). + ih¯ dt

(22.185)

Therefore the transformed Hamiltonian is actually HT (t) = exp[T (t)]H (t) exp[−T (t)] − ih¯ exp[T (t)] =

∞ ∞ 1 n 1 n [T (t), H (t)] − ih¯ [T (t), d/dt] n! n! n=0

=

n=0

∞ ∞ 1 n 1 n−1 [T (t), H (t)] + ih¯ [ T (t), dT (t)/dt] n! n! n=0

=

d exp[−T (t)] dt

∞ n=0

n=1

∞ 1 n iq h¯ 1 n−1 ˙ [T (t), H (t)] − [ T (t), γ · A(t)]. (22.186) n! 2mc n! n=1

We also wish to expand the Hamiltonian up to terms of order (E/mc2 )3 , where E contains contributions from the kinetic energy of the particle and from its interactions with the electromagnetic fields. Equation (22.184) implies HT (t) = (γ 0 − 1)mc2 + q(t) + mc2

4 1 n [T (t), γ 0 ] n! n=2

+

3 n=1

−

1 n iq h¯ ˙ γ · A(t) [T (t), q(t) + K(t)] − n! 2mc

  2 n 1 iq h¯ E 4 ˙ [T (t), γ · A(t)] +O . (22.187) 2mc (n + 1)! mc2 n=1

The relevant commutators are

22.5 The Non-relativistic Limit of the Dirac Equation

[T (t), q(t)] −

615

iq h¯ iq h¯ ˙ γ · A(t) = γ · E(x, t), 2mc 2mc

(22.188)

iq h¯ iq h¯ ˙ [T (t), γ · A(t)] = [T (t), γ · E(t)] 2mc 2mc   & σ 0 q h¯ 2 q h¯ 2 % =− ∇ · E(x, t) − i 2 2 ∇ × E(x, t) · 0 σ 4m2 c2 4m c   & σ 0 q h¯ % , (22.189) − E(x, t) × [p − qA(x, t)] · 2 2 0 σ 2m c

2

[T (t), q(t)] −

  γ0 2 γ0 q h¯ σ 0 2 [T (t), K(t)] = , [p − qA(x, t)] − B(x, t) · K (t) = 0 −σ m m mc2 2

[T (t), K(t)] = −

3

1 K 3 (t), m2 c 4

[T (t), K(t)] = −

γ0 K 4 (t). m3 c 6

(22.190)

and 2

mc2 [T (t), γ 0 ] = [K(t), T (t)] = − 3

mc2 [T (t), γ 0 ] =

1 K 3 (t), m2 c 4

4

γ0 2 K (t), mc2

mc2 [T (t), γ 0 ] =

γ0 K 4 (t). m3 c 6

(22.191)

(22.192)

We don’t need to evaluate the final higher order commutator 3

(3)

Codd (t) =[T (t), q(t)] −

iq h¯ 2 ˙ [T (t), γ · A(t)], 2mc

(22.193)

because this is an odd term of order (E/mc2 )3 , which is eliminated in the next step through a unitary transformation, to which it contributes in order (E/mc2 )4 . We (3) only need to observe that Codd (t) contains one term proportional to γ , and other terms proportional to 

σj 0 γ · 0 σj i



 =δ

ij

0 1 −1 0

 + i ij k γk ,

(22.194)

(3) such that {γ 0 , Codd (t)} = 0. This will become relevant for the elimination of (3) Codd (t) in the next step. However, for now our transformed Hamiltonian is

HT (t) = (γ 0 − 1)mc2 + q(t) +

γ0 γ0 2 K (t) − K 4 (t) 2mc2 8m3 c6

616

22 Relativistic Quantum Fields

  & σ 0 q h¯ 2 q h¯ 2 % ∇ × E(t) · ∇ · E(t) − i 0 σ 8m2 c2 8m2 c2   & σ 0 q h¯ % − E(t) × [p − qA(x, t)] · 2 2 0 σ 4m c   E 4 iq h¯ 1 1 (3) 3 + γ · E(t) − K (t) + Codd (t) + O . 2mc 6 3m2 c4 mc2

−

The last line contains three odd contributions L(t) =

iq h¯ 1 1 (3) γ · E(t) − K 3 (t) + Codd (t), 2 4 2mc 6 3m c

(22.195)

which we can eliminate exactly as in the previous step by using a unitary transformation |ϒW T (t) = exp[W (t)]|ϒT (t) with W (t) =

  γ0 iq h¯ γ0 E 4 3 L(t) = α · E(t) − K (t) + O . 2mc2 4m2 c3 6m3 c6 mc2

(22.196)

This yields a new Hamiltonian HW T (t) = exp[W (t)]HT (t) exp[−W (t)] − ih¯ exp[W (t)] =

d exp[−W (t)] dt

∞ ∞ 1 n 1 n−1 [W (t), HT (t)] + ih¯ [ W (t), dW (t)/dt], (22.197) n! n! n=0

n=1

which is in the required order γ0 γ0 2 K (t) − K 4 (t) 2mc2 8m3 c6   & σ 0 q h¯ 2 % q h¯ 2 ∇ × E(t) · ∇ · E(t) − i − 0 σ 8m2 c2 8m2 c2   & σ 0 q h¯ % − E(t) × [p − qA(x, t)] · 0 σ 4m2 c2

HW T (t) = (γ 0 − 1)mc2 + q(t) +

qγ 0 iq h¯ [K 3 (t), (t)] + [α · E(t), γ 0 K 2 (t)] 3 6 6m c 8m3 c5   dW (t) E 4 . (22.198) + ih¯ +O dt mc2

−

This contains again an odd piece

22.5 The Non-relativistic Limit of the Dirac Equation

M(t) = ih¯

qγ 0 dW (t) iq h¯ − 3 6 [K 3 (t), (t)]+ 3 5 [α ·E(t), γ 0 K 2 (t)] dt 6m c 8m c

617

(22.199)

which is eliminated by another unitary transformation of the form |ϒF W T (t) = exp[F (t)]|ϒW T (t) with   γ0 q h¯ 2 E 4 ˙ F (t) = M(t) = − 3 5 γ · E(t) + O . 2mc2 8m c mc2

(22.200)

The resulting Hamiltonian after this transformation still contains an odd piece N(t) = − i

q h¯ 3 ¨ γ · E(t) 8m3 c5

(22.201)

which is eliminated in a final transformation   E 4 γ0 G(t) = N(t) = O . 2mc2 mc2

(22.202)

Therefore up to terms of order O(E/mc2 )4 , we finally find an equation which is diagonal in upper and lower 2-spinors |ϒF W T (t) = exp[F (t)] exp[W (t)] exp[T (t)]|ϒ(t), ih¯

d |ϒF W T (t) = HF W T (t)|ϒF W T (t) dt

(22.203) (22.204)

with γ0 γ0 2 K (t) − K 4 (t) 2mc2 8m3 c6   & σ 0 q h¯ 2 % q h¯ 2 ∇ × E(t) · ∇ · E(t) − i − 0 σ 8m2 c2 8m2 c2   & σ 0 q h¯ % . (22.205) − E(t) × [p − qA(x, t)] · 0 σ 4m2 c2

HF W T (t) = (γ 0 − 1)mc2 + q(t) +

The transformation (22.203) and (22.205) is known as a Foldy-Wouthuysen transformation [56]. The Hamiltonian acting on the upper 2-spinor is H (t) =

[p − qA(x, t)]2 q h¯ q h¯ 2 + q(x, t) − B(x, t) · σ − ∇ · E(x, t) 2m 2m 8m2 c2 & q h¯ % ih∇ − ¯ × E(x, t) + 2E(x, t) × [p − qA(x, t)] · σ 2 2 8m c

618

22 Relativistic Quantum Fields

−

2 1

2 [p − qA(x, t)] − q hB(x, t) · σ . ¯ 8m3 c2

(22.206)

The first three terms comprise the Pauli Hamiltonian (22.163). It is of interest to write some of the higher order terms in the Hamiltonian (22.206) also in terms of the charge density (x, t) which generates the electromagnetic fields. The term −

q h¯ 2 q h¯ 2 ∇ · E(x, t) = − (x, t) 8m2 c2 8m2 c2 0

(22.207)

amounts to a contact interaction between the particles described by Eq. (22.204) (e.g. electrons) and the particles which generate the electromagnetic fields. This term is known as the Darwin term. The contact interaction has the counter-intuitive property to lower the interaction energy between like charges, but recall that it emerged from eliminating the anti-particle components up to terms of order O(E/mc2 )4 . It should not surprise us that a positronic component in electron wave functions contributes an attractive term to the electron-electron interaction. The Hamiltonian (22.206) is in excellent agreement with spectroscopy if radiative corrections are also taken into account, see e.g. [87]. The term −i

& q h¯ 2 % q h¯ ˙ ∇ × E(x, t) · σ = i 2 2 B(x, t) · S 2 2 8m c 4m c

(22.208)

is apparently a coupling between spin S = hσ ¯ /2 and induced potentials from timedependent charge-current distributions. In the static case we can write the E(x) × p term in (22.206) in the form −

& q % μ0 q E(x) × p · S = − 2 2 2m c 8π m2



d 3x

(x  ) M(x − x  , p) · S. |x − x  |3

Here M(x − x  , p) = (x − x  ) × p

(22.209)

is the orbital angular momentum operator with respect to the point x  , and the E(x) × p term apparently contains a charge weighted sum over angular momentum operators. The E(x) × p term is therefore the origin of spin-orbit coupling. In particular, for a radially symmetric charge distribution E(x) = − one finds

x d(r) r dr

(22.210)

22.6 Covariant Quantization of the Maxwell Field

−

& d(r) q % q M · S. E(x) × p · S = 2m2 c2 2m2 c2 r dr

619

(22.211)

This implies Eq. (8.33) for spin-orbit coupling in hydrogen atoms. So far we have emphasized the emergence of M ·S terms from the E(x)×p term, and historically the coupling of spin and orbital angular momentum had provided the initial motivation for the designation as spin-orbit coupling term. However, the direct coupling of orbital momentum p and spin provides just as good a reason for the name spin-orbit coupling, and another important special case of the E(x) × p term arises for a uni-directional electric field e.g. in z direction. In this case the term takes the form −

% & & q % q Ez (z) px Sy − py Sx . E(x) × p · S = − 2 2 2 2 2m c 2m c

(22.212)

For homogeneous electric field this yields a spin-orbit coupling term of the form αR (px Sy − py Sx ) with constant αR . This particular form of a spin-orbit coupling term is known as a Rashba term [22, 141]. Spin-orbit coupling was always relevant not only for atomic and molecular spectroscopy, but also for electronic energy band structure in materials where they are often significantly enhanced e.g. due to low effective masses. In recent years spin-orbit coupling terms in low-dimensional systems, and Rashba terms in particular, have also attracted a lot of interest because of their relevance for spintronics (see e.g. [24, 69, 72, 125, 134, 157, 159]).

22.6 Covariant Quantization of the Maxwell Field We have seen in Sect. 18.2 how to quantize the Maxwell field and describe photons in Coulomb gauge. This is useful if our problem contains non-relativistic charged particles, since the Hamiltonian in Coulomb gauge conveniently describes the electromagnetic interaction between the charged particles through Coulomb terms. The free interaction picture photon operators A(x, t) or the corresponding Schrödinger picture operators A(x) are then only needed for the calculation of absorption, emission or scattering of external photons. Exchange of virtual photons provides only small corrections to Coulomb interactions for non-relativistic charged particles. The relevant Hamiltonian is (22.173) with Schrödinger fields and Coulomb terms for all the different kinds of charged particles. Coulomb gauge can also be used for problems involving relativistic fermions. These can be described by the Hamiltonian (22.155) including Dirac fields and Coulomb interaction terms for all the different kinds of spin-1/2 particles in the problem. Indeed, we will calculate basic scattering processes involving relativistic charged particles in Sects. 23.2 and 23.4 in Coulomb gauge, and the calculations will explicitly show how the Coulomb interaction terms between charged particles dominate over photon exchange terms if the kinetic energies of the charged particles are small compared to their rest energies, see in particular Eq. (23.52).

620

22 Relativistic Quantum Fields

However, if the problem indeed contains relativistic charged particles, then the interaction of those particles with other charged particles is more conveniently described through a covariant quantization of photons in Lorentz gauge, ∂μ Aμ (x) = 0.

(22.213)

Suppose the potential Aμ (x) does not satisfy the Lorentz gauge condition. We can construct the Lorentz gauge vector potential Aμ (x) by performing a gauge transformation Aμ (x) = Aμ (x) + ∂μ f (x)

(22.214)

with  f (x) =  =

  (r) d 4 x  Gd (x − x  ) d 3x

m=0

∂μ Aμ (x  )

 1  μ   ∂ A (x ) .   μ ct =ct−|x−x  | 4π |x − x  |

(22.215)

Here   (r) Gd (x)

m=0

=

 1 (r) 1  Gd (x, t) δ(r − ct) = m=0 c 4π r

(22.216)

is the retarded massless scalar Green’s function, cf. (J.83) and (J.115). This also helps us to solve Maxwell’s equations in Lorentz gauge, ∂μ ∂ μ Aν (x) = − μ0 j ν (x)

(22.217)

in the form μ

μ

Aμ (x) = ALW (x) + AD (x),

(22.218)

where the Liénard-Wiechert potentials (see also Eqs. (J.150) and (J.151)) 

μ

ALW (x) = μ0 μ0 = 4π



  (r) d 4 x  Gd (x − x  ) d 3x

m=0

j μ (x  )

& % 1 j μ x  , ct − |x − x  |  |x − x |

(22.219)

solve the inhomogeneous Maxwell equations (22.217) and satisfy the Lorentz gauge μ condition due to charge conservation. The remainder AD (x) must therefore satisfy ∂μ ∂ μ AνD (x) = 0,

μ

∂μ AD (x) = 0.

(22.220)

22.6 Covariant Quantization of the Maxwell Field

621

To quantize this, we observe that Maxwell’s equations in Lorentz gauge follow from the Lagrange density of electromagnetic fields (18.3) if we take into account the Lorentz gauge condition, L = Aμ j μ −

1 ∂ν Aμ · ∂ ν Aμ . 2μ0

(22.221)

This yields canonically conjugate momentum fields for all components of the vector potential, "μ =

∂L = 0 A˙ μ . ∂ A˙ μ

(22.222)

The principles of canonical quantization and the Lorentz gauge condition then motivate the following quantization condition for electromagnetic potentials in Lorentz gauge (with k 0 = |k|), ih¯ (2π )3 0    kμ kν exp[ik · (x − x  )]. (22.223) × d 3 k ημν − 2 k − i

[Aμ (x, t), A˙ ν (x  , t)] =

Here and in the following equations, the limit k 2 → 0 is only taken after complete evaluation of any calculations in which terms like [Aμ (x, t), A˙ ν (x  , t)] would appear. The general solution of Eq. (22.220), μ

AD (x) = x, μ|AD  =  ×

h¯ μ0 c (2π )3

 d 3k μ

α (k) aα (k) exp(ik · x) + aα+ (k) exp(− ik · x) , √ 2|k| α=1 3

(22.224)

with k · α (k) = 0 and 3

αμ (k)αν (k) = ημν −

α=1

kμkν , k 2 − i

(22.225)

satisfies the quantization condition if [aα (k), aβ+ (k  )] = δαβ δ(k − k  ) and the other commutators vanish.

(22.226)

622

22 Relativistic Quantum Fields μ

μ

A possible choice for the polarization vectors α (k) is e.g. to choose 1 (k) and as spatial orthonormal vectors without time-like components and perpendicular to k such that μ 2 (k)

2

ˆ  α (k) ⊗  α (k) = 1 − kˆ ⊗ k,

(22.227)

α=1

and choose the third polarization vector as ˆ (|k|, k 0 k) . 3 (k) = √ − k 2 + i

(22.228)

This formalism can be motivated as a limiting case of the quantization of massive vector fields where we would have k 2 = − (mc/h) ¯ 2 . However, the singularity of the polarization vector (22.228) on the photon mass shell already indicates that it cannot appear as the polarization of a free photon state. Formulating quantum electrodynamics in Lorentz gauge with the virtual polarization vector (22.228) has the advantage of faster and easier calculation of scattering amplitudes involving electromagnetic interactions of relativistic particles, because there are no separate amplitudes for photon exchange and Coulomb interactions which need to be added up to give the full scattering amplitude. The spatially longitudinal photons generated by a3+ (k), k ·  3 (k) = k 0 30 (k) = 0, incorporate the contributions from the Coulomb interactions.9 Why then don’t we see photon states a3+ (k)|0? These photon states are spurious gauge degrees of freedom. We could perform another gauge transformation Aμ (x) → A˜ μ (x) = Aμ (x) + ∂μ g(x)

(22.229)

with  g(x, t) =

d 3x

∇  · A(x  , t) , 4π |x − x  |

(22.230)

˙ which we can rewrite as a time integral, using that in any gauge ∇ · A(x, t) = − (x, t) − (x, t)/0 ,  g(x, t) =

9 Of

t −∞

  dt  (x, t  ) − d 3 x 

(x  , t  ) 4π 0 |x − x  |



course, this implies that one cannot naively invoke Hamiltonians with Coulomb interaction terms if we describe photons in Lorentz gauge. Otherwise we would overcount interactions. Remember that the Coulomb interaction terms came from the contributions to Hamiltonians from electromagnetic fields in Coulomb gauge, see Sect. 22.4.

22.6 Covariant Quantization of the Maxwell Field

 +

d 3x

623

∇  · A(x  , −∞) . 4π |x − x  |

(22.231)

This takes us right back to Coulomb gauge, ˜ (x) =



d 3x

(x  , t) , 4π 0 |x − x  |

˜ ∇ · A(x) = 0,

(22.232)

without any freely oscillating time-like component. Since a3+ (k)|0 was the only photon state with a time-like component, (22.229) has removed that photon state. We can think of the photons with longitudinal spatial components and corresponding time-like components as virtual place holders for the Coulomb interaction. The interaction Hamiltonian for spinor QED in Lorentz gauge is HI = − j μ Aμ = − cq · γ μ Aμ · ,

(22.233)

see Problem 22.28. It is also useful to know that in the calculation of scattering matrix elements, Eq. (22.225) can effectively be replaced with 3

αμ (k)αν (k) → ημν .

(22.234)

α=1

The reason for this is that the polarization vectors α (k) always enter through products γ · α (k) into scattering matrix elements, and the k μ terms from Eq. (22.225) then always contribute terms of the form10   u(p 1 , s1 ) · γ · hk ¯ · u(p2 , s2 ) h¯ k=p2 −p1   v(p 1 , s1 ) · γ · hk ¯ · v(p 2 , s2 ) h¯ k=p2 −p1   v(p 1 , s1 ) · γ · hk ¯ · u(p2 , s2 ) h¯ k=p2 +p1   u(p 1 , s1 ) · γ · hk ¯ · v(p 2 , s2 )

h¯ k=p2 +p1

= 0,

(22.235)

= 0,

(22.236)

= 0,

(22.237)

= 0.

(22.238)

The vanishing of all these terms is a consequence of the fact that the u and v spinors satisfy Dirac equations

· h¯ k also has the form (± mc −γ ·p)/(m2 c2 +p 2 − i). These terms can be reduced to terms of the form (22.235)–(22.238) through the Eqs. (22.101) or (22.103).

10 Often the factor on the left or on the right of γ

624

22 Relativistic Quantum Fields

(mc + γ · p)u(p, s) = 0,

u(p, s)(mc + γ · p) = 0,

(22.239)

(mc − γ · p)v(p, s) = 0,

v(p, s)(mc − γ · p) = 0.

(22.240)

22.7 Problems 22.1 Show that for an appropriate class of integration contours C in the complex k 0 plane the scalar propagator (22.15) can be written in the form K(x) = −

 * exp(ik · x) 1 0 dk . d 3k 2 4 (2π ) c C k + (mc/h) ¯ 2

(22.241)

22.2 Show that we can write the free Klein-Gordon equation (22.3) in the form of a Schrödinger equation ih¯

d |(t) = H |(t) dt

(22.242)

with a state vector  |(t) =

|φ(t) |χ (t)

 (22.243)

and a Hamiltonian  H =

  2 + m2 c 2 0 c p  , c p2 + m2 c2 0

if we define the action of the operator k|



(22.244)

p2 + m2 c2 on states through



 p2

+ m2 c2 |φ(t)

=

h¯ 2 k 2 + m2 c2 k|φ(t).

(22.245)

Derive the representation (22.14)–(22.17) of the time evolution of the free KleinGordon field from integration of Eq. (22.242) through the time evolution operator U (t, t  ) = exp[− iH (t − t  )/h]. ¯

(22.246)

22.3 We have discussed the non-relativistic limit of the Klein-Gordon field in the case |b+ (k)| % |a(k)|. However, there must also exist a non-relativistic limit for the anti-particles. How does the non-relativistic limit work in the case of negligible particle amplitude |a(k)| % |b+ (k)|?

22.7 Problems

625

22.4 Derive the energy density H and the momentum density P for the real KleinGordon field. 22.5 Calculate the non-relativistic limits for the Hamilton operator H and the momentum operator P of the real Klein-Gordon field. 22.6 Derive the energy-momentum tensor for QED with scalar matter (22.4), Tν

μ

   m2 c 4 + q + q ρ + ρ ∂ρ φ + i φ Aρ ∂ φ − i A φ − ην μ = − ην hc φ φ ¯ h¯ h¯ h¯    1 q + q μ 2 + μ ∂ ∂ Fρσ F ρσ + hc φ + i A φ − i φ − ην μ φ A ¯ ν ν h¯ h¯ 4μ0    q q 1 2 Fνρ F μρ . (22.247) + hc ∂ μ φ + + i φ + Aμ ∂ν φ − i Aν φ + ¯ μ0 h¯ h¯ μ

2

The corresponding densities of energy, momentum, and energy current are    0 2 1 2 m2 c 4 + q + q + ˙ ˙ H=T = E + B + φ φ + h¯ φ − i φ  φ + i φ 2 2μ0 h¯ h¯ h¯    q q 2 ∇φ + + i φ + A · ∇φ − i Aφ , (22.248) + hc ¯ h¯ h¯    1 q q P = ei T i0 = 0 E × B − h¯ φ˙ + − i φ +  ∇φ − i Aφ c h¯ h¯    q q (22.249) φ˙ + i φ , − h¯ ∇φ + + i φ + A h¯ h¯ 00

S = cei T 0i = c2 P. Solution The Lagrange density for quantum electrodynamics with scalar matter is    m2 c 4 + q + q 1 2 + ∂φ φ Aφ − φ ·φ− + i A · ∂φ − i Fμν F μν . L = − hc ¯ 4μ0 h¯ h¯ h¯ This yields according to (16.31) a conserved energy-momentum tensor ∂L ∂L ∂L − ∂μ φ − ∂μ Aλ ∂(∂ν φ + ) ∂(∂ν φ) ∂(∂ν Aλ )     q q + ν 2 ν + ∂ ∂μ φ φ + i A φ = ημ ν L + h¯ c2 ∂μ φ + ∂ ν φ − i Aν φ + hc ¯ h¯ h¯ 1 + ∂μ Aλ · F νλ . (22.250) μ0

μ ν = ημ ν L − ∂μ φ +

626

22 Relativistic Quantum Fields

To find a gauge invariant energy-momentum tensor we add the identically conserved improvement term −

& 1 % 1 1 ∂λ Aμ F νλ = − ∂λ Aμ F νλ − Aμ ∂λ F νλ μ0 μ0 μ0

=−

% & 1 q 2 c2 + ∂λ Aμ F νλ + iqc2 φ + Aμ · ∂ ν φ − ∂ ν φ + · Aμ φ + 2 φ Aμ Aν φ, μ0 h¯

where Maxwell’s equations ∂μ F μν = − μ0

& ∂L q% + ν q2 + ν =i φ · ∂ φ − ∂ ν φ+ · φ + 2 φ A φ ∂Aν 0 0 h¯

(22.251)

were used. This yields the gauge invariant tensor (22.247) from Tμ ν =  μ ν −

& 1 % ∂λ Aμ F νλ . μ0

(22.252)

22.7 If we write the solution (22.9) of the free Klein-Gordon equation as the sum of the positive and negative energy components, φ(x) = φ+ (x) + φ− (x), φ+ (x) = √

φ− (x) = √



1 2π



1 2π

3

3

(22.253)

d 3k a(k) exp[i(k · x − ωk t)], √ 2ωk

(22.254)

d 3k + b (k) exp[− i(k · x − ωk t)], √ 2ωk

(22.255)

+ + ˙ the charge densities ± = − iq(φ˙ ± · φ± − φ± · φ± ) are separately conserved, and therefore we can also identify conserved particle and anti-particle numbers

1 Q± =± N± = ± q q

 d 3 x ± (x, t).

(22.256)

22.7a Show that the conserved current density for QED with scalar matter (22.4) is   q jμ = iqc2 ∂μ φ + · φ − φ + · ∂μ φ + 2i φ + · Aμ · φ . h¯

(22.257)

Note that charge conjugation (22.5) is equivalent to the substitution q → − q in j μ , as it should. 22.7b Why is it not possible to derive separately conserved (anti-)particle numbers N± for the scalar particles in the interacting theory (22.4)?

22.7 Problems

627

22.8 Show that the junction conditions (22.57) are necessary and sufficient to ensure that the Klein-Gordon equation holds at the step of the potential. 22.9 Generalize the reasoning from Sect. 22.2 to the case of oblique incidence against the potential step, e.g. by considering a scalar boson running against the potential (22.48) with initial momentum components hk ¯ x > 0 and hk ¯ y > 0. Remarks on the Solution The ansatz for the Klein-Gordon wave function which complies with the boundary conditions on the incoming particle and the requirement of smoothness for all times t and values of y along the interface x = 0 is φ(x, y, t) =

" ! exp(ikx x) + β exp(− ikx x) exp[i(ky y − ωt)], x < 0, x > 0. θ exp[i(κx x + ky y − ωt)],

The frequency follows again from the solution of the Klein-Gordon equation in the two domains, ω = c kx2 + ky2 +

m2 c 2 h2 ¯

=

V m2 c 2 ± c κx2 + ky2 + 2 . h¯ h¯

(22.258)

All other pertinent results follow also exactly as in Sect. 22.2 if we make the substitutions k → kx , κ → κx and mc → m2 c2 + h¯ 2 ky2 . This applies in particular also to Table 22.1 and Fig. 22.1. In particular, we have generation of pairs of particles and anti-particles in the energy range    2 k 2 + h2 k 2 + m2 c2 < V − c h2 k 2 + m2 c2 c h¯ 2 ky2 + m2 c2 < hω = c h ¯ ¯ x ¯ y ¯ y if the height of the potential step satisfies  V > 2c h¯ 2 ky2 + m2 c2 .

(22.259)

The wave number κx in this energy range is κx = −

1 h¯

(hω ¯ − V )2 − m2 c2 − h¯ 2 ky2 , c2

(22.260)

and writing the solution for x > 0 as φ(x, y, t) = θ exp[− i(−κx x − ky y + ωt)],

x > 0,

(22.261)

shows that it is an anti-particle solution with energy E p = −hω ¯ and momentum components −hκ ¯ x > 0, −hk ¯ y < 0. The kinetic+rest energy of the generated antiparticles in the region x > 0 is

628

22 Relativistic Quantum Fields

 Kp = c h¯ 2 κx2 + h¯ 2 ky2 + m2 c2 = V − hω, ¯

(22.262)

and the energy of the anti-particles is just the sum of their kinetic+rest energy and their potential energy, E p = Kp − V . 22.10 Calculate the boson number operator  Nb =

% & d 3 k a + (k)a(k) + b+ (k)b(k)

(22.263)

for the free Klein-Gordon field in x representation. 22.11 Show that scattering of a Klein-Gordon field off a hard sphere yields the same result√(11.68) as the non-relativistic Schrödinger theory, except that the definition k = 2mE/h¯ (where E is the kinetic energy of the scattered particle) is replaced by k=

 1 h¯ 2 ω2 − m2 c4 . hc ¯

(22.264)

The hard sphere is taken into account through a boundary condition of vanishing Klein-Gordon field on the surface of the sphere, like the condition on the Schrödinger wave function in Sect. 11.3, i.e. we do not model it as a potential. We could think of the hard sphere in this case as arising from a hypothetical interaction which repels particles and anti-particles alike (just like gravity is equally attractive for particles and anti-particles). 22.12 You could also model an impenetrable wall for a Klein-Gordon field in the manner of the hard sphere of Problem 22.11. Which wave function for the KleinGordon field do you get if the impenetrable wall prevents the field from entering the region x > 0? Why does this result not contradict the Klein paradox? 22.13 Calculate the fermion number operator  Nf =

d 3k

! " bs+ (k)bs (k) + ds+ (k)ds (k)

(22.265)

s∈{↓,↑}

for the free Dirac field in x representation. 22.14 Calculate the reflection and transmission coefficients for a Dirac field of charge q in the presence of a potential step q(x) = V (x) = V (x). How do your results compare with the results for the Klein-Gordon field in Sect. 22.2? 22.15 Spinor momentum eigenstates 22.15a What are the non-relativistic limits of the spinor plane waves (22.117)?

22.7 Problems

629

22.15b Verify the relations (22.118) and (22.119) for the relativistic spinor plane wave states. 22.16 Suppose A(x) and B(x) are two operators with the property that their commutator or their anti-commutator closes into a function f (i.e. it becomes the unit operator times f ), [A(x), B(x  )]± = f (x, x  ).

(22.266)

We also assume that A(x) and B(x) are products of quantized fields like ψ(x), φ(x), ψ(x) etc. 22.16a The momentum operators Pμ in quantum field theory satisfy [Pμ , A(x)] = ih∂ ¯ μ A(x),

(22.267)

if A(x) is a product of quantized fields. Use this property and the identity [Pμ , [A(x), B(x  )]± ] = [[Pμ , A(x)], B(x  )]± + [A(x), [Pμ , B(x  )]]±

(22.268)

to prove f (x, x  ) = f (x − x  ). 22.16b Show in particular that the commutator of free Klein-Gordon fields yields the time-evolution kernel (22.15), [φ(x), φ + (x  )] = − iK(x − x  ),

(22.269)

and that the anti-commutator of free Dirac fields yields the time-evolution kernel (22.108), {ψ(x), ψ + (x  )} = W(x − x  ).

(22.270)

22.17 Angular momentum in relativistic field theory 22.17a Show that if Tμν is a symmetric conserved energy momentum tensor, then the currents Mαβ μ =

& 1% xα Tβ μ − xβ Tα μ c

(22.271)

are also conserved: ∂μ Mαβ μ = 0.

(22.272)

22.17b The quantities Mαβ μ have the properties Mαβ 0 = xα Pβ − xβ Pα and are therefore associated with angular momentum conservation and conservation of the

630

22 Relativistic Quantum Fields

center of energy motion (18.227) and (18.228) in relativistic field theories. Show that invariance of the relativistic field theory    Q + Q μ m2 c 4 + + μ ∂μ φ + i φ Aμ · ∂ φ − i A φ − φ ·φ L = − hc ¯ h¯ h¯ h¯     ih¯ ↔ 1 + c γ μ ∂μ + qAμ − mc  − Fμν F μν . (22.273) 2 4μ0 2

under rotations and Lorentz boosts  μ = − δx μ = − ϕ μν xν , δφ(x) ≡ φ  (x  ) − φ(x) = 0, δ(x) =

i αβ ϕ Sαβ · (x), 2

ϕ μν = − ϕ νμ , δAμ (x) = ϕ μν Aν (x),

i δ(x) = − ϕ αβ (x) · Sαβ , 2

(22.274) (22.275) (22.276)

yields the conservations laws (22.272) from the results of Sect. 16.2 if proper improvement terms are added. The Lorentz generators Sαβ in the spinor representation are defined in Eq. (H.18). 22.17c We have seen in the previous problem that invariance of the relativistic theory (22.273) under the rotations δx i = ϕ ij xj =  ij k xj ϕk yields densities of conserved charges Mij 0 which we can express in vector form through Mij 0 = ij k Jk , Ji = ij k Mj k 0 /2, i.e. J = x × P. On the other hand, we have seen in Problem 16.6 that the total angular momentum density J of non-relativistic fermions contains a spin term which is not proportional to any spacetime coordinates xα , and yet we have also seen in Problem 16.8 that only the combination of both terms in (16.49) yields the density of a conserved quantity in the presence of spinorbit coupling. How can that be? Replace time derivatives on spinor fields in the momentum density using the Dirac equation. This yields spin contributions to the momentum density. Show that partial integration of the resulting spin contributions to the angular momentum density yields spin terms which reduce to the spin term in Eq. (16.49) in the nonrelativistic limit. Solution for 22.17b. The electric current density for the Lagrange density (22.273) is ↔  ∂L Q jqμ = = − iQc2 φ + ∂ μ − 2i Aμ φ + qcγ μ . (22.277) ∂Aμ h¯ Addition of the identically conserved improvement term −

% & 1 1 1 ∂ν xα Aβ F μν = − Aβ F μ α − xα ∂ν Aβ · F μν − xα Aβ jqμ μ0 μ0 μ0

(22.278)

22.7 Problems

631

to the conserved current (16.26) for the transformation (22.274)–(22.276) yields the gauge invariant conserved current     Q Q 2 ∂ μ φ + + i φ + Aμ ∂β φ − i Aβ φ j μ = − ϕ μν xν L + ϕ αβ xα hc ¯ h¯ h¯    Q Q 2 ∂β φ + + i φ + Aβ ∂ μ φ − i Aμ φ + hc ¯ h¯ h¯    ↔ hc q 1 ¯ μ μν ∂β − 2i Aβ  + Fβν F − i γ 2 μ0 h¯ & h¯ c % μ + ϕ αβ (22.279)  γ Sαβ + Sαβ γ μ . 4 The divergence of the spinor contributions in the last two lines of this current density is   ↔ & " c αβ ¯c ! % μ μ αβ h μ ∂μ j = ϕ ∂μ  γ Sαβ + Sαβ γ  − ϕ γα ih¯ ∂β + 2qAβ  4 2     ↔ c (22.280) − ϕ αβ xα ∂μ γ μ ih¯ ∂β + 2qAβ  . 2 The relation (22.143) implies that on-shell ϕ αβ

  ↔ ! % & " c hc ¯ ∂μ  γ μ Sαβ + Sαβ γ μ  − ϕ αβ γα ih¯ ∂β + 2qAβ  = 0, 4 2

and comparison of (22.140) and (22.145) implies that we can write the remaining μ part of ∂μ j in the form     ↔   ↔ c μ ∂μ j = − ϕ αβ xα ∂μ γ μ ih¯ ∂β + 2qAβ  + γβ ih¯ ∂ μ + 2qAμ  4     ↔   ↔ c αβ μ μ μ ih¯ ∂β + 2qAβ  + xα γβ ih¯ ∂ + 2qA  . = − ϕ ∂μ xα γ 4 The conserved current (22.279) is therefore equivalent to the conserved current jμ =

& c 1 αβ % ϕ xα Tβ μ − xβ Tα μ = ϕ αβ Mαβ μ , 2 2

(22.281)

with the symmetric stress-energy tensor for the Lagrange density (22.273) (cf. (22.146) and (22.247))   ↔ 1 c ih¯ ν ↔ ih¯ Tμ ν = ημ ν L −  γ ∂μ + γμ ∂ ν + qγ ν Aμ + qγμ Aν  + Fμλ F νλ 2 2 2 μ0

632

22 Relativistic Quantum Fields

   Q + Q ν + ν + hc ∂μ φ + i φ Aμ ∂ φ−i A φ ¯ h¯ h¯    Q + ν Q 2 ν + φ A + hc ∂ ∂ φ + i A φ − i φ . ¯ μ μ h¯ h¯ 2

(22.282)

Solution for 22.17c. We discuss the angular momentum densities in vector form, Ji = ij k Mj k 0 /2 = ij k xj Pk , with the momentum densities Pk = Tk 0 /c (cf. (22.148) and (22.249)), P=

    1 + h¯ ↔ 1 ih¯ ↔ q  ∇ − qA  + γ ∂0 −   +  0 E × B 2 2i 2 2 c    Q Q − h¯ φ˙ + − i φ +  ∇φ − i Aφ h¯ h¯    Q Q (22.283) − h¯ ∇φ + + i φ + A φ˙ + i φ . h¯ h¯

The γ matrices satisfy  γi · γj = − δij

10 01



1 + ij k kmn γm · γn = − δij 2



10 01

 +

2 ij k Sk ih¯

with the vector of 4 × 4 spin matrices S = ihγ ¯ × γ /4, cf. (22.126). The Dirac equation then implies  γ

   ih¯ ↔ q h¯ ↔ ∂0 −   =  + ∇ − qA  + ∇ × ( + · S · ), 2 c 2i

(22.284)

and we can write the momentum density in the form 

 1 h¯ ↔ ∇ − qA  + ∇ × ( + · S · ) + 0 E × B 2i 2    Q + Q + ˙ − h¯ φ − i φ  ∇φ − i Aφ h¯ h¯    Q Q − h¯ ∇φ + + i φ + A φ˙ + i φ . h¯ h¯

P = +

(22.285)

The spin term P S = ∇ × ( + · S · )/2 in the momentum density contributes a term to the angular momentum of the form  JS =

1 d 3 x x × [∇ × ( + · S · )] = 2



d 3 x  + · S · ,

(22.286)

22.7 Problems

633

such that we can write the total angular momentum density also in the form J = M + S,

(22.287)

with a spin contribution S =  + · S · , and an orbital angular momentum M = x × P O with the orbital momentum density: 

    h¯ ↔ Q Q ∇ − qA  − h¯ φ˙ + − i φ +  ∇φ − i Aφ h¯ h¯ 2i    Q Q (22.288) − h¯ ∇φ + + i φ + A φ˙ + i φ + 0 E × B. h¯ h¯

PO = +

Substitution of the non-relativistic approximations in the Dirac representation of the γ matrices (cf. (22.41) and (22.159)),  φ(x, t) →

  mc2 h¯ φ(x, t) exp − i t , h¯ 2mc2 

(x, t) =

   mc2 ψ(x, t) t exp − i χ (x, t) h¯

(22.289)

(22.290)

into (22.285) yields after neglecting the subleading χ components (cf. (22.162)) PO = ψ

+



   h¯ ↔ ¯ ↔ + h ∇ − qA ψ + φ ∇ − QA φ + 0 E × B 2i 2i

(22.291)

and S=

h¯ + ψ · σ · ψ. 2

(22.292)

If we would not have used the equations and motion and Eq. (22.143) to write the conserved current in the form (22.281), the spin term S =  + · S ·  would have come from the last line in Eq. (22.279). 22.18 New charges from local phase invariance? We have derived expressions for charge and current densities from phase invariance q δ(x) = i ϕ(x), h¯

q δ + (x) = − i ϕ + (x), h¯

(22.293)

of Lagrange densities, see e.g. (16.61) and (16.62) for the charge and current densities of non-relativistic charged matter fields, and (22.277) for relativistic charged matter fields. In the final expressions we always divided out the irrelevant

634

22 Relativistic Quantum Fields

constant parameter ϕ. However, introduction of the electromagnetic potentials rendered the Lagrange densities invariant under local phase transformations q δ(x) = i ϕ(x)(x), h¯

q δ + (x) = − i ϕ(x) + (x), h¯

δAμ (x) = ∂μ ϕ(x).

In this case we cannot discard the phase parameter ϕ(x) from the current densities for the local symmetry. Does this provide us with additional useful notions of conserved charges for quantum electronics and quantum electrodynamics? 22.18a Show that application of the result (16.26) to local phase transformations yields current densities J μ = ϕj μ +

1 μν F ∂ν ϕ, μ0

(22.294)

where j μ are the current densities which were derived for constant phase parameter ϕ, e.g. (16.61), (16.62) or (22.277). Show also that ∂μ J μ = ϕ∂μ j μ ,

(22.295)

i.e. local charge conservation for QV = V d 3 xj 0 /c is equivalent to local charge conservation for Qϕ,V = V d 3 xJ 0 /c: The fact that QV can only change through a current j flowing through the boundary of V is equivalent to the fact that Qϕ,V can only change through a current J flowing through the boundary of V . Equation (22.295) already tells us that we cannot learn anything substantially new from the generalized currents J μ . However, we would still like to understand the relation between the ordinary charges Q and the generalized charges Qϕ . 22.18b Show that the currents J μ can also be written in the strong form11 Jμ =

% & 1 ∂ν ϕF μν . μ0

(22.296)

Show in particular that the charge density Qϕ = J 0 /c can be written in the form Qϕ = 0 ∇ · (ϕE),

(22.297)

and that the current density is J =

11 A current J μ

is an identity.

1 ∇ × (ϕB) − 0 ∂t (ϕE). μ0

(22.298)

is sometimes denoted as strongly conserved if the local conservation law ∂μ J μ = 0

22.7 Problems

635

22.18c Any local conservation law ∂μ J μ implies a global conservation law for the corresponding charge with density J 0 /c only if the current J does not yield radial fluxes at spatial infinity. In the current problem this implies for the a priori timedependent quantity  Qϕ (t) ≡

d 3 x Qϕ (x, t)

(22.299)

the statement * lim

|x|→∞

d 2  |x|x · J (x, t) = 0 d Qϕ (t) = 0. dt

⇔

(22.300) (22.301)

Show that the condition (22.300) implies that the average value of ϕ(x, t) over a sphere at infinity must be time-independent, 1 |x|→∞ 4π

ϕ ≡ lim

*

d ϕ = 0. dt

d 2  ϕ(x, t),

(22.302)

Furthermore, show that this implies that the corresponding charge Qϕ differs from the standard electric charge  Q=

d 3 x (x, t)

(22.303)

only by a constant factor Qϕ = ϕQ.

(22.304)

Promotion of global phase invariance to local phase invariance therefore does not create substantially new charges. Hint: The radial electric field Er (x, t) for a localized charge distribution (x, t) = j 0 (x, t)/c has a static limit at r ≡ |x| → ∞, Q , lim Er (x, t) = |x|→∞ 4π 0 r 2

 Q=

d 3 x (x, t).

(22.305)

Furthermore, you can use Eq. (5.100) (with azimuthal angle φ to avoid confusion with the current phase factor ϕ(x, t)) to show that the magnetic contribution to the current J (22.298) does not contribute to radial fluxes, * d 2  |x|x · [∇ × (ϕB)] = 0. (22.306)

636

22 Relativistic Quantum Fields

Remarks 1. Equation (22.305) is a consequence of lim|x|→∞ Ar (x, t) = 0 for spatially localized currents, which in turn is a consequence of d 3 x jr (x, t) = 0 for those currents. 2. It was clear from Eqs. (22.294) and (22.296) that Qϕ /Q must be a functional with respect to the spatial coordinates of ϕ, ϕ(x, t) → F [ϕ](t), and that this functional must be time-independent for Qϕ to be conserved. The derivation of Eq. (22.304) clarifies which functional of ϕ describes the relationship between Qϕ and Q. 22.19 Show for a free electron that Eq. (22.162) implies that a positron component φ in the wave function is not negligible any more relative to the electron wave function ψ at a distance of order d

4mc x 2  104 nm−1 x 2 = 1011 cm−1 x 2 . h¯

(22.307)

This implies that we cannot use the wave packet for a strongly localized free electron with x = 1 Å beyond a distance of about 0.1 µm from the center. However, for a free electron wave packet with x = 1 mm the limit (22.307) is much larger than the confines of any physics or chemistry lab and therefore of no concern. Show also that at time t, the estimate for the usable range of the wave packet is d(t) 

4mc x(0)x(t). h¯

(22.308)

22.20 When we apply charge conjugation (22.130) to products of fermion operators, we have to take into account their anti-commutation properties to find how these products behave under charge reversal. For example ()c = ( c )+ γ 0  c =  T iγ 2 γ 0 iγ 2  ∗ = −  T γ 0  ∗ =  + γ 0  = .

(22.309)

Show that charge conjugation indeed reverses the signs of the current densities j μ = qγ μ , (γ μ )c = ( c )+ γ 0 γ μ  c = − γ μ .

(22.310)

22.21 Show that under a unitary transformation from the Dirac or Weyl basis into another basis of Dirac matrices, γμ = U γμ U + ,

  = U ,

(22.311)

the property (22.127) and the charge conjugation law (22.130) generalize to

22.7 Problems

637

γμ∗ = − Cγμ C −1 ,

C = U ∗ γ2 U + ,

(22.312)

and  c = U  c = − iC ∗  ∗ ,

C ∗ = C −1 = − U γ2 U T .

(22.313)

Note also that for an orthogonal transformation, U + = U T , we find C = − C ∗ = γ2 , i.e. we recover again the relations (22.127) and (22.130) in the transformed basis. 22.22 Show that the Foldy-Wouthuysen transformation (22.203) can also be written as |ϒF W T (t) = exp[F (t) + W (t) + T (t) + C(t) + O(E/mc2 )4 ]|ϒ(t), with C(t) =

% & q h¯ 0 γ h∇ · E(x, t) + 2iE(x, t) · [p − qA(x, t)] ¯ 16m3 c4   & σ 0 q h¯ 2 % . (22.314) ∇ × E(x, t) · +i 0 −σ 16m3 c4

22.23 We have derived Eqs. (12.58) and (12.60) for particles which satisfy the relativistic dispersion equation. In the meantime, we have seen that at the quantum level these particles are described by scalar fields φ, spinors ψ, or vector fields Aμ . The factor g counts spin and internal symmetry degrees of freedom and has the form g = gs × drep(G) , where drep(G) is the dimension of the representation of the internal symmetry group G under which the fields transform. 22.23a Scalar fields have gs = 1. Show that in d + 1 spacetime dimensions, gs = 2-(d−1)/2. for Dirac fields and gs = d − 1 for massless vector fields. Hint: A Dirac spinor in d + 1 spacetime dimensions has 2-(d+1)/2. components, see Appendix G (note that there d denotes the number of spacetime dimensions). 22.23b We have seen that scalar fields can be either real or complex (and similar remarks apply to spinor and vector fields if we go beyond quantum electrodynamics into the standard model of particle physics). However, a complex field has twice as many degrees of freedom as a real field. Should g therefore not include an additional factor gc with gc = 2 for complex fields and gc = 1 for real fields? 22.24 Formulate the basic relations for basis kets |x, μ, |k, α for the potentials |AD  in Lorentz gauge in analogy to the corresponding relations (18.37)–(18.41) in Coulomb gauge. 22.25 Show that the gauge transformation (22.229) and (22.231) takes us from any vector potential Aμ (x) which satisfies Maxwell’s equations into the vector potential in Coulomb gauge.

638

22 Relativistic Quantum Fields

22.26 Substitute the Liénard-Wiechert potential (22.219) for μ = Eq. (22.231) to show that this equation can also be written in the form  g(x, t) =

d 3x

0 into

   t 1       ∇ · A(x , −∞) − dt (x , t ) . 0 4π 0 |x − x  | t−|x−x  |/c

22.27 We have seen that Lorentz gauge corresponds to an effective Lagrange density (22.221) which allows for all four components Aμ of the vector potential to appear in the canonical quantization condition (22.223), whereas in Coulomb gauge we eliminated  = cA0 by solving Eq. (18.6). Could it be possible that we could somehow also find a way to include  with a canonical quantization condition in Coulomb gauge? Show that Maxwell’s equations (18.6) and (18.7) with ∇(∇ · A) = 0 follow from the two equivalent Lagrange densities L1 = j · A −  +

0 1 1 ∂μ A · ∂ μ A + ∇ · ∇ − ∇ · ∂0 A, 2 2μ0 μ0 c

L2 = j · A −  +

1 ∂ 0 ∇ · ∇ − ∇ · A. ∂μ A · ∂ μ A +  0 2 2μ0 ∂t

Does any of these yield a conjugate momentum " which would allow for canonical quantization of the potential  in Coulomb gauge? Remark The full Maxwell’s equations (18.6) and (18.7) without any gauge condition also follow from the Lagrange density L = j · A −  +

0 0 ∂A ∂A 1 ∂ ∇ · ∇ + 0 ∇·A+ · − (∇ × A)2 . 2 ∂t 2 ∂t ∂t 2μ0

This Lagrange density is (of course) equivalent to the standard Lagrange density (18.3), but it yields " = 0 ∇ · A as a conjugate momentum for . If we wish to minimize the number of quantized photon operators, then this leads us back to Coulomb gauge. On the other hand, if we wish to treat A0 = − /c relativistically symmetric with the other potentials Ai , then this leads us back to Lorentz gauge, ∇ · A = − c−2 ∂/∂t. 22.28 We have derived the general gauge invariant symmetric energy-momentum tensor for spinor QED and the corresponding Hamiltonian in equations (22.146) and (22.147), respectively. Using the equation of motion for  = cA0 and the gauge condition, show that in Lorentz gauge (up to boundary terms at spatial infinity) & 0 2 1 2 1 % E + ∂0 Aν · ∂0 Aν + ∇Aν · ∇Aν . B =  + 2 2μ0 2μ0

(22.315)

Together with the interaction term in (22.147), this yields a total interaction term

22.7 Problems

639

HI = − qc · γ μ Aμ ·  = −j μ Aμ

(22.316)

in spinor QED in Lorentz gauge. Note that the Lorentz invariant form of the interaction Hamiltonian (22.316) is an accidental consequence of using the gauge condition and equations of motion to extract an interaction piece from the energy density of the electromagnetic fields. The free part H0 is not Lorentz invariant and H = − T0 0 is a tensor component anyway. 22.29 We have derived the general gauge invariant symmetric energy-momentum tensor for scalar QED and the corresponding Hamiltonian in equations (22.247) and (22.248), respectively. Using the same techniques as in the previous problem, show that the Hamiltonian in Lorentz gauge for QED with scalar matter is H=

& 1 % + 2 + ∂0 Aν · ∂0 Aν + ∇Aν · ∇Aν + h∂ ¯ t φ · ∂t φ + hc ¯ ∇φ · ∇φ 2μ0 +

↔ q2 m2 c 4 + φ φ + iqc2 A · (φ + ∇ φ) + c2 φ + Aμ Aμ φ. h¯ h¯

(22.317)

You can verify independently that this is the correct Hamiltonian for QED with scalar matter in Lorentz gauge by showing that it yields the correct Heisenberg evolution equations. However, you have to substitute the canonical momenta for the time derivatives and use the canonical commutation relations for the scalar fields, e.g. "φ =

∂L = h¯ φ˙ + − iqφ + , ∂ φ˙

 ["φ (x  , t), φ(x, t)] = − ihδ(x − x), ¯

(22.318)

to verify the Heisenberg evolution equations. 22.30 Show that the free photon part in Eqs. (22.315) and (22.317) yields the correct photon terms for the Hamiltonian in Lorentz gauge,  Hγ =

d 3x 

=

d 3k

" 1 ! ∂0 Aν (x, t) · ∂0 Aν (x, t) + ∇Aν (x, t) · ∇Aν (x, t) 2μ0 3

h¯ c|k|aα+ (k)aα (k),

(22.319)

α=1

where normal ordering was used. 22.31 Edelstein effects 22.31a Consider the two-dimensional Hamiltonian for electrons in the (x, y) plane with a Rashba term,

640

22 Relativistic Quantum Fields

Fig. 22.2 The dispersion relation E(k) for motion in the (x, y) plane with the Hamiltonian (22.320). The red arrows show the spin orientation of the electrons for k vectors on the inner or outer surface if vR > 0

H =

p2 + vR (p × σ ) · zˆ . 2m

(22.320)

Calculate the energy eigenvalues and eigenstates for the electrons moving in the (x, y) plane. 22.31b Show that the energy eigenvalues E± (k) =

h¯ 2 k 2 ± hv ¯ Rk 2m

(22.321)

correspond to the dispersion relation displayed in Fig. 22.2, where the arrows show the spin orientation of the electrons with that particular energy and momentum if vR > 0. Hint: First plot the energy dispersion relations E± (kx ) for ky = 0. Which spin orientations do the corresponding eigenspinors have for E+ (kx ) and for E− (kx )? Equivalently, you can also use the fact that the eigenspins of a matrix a · σ point in directions ±a. The effect that electrons for given energy E(k) and smaller momentum hk ¯ have opposite spin polarization in the plane than the electrons with the same energy and larger momentum hk ¯ (in the same direction of momentum) is denoted as chiral spinmomentum locking, since it produces the picture of opposite rotations of spins when moving around the inner or outer surface in Fig. 22.2. 22.31c It is convenient to parametrize the dispersion relation (22.321) depicted in Fig. 22.2 in the form

22.7 Problems

641

E± (k) =

h¯ 2 k 2 ± h¯ |vR |k, 2m

(22.322)

such that E± (k) → E± (k) if vR > 0 and E± (k) → E∓ (k) otherwise. With the representation (22.322), the inner cone corresponds to E+ (k) and the outer cone to E− (k). Calculate the two-dimensional densities ± (E) of states in the energy scale for the two dispersion relations (22.322). Show that the density of states on the outer cone always exceeds the density of states on the inner cone for the same energy, − (E) > + (E). 22.31d Now assume that we apply an electric field E = E xˆ in x direction. This will move electrons in −x direction, i.e. it will increase electron population for momenta kx < 0 and decrease population with momenta kx > 0. Show that this will produce a spin current in −x direction with net polarization in +y direction if vR > 0. This is the Edelstein effect [46]. 22.31e Now suppose that we inject electrons with spin polarization in y direction into the surface and extract electrons with spin polarization in −y direction from the surface. If we inject and extract equal numbers of electrons for both polarizatons from the same side of the surface, e.g. from above, then this does not correspond to a net electric current in the z direction. However, show that this procedure leads to an increase of electrons with kx < 0 and a decrease of electrons with kx > 0 if vR > 0, i.e. it produces a net electron current in −x direction, which is an electric current in +x direction. This effect of creating an electric current by inducing a spin polarization in the surface is the inverse Edelstein effect.

Chapter 23

Applications of Spinor QED

We have seen in Chap. 18 that inclusion of the quantized Maxwell field did not change the basic formalism of time-dependent perturbation theory, see Eqs. (18.105)–(18.107), and this property also persists after promotion of the matter fields in the Hamiltonian to relativistic Klein–Gordon or Dirac fields. In the following sections we will use the Hamiltonian (22.155) of spinor quantum electrodynamics for the calculation of scattering processes. However, we first need to generalize our previous results for the scattering matrix to the case of two free particles in the initial and final state.

23.1 Two-Particle Scattering Cross Sections We have discussed events with one free particle in the initial or final state of a scattering event in the framework of potential scattering theory in Chaps. 11 and 13, or in photon emission, absorption or scattering off bound electrons in Sects. 18.6– 18.9. The techniques that we have discussed so far cover many applications of scattering theory, but eventually we also wish to understand scattering involving two (quasi-)free particles in the initial and final states. Electron scattering off atomic nuclei, electron-electron scattering, electron-photon scattering, or electron-phonon scattering provide examples of these kinds of scattering events which happen all the time in materials. In these cases we are discussing scattering events with two particles in the initial or final states. We should therefore address the question how to generalize the equations from Sects. 13.6 and 18.9, which dealt with the case of one free particle in the initial and final state. Let us recall from Sect. 13.6 that with a free particle with wave vectors k and k  in the initial and final state, the scattering matrix element for a monochromatic perturbation W (t) ∼ exp(− iωt)

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 R. Dick, Advanced Quantum Mechanics, Graduate Texts in Physics, https://doi.org/10.1007/978-3-030-57870-1_23

643

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23 Applications of Spinor QED

Sk  ,k = k  |UD (∞, −∞)|k = − iMk  ,k δ(ω(k  ) − ω(k) − ω)

(23.1)

has dimension length3 due to the length dimensions of the external states, and yields a differential scattering cross section dσk→k

  S  2 k ,k =d k T dj (k)/d 3 k   M  2 k ,k 3  =d k δ(ω(k  ) − ω(k) − ω). 2π dj (k)/d 3 k 3 

(23.2)

Here we substituted the more precise notation dj (k)/d 3 k for the incoming current density jin per k space volume. Substitution of the current density for a free particle of momentum hk, ¯ dj (k)/d 3 k = v/(2π )3 , yields  2 vdσk→k = 4π 2 Mk  ,k  δ(ω(k  ) − ω(k) − ω)d 3 k  ,

(23.3)

see also Eq. (18.191), where we found this equation for photon scattering off atoms or molecules. Now suppose that we have two free particles with momenta k and q in the initial state, and they scatter into free particles with momenta k  and q  in the final state. We also assume that the scattering preserves total energy and momentum. The corresponding scattering matrix element Sk  ,q  ;k,q = k  , q  |UD (∞, −∞)|k, q = − iMk  ,q  ;k,q δ(k  +q  −k−q)

(23.4)

 2 has dimension length6 . This is consistent with the fact that Sk  ,q  ;k,q  is a transition probability density per volume units d 3 k  d 3 q  d 3 kd 3 q in wave vector space. For ease of the present discussion, we also assume that the scattering particles are different, like in electron-photon or electron-phonon scattering, and we will use markers e and γ to label quantities refering to the different particles. The notation is motivated from electron-photon scattering, but we will develop the formalism in this section with general pairs of particles of masses me and mγ in mind. Suppose the two particles have momentum 4-vectors pe = hk ¯ = h(ω ¯ e /c, k),

pγ = h¯ q = h¯ (ωγ /c, q)

(23.5)

relative to the laboratory frame in which we observe the collisions. The scattering rate will be proportional to the product d e (k) d γ (q) d e (k) djγ (q) = v˜eγ , 3 3 d k d q d 3k d 3q where

(23.6)

23.1 Two-Particle Scattering Cross Sections

v˜eγ

645

  k q  =c  − ωe ωγ  2

(23.7)

is the relative speed between the two particles that we assign from the point of view of our laboratory frame. The speed v˜eγ is usually replaced with another measure for relative speed between the two particles, veγ =

c3 ωe ωγ

(k · q)2 −

m2e m2γ c4 h¯ 4

(23.8)

,

which agrees with v˜eγ in laboratory frames in which the two momentum vectors p e and p γ are parallel or anti-parallel, or where the laboratory frame coincides with the rest frame of one of the two particles: 2 2 − veγ = v˜eγ

 c6 2 2 2 |k| |q| 1 − cos ϑ . ωe2 ωγ2

(23.9)

Here ϑ is the angle between pe and pγ . E.g. in the rest frame of the e-particle, hk ¯ e = h¯ (ωe /c, 0) = (me c, 0), we find veγ =

c ωγ

ωγ2 −

m2γ c4 h2 ¯

= c2

|pγ | Eγ

= vγ ,

(23.10)

and in the center of mass frame of the two particles, k = −q, we also find the difference of particle velocities, 2 veγ = c2

= c4

h¯ 4 (ωe ωγ + c2 k 2 )2 − m2e m2γ c8 h¯ 4 ωe2 ωγ2 2h¯ 2 c2 k 4 + 2h¯ 2 ωe ωγ k 2 + c4 k 2 (m2e + m2γ )



k q = c − c2 ωe ωγ 2

2

h¯ 2 ωe2 ωγ2

= c4 k 2

ωe2 + ωγ2 + 2ωe ωγ ωe2 ωγ2 (23.11)

.

As a byproduct we also find another useful formula for the relative speed in the center of mass frame, veγ = c2 |k|

ωe + ωγ . ωe ωγ

(23.12)

Please keep in mind that (23.12) is the relative particle speed assigned to the two colliding particles by an observer at rest in the center of mass frame, but not the

646

23 Applications of Spinor QED

speed of one particle relative to the other particle as measured in the rest frame of one of the particles. The differential cross section for two-particle scattering can then be defined through the equation 3  3 

vdσk,q→k ,q  = vd k d q 3  3 

=d kd q

 S

2  k  ,q  ;k,q 3 V T (d e /d k)(djγ /d 3 q)   S   2 k ,q ;k,q

V T (d e /d 3 k)(d γ /d 3 q)

.

(23.13)

 2 In words, we divide the scattering rate d 3 k  d 3 q  Sk  ,q  ;k,q  d 3 kd 3 q/T between wave vector volumes d 3 kd 3 q → d 3 k  d 3 q  by the number of scattering centers V d e in the phase space volume V d 3 k and the incoming particle flux djγ in the wave vector volume d 3 q to calculate dσk,q→k  ,q  . If we substitute the scattering amplitude Mk  ,q  ;k,q for the scattering matrix element and use for the four-dimensional δ function in momentum space the equation 1 k→0 (2π )4

δ 4 (0) = lim

 d 4 x exp(ik · x) =

cV T , (2π )4

(23.14)

we find 3  3 

vdσk,q→k  ,q  = d k d q

 2 c Mk  ,q  ;k,q  (2π )4 (d e /d 3 k)(d γ /d 3 q)

δ 4 (k  + q  − k − q).

(23.15) The density per x space volume and per unit d 3 k of k space volume for momentum eigenstates is 1 d = . 3 d k (2π )3

(23.16)

This yields  2 vdσk,q→k ,q  = 4π 2 c Mk  ,q  ;k,q  δ 4 (k  + q  − k − q)d 3 k  d 3 q  .

(23.17)

Note from Eq. (23.4) that the two-particle scattering amplitude Mk  ,q  ;k,q has the dimension length2 while the single-particle scattering amplitude Mk  ,k has dimension length3 /time due to the use of a δ function in frequencies rather than wave numbers in the single-particle case. We can derive a single-particle scattering cross section from (23.17) by integrating over the final wave number of one of the two particles, e.g. q  , while considering its initial wave number fixed, e.g. q = 0. This yields

23.1 Two-Particle Scattering Cross Sections

 vdσk→k  = v

d 3q 

647

dσk,0→k  ,q 

d 3q   2 = 4π 2 c2 Mk  ,k−k  ;k,0  δ(ω(k  ) − ω(k) − ωq )d 3 k  ,

(23.18)

with ωq = ω(q = 0) − ω(q  = k − k  )

(23.19)

and a resulting single-particle scattering amplitude Mk  ,k = cMk  ,k−k ;k,0 .

Measures for Final States with Two Identical Particles To explain the necessary modifications of the previous results if we have two identical particles in the final state, we first consider decay of a normalizable state |i into two identical particles with momenta hk ¯ 1 and h¯ k 2 . The initial state belongs to a set of orthonormal states, i|j  = δij . For the final states with two identical particles, we have to take into account that the decomposition of unity on identical two-particle states is 1identical 2−particle states

1 = 2 =

1 2





d 3 k 2 |k 1 , k 2 k 1 , k 2 |

3

d k1 

 d 3k1

d 3 k 2 a + (k 1 )a + (k 2 )|00|a(k 2 )a(k 1 ).

If the scattering matrix allows only for decay into the two-particle states, unitarity UD+ UD = 1 (or equivalently S + S = 1) implies 1 2



 d 3k1

d 3 k 2 |k 1 , k 2 |UD (∞, −∞)|i|2 = 1.

(23.20)

The proper probability density for the transition |i → |k 1 , k 2  is therefore wi→k 1 ,k 2 =

1 3 d k 1 d 3 k 2 |k 1 , k 2 |UD (∞, −∞)|i|2 . 2

(23.21)

Equation (23.17) for the two-particle scattering cross section must therefore be modified if the final state contains two identical particles, 2    vdσk 1 ,k 2 →k 1 ,k 2 = 4π 2 c Mk 1 ,k 2 ;k 1 ,k 2  δ 4 (k1 + k2 − k1 − k2 ) 1 × d 3 k 1 d 3 k 2 , 2

(23.22)

648

23 Applications of Spinor QED

and the total two-particle cross section is  σ =

d 3 k 1



d 3 k 2

dσk 1 ,k 2 →k1 ,k 2

d 3 k 1 d 3 k 2   2 c  1  = d 3 k 1 d 3 k 2 4π 2 Mk 1 ,k 2 ;k 1 ,k 2  δ 4 (k1 + k2 − k1 − k2 ).(23.23) 2 v

However, if we want to derive an effective single-particle differential scattering cross section dσ/d from dσk 1 ,k 2 →k1 ,k 2 by integrating over the momentum of one particle and the magnitude of momentum of the second particle using the energymomentum conserving δ function, we have to take into account that the particle observed in direction d can be either one of the two scattered particles: dσ = d



d 3 k 1 

+



∞

0

d 3 k 2

d|k 2 | |k 2 |2

 0

∞

dσk 1 ,k 2 →k1 ,k 2

d|k 1 | |k 1 |2

d 3 k 1 d 3 k 2

dσk 1 ,k 2 →k1 ,k 2 d 3 k 1 d 3 k 2

.

(23.24)

In the center of mass frame this reduces to a factor of 2, dσ =2 d



d 3 k 2

 0

∞

d|k 1 | |k 1 |2

dσk 1 ,k 2 →k1 ,k 2 d 3 k 1 d 3 k 2

.

(23.25)

If we then wish to calculate the total two-particle scattering cross section (23.23) from the single-particle differential cross section (23.25), we have to compensate with a factor 1/2,  1 dσ . (23.26) σ = d 2 d In practice one is often only interested in the effective single-particle differential cross section (23.25) and the total two-particle scattering cross section σ . The factor 1/2 is then usually neglected in the differential two-particle cross section dσk 1 ,k 2 →k 1 ,k 2 /(d 3 k 1 d 3 k 2 ), so that the factor of 2 is not needed in the calculation of dσ/d (23.25) because it has been absorbed in dσk 1 ,k 2 →k1 ,k 2 /(d 3 k 1 d 3 k 2 ). However, the factor 1/2 is still needed in the calculation of the total two-particle cross section σ from dσ/d according to Eq. (23.26).

23.2 Electron Scattering off an Atomic Nucleus

649

23.2 Electron Scattering off an Atomic Nucleus As a first application of two-particle scattering, we discuss scattering of an electron off an atomic nucleus. We assume that the nucleus is also a fermion and that the electrons are not energetic enough to resolve the internal structure of the nucleus. In that case we can use an effective description of the nucleus through Dirac field operators for a particle of charge Ze and mass M for the nucleus. The Coulomb gauge Hamiltonian (22.155) has the form H = H0 + Heγ + HN γ + HC ,

(23.27)

where the free part H0 contains the kinetic and mass terms and we have separated the different interaction terms. The electron-photon and nucleus-photon interaction terms are  Heγ = ec

d 3 x ψ(x, t)γ · A(x, t)ψ(x, t)

(23.28)

and  HN γ = −Zec

d 3 x (x, t)γ · A(x, t)(x, t),

(23.29)

respectively. The relevant part of the Coulomb interaction term is the term describing the interaction of the electron and the nucleus, HeN = −Z

  ψc+ (x, t)c+ (X, t)c (X, t)ψc (x, t) e2 , d 3x d 3X 4π 0  |x − X| cc

where the sum is over 4-spinor indices. The relevant leading order matrix element contains two terms, Sf i = − iMf i δ(k + K − k  − K  ) = Sf i + Sf i , (γ )

(C)

(23.30)

which correspond to photon exchange, Sf i = K  , S  ; k  , s  | (γ )

Ze2 h¯ 2

  T d 4 x d 4 X ψ(x)γ · A(x)ψ(x)

×(X)γ · A(X)(X)|K, S; k, s, or Coulomb scattering, Sf i = K  , S  ; k  , s  | (C)

iZe2 μ0 c 4π h¯



 d 4x

d 3X

(23.31)

650

23 Applications of Spinor QED

×

ψc+ (x, t) + (X, t)c (X, t)ψc (x, t) c

|x − X|

cc

|K, S; k, s. (23.32)

We first calculate the Coulomb contribution to the scattering amplitude. Evaluation of the operators yields + + Ze2 μ0 c us  (k  ) · us (k)uS  (K  ) · uS (K)  8(2π )7 h¯ Ee (k  )Ee (k)EN (K  )EN (K)   exp[i(K − K  ) · X + i(k − k  ) · x] 4 × d x d 3X |x − X|

(C)

Sf i = i

× exp[− i(ωe (k) + ωN (K) − ωe (k  ) − ωN (K  ))t].

(23.33)

In the next step we use the Fourier decomposition of the Coulomb potential  d 3x

4π exp(− iq · x) = 2 |x| q

(23.34)

to find (C)

Sf i = i

+ + Ze2 μ0 c us  (k  ) · us (k)uS  (K  ) · uS (K)  4(2π )2 h¯ Ee (k  )Ee (k)EN (K  )EN (K)

×

δ(k + K − k  − K  ) . |k − k  |2

(23.35)

For the evaluation of the photon exchange contribution (23.31), we first note that the photon operators between the photon vacuum states yield  3 d q hμ ¯ 0c  α (q) ⊗  α (q) 0|TA(x) ⊗ A(X)|0 = 3 2|q| α (2π )   × (t − T ) exp[iq · (x − X)] + (T − t) exp[− iq · (x − X)] =

hμ ¯ 0c 1 (2π )3 2π i



  exp[iq · (x − X)] q ⊗q . 1 − d 4q q 2 − i q2

q 0 =ωγ (q)/c=|q|

(23.36)

Evaluation of the fermion operators yields K  , S  ; k  , s  |ψ(x)γ ψ(x) ⊗ (X)γ (X)|K, S; k, s =

us  (k  )γ us (k) ⊗ uS  (K  )γ uS (K)  exp[i(k − k  ) · x + i(K − K  ) · X]. 4(2π )6 Ee (k  )Ee (k)EN (K  )EN (K)

23.2 Electron Scattering off an Atomic Nucleus

651

Assembling the pieces yields (γ )

Sf i =

δ(k + K − k  − K  ) Ze2 μ0 c 1  4i(2π )2 h¯ Ee (k  )Ee (k)EN (K  )EN (K) (k − k  )2 − i   (k − k  ) ⊗ (K − K  )  ×us  (k )γ us (k) 1 + uS  (K  )γ uS (K), |k − k  |2

where k−k  = −(K −K  ) from momentum conservation was used in the projection term. Substitution of the free Dirac equation for the external fermion states yields (γ )

Sf i =

Ze2 μ0 c δ(k + K − k  − K  ) 1  2  4i(2π ) h¯ Ee (k )Ee (k)EN (K  )EN (K) (k − k  )2 − i

× us  (k  )γ us (k) · uS  (K  )γ uS (K)  + u+ s  (k )us (k)

[Ee (k) − Ee (k  )][EN (K) − EN (K  )] h¯ 2 c2 |k − k  |2

 u+ S  (K )uS (K)



and energy conservation yields finally (γ )

Sf i =

Ze2 μ0 c δ(k + K − k  − K  ) 1  2   4i(2π ) h¯ Ee (k )Ee (k)EN (K )EN (K) (k − k  )2 − i

× us  (k  )γ us (k) · uS  (K  )γ uS (K) − us  (k  )γ 0 us (k) (γ )

(Ee (k) − Ee (k  ))2 h¯ 2 c2 |k − k  |2

 uS  (K  )γ 0 uS (K) . (23.37)

(C)

The sum Sf i = Sf i + Sf i contains a term 1 |k − k  |2

  (Ee (k) − Ee (k  ))2 1 1+ 2 . = (k − k  )2 − i h¯ c2 (k − k  )2 − i

(23.38)

This yields finally the scattering matrix element Sf i = − iMf i δ(k + K − k  − K  ) =

ZαS us  (k  )γ μ us (k)uS  (K  )γμ uS (K) δ(k + K − k  − K  )  , (23.39) 4π i (k − k  )2 − i Ee (k  )Ee (k)EN (K  )EN (K)

where the fine structure constant αS = μ0 ce2 /4π h¯ (7.149) was substituted. The differential scattering cross section for electron-nucleus scattering is then given by (23.17)

652

23 Applications of Spinor QED

veN dσ = 4π 2 c|Mf i |2 δ(k + K − k  − K  )d 3 K  d 3 k 

(23.40)

with the relativistic expression for the relative velocity of the electron and the nucleus, veN =

 c3 h¯ 4 (K · k)2 − m2 M 2 c4 , Ee (k)EN (K)

(23.41)

where m is the electron mass. Integration over the momentum K  of the scattered nucleus and the magnitude k  of the scattered electron momentum yields   dσ k 2 2 2  = 4π c|Mf i | , veN   dk  |∂f (k )/∂k | f (k  )=0

(23.42)

where k  has to satisfy the condition f (k  ) =

  (k  − K − k)2 + (Mc/h) ¯ 2 + k 2 + (mc/h) ¯ 2   2 − k 2 + (mc/h)2 = 0. − K 2 + (Mc/h) ¯ ¯

(23.43)

Usually we are not interested in the scattering with fixed initial and final spin polarizations. Therefore we average over initial spins and sum over final spins to calculate the unpolarized scattering cross section, |Mf i |2 →

1 |Mf i |2 . 4  

(23.44)

s,s ,S,S

We will further evaluate these expressions for the case of negligible momenta and momentum transfers compared to Mc. We can take this into account through the limit M → ∞. The condition (23.43) then reduces to the elastic electron scattering condition k = k  which yields  k 2 k 2 + (mc/h) = k ¯ 2. |∂f (k  )/∂k  |

(23.45)

Furthermore, using the Dirac representation (22.67) of the γ matrices yields with Eqs. (22.83), (22.84), and (22.93) for the 4-spinors the limit uS  (K  )γ μ uS (K) lim  = 2δSS  ημ 0 . M→∞ EN (K  )EN (K)

(23.46)

Note that this equation is invariant under similarity transformations of the γ matrices and therefore holds in every representation. Furthermore, the speed veN

23.2 Electron Scattering off an Atomic Nucleus

653

(23.41) becomes the electron speed in the rest frame of the heavy nucleus, veN = c2 hk/E ¯ e (k). The spin polarized scattering amplitude following from (23.39) in the heavy nucleus limit is Mf i = −

ZαS us  (k  )γ 0 us (k) δSS   .  2π Ee (k )Ee (k) (k − k  )2 − i

(23.47)

The remaining electron spin averaging is easily accomplished using Eq. (22.101), 2  1  c2   0  0 us  (k  )γ 0 us (k) = tr (mc − hk ¯ · γ )γ (mc − hk ¯ · γ )γ . 2  2

(23.48)

ss

The trace theorems for products of γ matrices in Appendix G, in particular Eqs. (G.55) and (G.56) in the form 

tr γκ γ 0 γμ γ 0 = 8ηκ 0 ημ 0 + 4ημκ ,

(23.49)

and the vanishing traces of odd numbers of γ matrices yield 2 1   us  (k  )γ 0 us (k) = 2m2 c4 + 4h¯ 2 c2 k 0 k 0 + 2h¯ 2 c2 k · k  2  ss

= 2c2 (h¯ 2 k 0 k 0 + h¯ 2 k · k  + m2 c2 ).

(23.50)

This yields the unpolarized differential scattering cross section for electrons in the field of a heavy nucleus, Z 2 αS2 2m2 c2 + h¯ 2 k 2 (1 + cos θ ) dσ = , dk  k 4 (1 − cos θ )2 2h¯ 2

(23.51)

where k ≡ |k|. The Rutherford scattering formula (11.82) follows for h¯ k % mc. Electron scattering off a heavy nucleus is equivalent to scattering in an external Coulomb field Ze/4π 0 r. This is known as Mott scattering [121]. From our calculation, we can easily understand why scattering off heavy nuclei is equivalent to scattering in an external Coulomb field. The Coulomb scattering matrix element (23.35) yields already the amplitude (23.47) in the heavy nucleus limit, while the photon exchange matrix element (23.37) vanishes  in that limit. If we take into account that terms of the form us  (k  )γ us (k)/ Ee (k  )Ee (k) are of order h¯ 2 |k||k  |/m2 c2 % 1 in the non-relativistic limit, we find that the ratio between the photon exchange amplitude and the Coulomb amplitude in the non-relativistic limit is of order

654

23 Applications of Spinor QED

   (γ )   Sf i  (Ee (k) − Ee (k  ))2 h¯ 2 (k 2 − k 2 )2     (C)  4m2 c2 (k − k  )2 h¯ 2 c2 (k − k  )2  Sf i  =

k − k  h¯ 2 (k + k  ) ⊗ (k + k  ) k − k  · % 1, · 4m2 c2 |k − k  | |k − k  |

(23.52)

If we denote the average velocity of the incoming and the scattered electron with v e , Eq. (23.52) tells us that photon exchange is suppressed by about p 2e /m2 c2 = v 2e /c2 compared to the Coulomb interaction in the non-relativistic limit. That is the reason why Coulomb gauge is convenient for the description of systems with nonrelativistic charged particles. We can use Coulomb potentials in the calculation of scattering events and bound states of the non-relativistic particles without worrying about photon exchange. The photon terms are only needed for photon absorption and emission, and for photon scattering. On the other hand, if we are primarily concerned with interactions of relativistic charged particles, then use of a Hamiltonian like (22.147) with covariantly gauged photons as in Sect. 22.6 is more efficient.

23.3 Photon Scattering by Free Electrons Photon scattering by free or quasifree electrons is also known as Compton scattering. The cross section for this process had been calculated in leading order by Klein and Nishina [98]. The electron-photon interaction term from (22.155) is Heγ = ecγ · A.

(23.53)

We denote the wave vectors of the incoming photon and electron with q and k, respectively. The relevant second order matrix element for scattering of photons by free electrons is Sf i = Sk  ,s  ;q  ,α  |k,s;q,α     ∞  t  e2 c2 i 3 3       d x d x =− 2 dt dt k , s ; q , α | exp H0 t h¯ h¯ −∞ −∞   i ×(x)γ · A(x)(x) exp − H0 (t − t  ) (x  )γ · A(x  )(x  ) h¯   i × exp − H0 t  |k, s; q, α h¯

23.3 Photon Scattering by Free Electrons

=−

e2



 d 4x

h¯ 2

655

d 4 x  (t − t  )k  , s  ; q  , α  |(x)γ · A(x)(x)

×(x  )γ · A(x  )(x  )|k, s; q, α.

(23.54)

Here A(x) ≡ AD (x, t) and (x) ≡ D (x, t) are the freely evolving field operators (18.32) and (22.94) in the interaction picture. We can insert a decomposition of unity between the two vertex operators γ ·A with a fermionic and a photon factor, 1 = 1f ⊗ 1γ .

(23.55)

The relevant parts in the photon factor have zero or two intermediate photons, 1γ ⇒ |00| +

1 2 



 d 3K

d 3 K  |K, β; K  , β  K, β; K  , β  |,

(23.56)

β,β

while for the intermediate fermion states only states with one intermediate electron or with two intermediate electrons and a positron contribute, 1f ⇒



d 3 κ bσ+ (κ)|00|bσ (κ)

σ

   1 3 3  d κ d κ d 3 λ bσ+ (κ)bσ+ (κ  )dν+ (λ)|0 + 2  σ,σ ,ν

×0|dν (λ)bσ  (κ  )bσ (κ).

(23.57)

The full photon matrix element is

hμ ¯ 0c   α  (q  ) ⊗  α (q) 16π 3 |q||q  | ! " ! " × exp i(q · x  − q  · x) +  α (q) ⊗  α  (q  ) exp i(q · x − q  · x  ) ,

q  , α  |A(x) ⊗ A(x  )|q, α =

(23.58)

where the first term arises from the term without intermediate photons and the second term arises from the term with two intermediate photons after integrating the intermediate photon momenta K and K  . Evaluation of the fermion matrix element with an electron in the intermediate state yields σ

k  , s  |(x)γ (x)bσ+ (κ)|0 ⊗ 0|bσ (κ)(x  )γ (x  )|k, s

656

23 Applications of Spinor QED

=

exp[iκ · (x − x  ) − ik  · x + ik · x  ]  us  (k  )γ uσ (κ) ⊗ uσ (κ)γ us (k). 6 4E(κ) E(k  )E(k) (2π ) σ

We can substitute the sum over intermediate u spinors using Eq. (22.101),

k  , s  |(x)γ (x)bσ+ (κ)|0 ⊗ 0|bσ (κ)(x  )γ (x  )|k, s

σ

exp[iκ · (x − x  ) − ik  · x + ik · x  ]  ei ⊗ ej (2π )6 4E(κ) E(k  )E(k)

 × us  (k  )γ i mc2 − h¯ cγ · κ − γ 0 E(κ) γ j us (k). =

(23.59)

Assembling the pieces so far then yields the amplitude with a single intermediate fermion, (1)



  e2 (t − t  ) d 3κ  d 4x d 4x  E(κ) 8(2π )9 0 hc ¯ |q||q  |E(k)E(k ) %  &

0 ×us  (k  )  α  (q  ) · γ mc2 − hcκ ¯ · γ − γ E(κ) ( α (q) · γ ) ! " × exp i(κ − k  − q  ) · x + i(k + q − κ) · x 

% & + ( α (q) · γ ) mc2 − h¯ cκ · γ − γ 0 E(κ)  α  (q  ) · γ ! " × exp i(κ − k  + q) · x + i(k − q  − κ) · x  us (k). (23.60)

Sf i = −

The fermion matrix element with three intermediate fermions is 1   k , s |(x)γ (x)bσ+ (κ)bσ+ (κ  )dν+ (λ)|0 2  σ,σ ,ν

⊗ 0|dν (λ)bσ  (κ  )bσ (κ)(x  )γ (x  )|k, s =−

ν

δ(κ  − k  )δ(κ − k)

exp[iλ · (x − x  ) + ik · x − ik  · x  ]  (2π )6 4E(λ) E(k  )E(k)

× v ν (λ)γ us (k) ⊗ us  (k  )γ vν (λ).

(23.61)

The last line has been simplified from an expression which is symmetric in the intermediate momenta κ and κ  by taking into account that those momenta will be integrated. We can substitute the sum over intermediate v spinors using Eq. (22.103). This yields

23.3 Photon Scattering by Free Electrons

657

1   k , s |(x)γ (x)bσ+ (κ)bσ+ (κ  )dν+ (λ)|0 2  σ,σ ,ν

⊗ 0|dν (λ)bσ  (κ  )bσ (κ)(x  )γ (x  )|k, s exp[iλ · (x − x  ) + ik · x − ik  · x  ]  (2π )6 4E(λ) E(k  )E(k)

 0 i × ei ⊗ ej us  (k  )γ j mc2 + hcγ ¯ · λ + γ E(λ) γ us (k).

= δ(κ  − k  )δ(κ − k)

(23.62)

The factor ei ⊗ ej maintains the correct order of the tensor product of the two spinor products in Eq. (23.61), which were combined into a single spinor product through Eq. (22.103). If we substitute λ → κ (after integration over the intermediate electron momenta) for the wave vector of the intermediate positron in Eq. (23.62), the contribution from three intermediate fermions to the scattering matrix element is (3)



  e2 (t − t  ) d 3κ  d 4x d 4x  E(κ) 8(2π )9 0 hc ¯ |q||q  |E(k)E(k ) %  &

0 ×us  (k  )  α  (q  ) · γ mc2 + hcκ ¯ · γ + γ E(κ) ( α (q) · γ ) ! " × exp i(κ + k + q) · x − i(κ + k  + q  ) · x 

% & + ( α (q) · γ ) mc2 + h¯ cκ · γ + γ 0 E(κ)  α  (q  ) · γ ! " × exp i(κ + k − q  ) · x − i(κ + k  − q) · x  us (k). (23.63)

Sf i = −

We can simplify the total scattering matrix element Sf i = Sf(1)i + Sf(3)i by swapping x ↔ x  in Sf i and taking into account that (3)



(mc2 − h¯ cγ · κ)f (κ)

exp(iκ · x) κ 2 + (m2 c2 /h¯ 2 ) − i   exp(iκ · x)(mc2 − h¯ cγ · κ)f (κ) exp(− iκ 0 ct) 3 = − d κ dκ 0 [κ 0 − (ω(κ)/c) + i][κ 0 + (ω(κ)/c) − i]  mc2 − hcκ ¯ · γ + γ 0 E(κ) f (κ) exp(− iω(κ)t) = 2π ic(t) d 3 κ exp(iκ · x) 2ω(κ)  mc2 − hcκ ¯ · γ − γ 0 E(κ) − 2π ic(− t) d 3 κ exp(iκ · x) f (κ) exp(iω(κ)t) −2ω(κ)   3  d κ  = iπ c(t) · κ)f (κ) exp(iκ · x) (mc2 − hcγ ¯  0 ω(κ) κ =ω(κ)/c d 4κ

658

23 Applications of Spinor QED

  d 3κ 2 (mc + hcγ . ¯ · κ)f (− κ) exp(− iκ · x) ω(κ) κ 0 =ω(κ)/c

 + iπ c(− t)

(23.64) This yields the total scattering matrix element in the form  Sf i =





4

4

d κ

d x

×us  (k  )

d 4x

ie2  4(2π )10 0 h¯ 2 c |q||q  |E(k)E(k  )

% &  α  (q  ) · γ

mc − hγ ¯ ·κ

+ (m2 c2 /h¯ 2 ) − i ! " × exp i(κ − k  − q  ) · x + i(k + q − κ) · x 

+ ( α (q) · γ )

κ2

mc − hγ ¯ ·κ + (m2 c2 /h2 ) − i

( α (q) · γ )

% &  α  (q  ) · γ

¯ " ! × exp i(κ − k  + q) · x + i(k − q  − κ) · x  us (k). κ2

(23.65)

After performing the trivial integrations, we find Sf i = δ(k  + q  − k − q) ×us  (k  )

ie2

16π 2 0 h¯ 2 c |q||q  |E(k)E(k  )

% &  α  (q  ) · γ

+ ( α (q) · γ )



mc − hγ ¯ · (k + q) (k + q)2 + (m2 c2 /h¯ 2 ) − i

mc − hγ ¯ · (k − q  ) (k

− q  )2 + (m2 c2 /h2 ) − i ¯

( α (q) · γ )

% &  α  (q  ) · γ us (k). (23.66)

The first contribution to the amplitude corresponds to absorption of a photon with wave vector q followed by emission of a photon with wave vector q  , see Fig. 23.1, while the second contribution to the amplitude corresponds to emission of the photon with wave vector q  before absorption of the photon with wave vector q as shown in Fig. 23.2. The denominators in (23.66) can be simplified by noting that

q k+q−q’ e−

k

k+q

q’

Fig. 23.1 Absorption of the incoming photon with momentum h¯ q before emission of the outgoing photon with momentum h¯ q  . The virtual intermediate electron has 4-momentum h¯ (k + q). The left panel uses particle labels and the right panel uses momentum labels

23.3 Photon Scattering by Free Electrons

659

Fig. 23.2 Emission of the outgoing photon with momentum h¯ q  before absorption of the incoming photon with momentum h¯ q. The virtual intermediate electron has 4-momentum h¯ (k − q  )

q

e−

k

k−q’

k+q−q’

q’

k 2 + (m2 c2 /h¯ 2 ) = 0,

q 2 = q 2 = 0,

(23.67)

and  k · q = k · q − |q| k 2 + (m2 c2 /h¯ 2 ) < 0.

(23.68)

This yields with the definition αS = e2 /(4π 0 hc) ¯ (7.149) of Sommerfeld’s fine structure constant the result Sf i = δ(k  + q  − k − q)



iαS

8π h¯ |q||q  |E(k)E(k  )

% & mc − hγ ¯ · (k + q)  α  (q  ) · γ ( α (q) · γ ) k·q & mc − hγ ¯ · (k − q  ) %   (q ) · γ − ( α (q) · γ )  us (k). α k · q

×us  (k  )

(23.69)

The spin and helicity polarized differential scattering cross section then follows from (23.17), vdσk,s;q,α→k  ,s  ;q  ,α  = cd 3 k  d 3 q 

αS2 δ(k  + q  − k − q)

16h¯ 2 |q||q  |E(k)E(k  )  %  & mc − h¯ γ · (k + q) × us  (k  )  α  (q  ) · γ ( α (q) · γ ) k·q

2  & mc − h¯ γ · (k − q  ) %  − ( α (q) · γ )  α  (q ) · γ us (k) .  k·q

(23.70)

Spin-polarized cross sections are usually of less physical interest than electronphoton cross sections which average over polarizations of initial electron states and

660

23 Applications of Spinor QED

sum over the polarizations of the final electron states,1 dσk;q,α→k  ;q  ,α  =

1    . dσ 2  k,s;q,α→k ,s ;q ,α

(23.71)

s,s

Use of the property (22.101)

  us (k)us (k) = mc2 − h¯ cγ μ kμ 

s

(23.72)

k 0 =ω(k)/c

and of the relations (u)+ = γ 0 u, γμ+ = γ 0 γμ γ 0 , yields vdσk;q,α→k  ;q  ,α  =

α 2 c3 δ(k  + q  − k − q) d 3kd 3q  S 2 32h |q||q  |E(k)E(k  )

 & % tr mc − h¯ γ · k 

¯  % & mc − hγ ¯ · (k + q) ×  α  (q  ) · γ ( α (q) · γ ) k·q  & mc − h¯ γ · (k − q  ) %   − ( α (q) · γ )  (q ) · γ (mc − hγ ¯ · k) α k · q  & mc − hγ ¯ · (k + q) %  α  (q  ) · γ × ( α (q) · γ ) k·q  % & mc − hγ ¯ · (k − q  ) . −  α  (q  ) · γ (q) · γ ) ( α k · q

(23.73)

This can be evaluated using the trace theorems for γ matrices from Appendix G. The full evaluation of dσk;q,α→k ;q  ,α  needs in particular the trace theorems (G.56)– (G.58) for products of 4, 6, and 8 γ matrices. We can simplify the evaluation in the rest frame of the initial electron, 



  1 . 0

(23.74)

(γ · q) ( α (q) · γ ) = − ( α (q) · γ ) (γ · q).

(23.75)

1 k= hc ¯

m2 c4 + h¯ 2 k 2 hck ¯

mc ⇒ h¯

We can also use that  α (q) · q = α (q) · q = 0 implies

This reduces products according to

1 However,

spin polarized cross sections for electron scattering will likely become important in the framework of spintronics and spin based quantum computing.

23.3 Photon Scattering by Free Electrons

661

(mc − h¯ γ · (k + q)) ( α (q) · γ ) (mc − hγ ¯ · k)

 

0 = mc + mcγ 0 − hγ ¯ · q ( α (q) · γ ) mc 1 + γ

 

= ( α (q) · γ ) mc − mcγ 0 + h¯ γ · q mc 1 + γ 0

 0 = mhc ¯ ( α (q) · γ ) (γ · q) 1 + γ .

(23.76)

The resulting cross section in the rest frame of the electron before scattering is dσ

0;q,α→k  ;q  ,α 

=

α 2 h¯ 2 δ(k  + q  − k − q) d 3kd 3q  S 32m2 c|q||q  |E(k  )

 tr

& %  mc − hγ ¯ ·k

   % & % & γ · q

γ ·q   0   + ( α (q) · γ )  α (q ) · γ 1 + γ ×  α (q ) · γ ( α (q) · γ ) |q| |q  |   % & γ · q % & γ ·q   . ×  (q ) · γ (q) · γ ( ) ( α (q) · γ )  α  (q  ) · γ + α α |q| |q  | Traces over products of an odd number of γ matrices vanish. The terms under the trace proportional to mc2 contain products of six γ matrices, but two of these products vanish due to (γ ·q)2 = −q 2 = 0 and (γ ·q  )2 = −q 2 = 0. The remaining two terms involving six γ matrices turn out to yield the same result, such that the contribution to the trace term from products of six γ matrices is tr6 =

8mc

q · q  − 2( α (q) ·  α  (q  ))2 q · q  |q||q  |

 + 2( α (q) ·  α  (q  ))( α (q) · q  )( α  (q  ) · q) .

(23.77)

For the traces over products of eight γ matrices, we observe that those which contain the products (γ · q)γ 0 (γ · q) or (γ · q  )γ 0 (γ · q  ) can be simplified to products of six γ matrices due to 

γ μ γ 0 γ ν qμ qν = − 2ημ0 γ ν − γ 0 γ μ γ ν qμ qν = − 2|q|γ ν qν .

(23.78)

This yields for the sum of those terms which contain the products (γ · q)γ 0 (γ · q) or (γ · q  )γ 0 (γ · q  ) the result tr8a =

 8h¯

2( α  (q  ) · k  )( α  (q  ) · q) − k  · q |q|  8h¯

+  2( α (q) · k  )( α (q) · q  ) − k  · q  , |q |

(23.79)

662

23 Applications of Spinor QED

and after substitution of k  = k + q − q  ,  8h¯

q · q  + 2( α  (q  ) · q)2 |q|

 8h¯ −  q · q  + 2( α (q) · q  )2 . |q |

tr8a = 16mc +

(23.80)

The traces over products of eight γ matrices which contain terms (γ · q)γ 0 (γ · q  ) or (γ · q  )γ 0 (γ · q) can also be reduced to traces over products of six γ matrices by using the fact that γ 0 can only by contracted with one of the three γ matrices in products with 4-vectors. This yields after a bit of calculation and after substitution of k  = k + q − q  ,

 8mc

q · q tr8b = 16mc 2( α (q) ·  α  (q  ))2 − 1 − |q||q  | − 2( α (q) ·  α  (q  ))2 q · q  + 2( α (q) ·  α  (q  ))( α (q) · q  )( α  (q  ) · q) − 16

h¯ h¯ ( α  (q  ) · q)2 + 16  ( α (q) · q  )2 . |q| |q |



(23.81)

The total trace term is therefore tr = tr6 + tr8a + tr8b = 32mc( α (q) ·  α  (q  ))2 + 8h¯



 1 1 −  q · q , |q| |q |

(23.82)

and combining all the terms yields αS2 h¯ 2 δ(k  + q  − k − q) 4m2 c|q||q  |E(k  )     1 1  2  − q ·q . × 4mc( α (q) ·  α  (q )) + h¯ |q| |q  |

dσ0;q,α→k  ;q  ,α  = d 3 k  d 3 q 

(23.83)

The product q · q  is directly related to the photon scattering angle, q · q  = − |q||q  | (1 − cos θ ).

(23.84)

However, energy and momentum conservation also imply & 1 1 m2 c 2 mc 0 mc %  k = |q | − |q| . − q · q  = − (q − q  )2 = − (k  − k)2 = 2 2 2 h¯ h¯ h¯ The relation between scattering angle and scattered photon wave number is therefore

23.3 Photon Scattering by Free Electrons

mc cos θ = 1 − h¯



 1 1 − , |q  | |q|

663

|q  | =

mc|q| . mc + h|q| ¯ (1 − cos θ )

(23.85)

This is of course nothing but the Compton relation (1.56) for the wavelength of the scattered photon in terms of the scattering angle, λ = λ +

h (1 − cos θ ). mc

(23.86)

The four-dimensional δ function      2 2 2 2 k + (m c /h¯ ) + |q | − (mc/h) δ(k + q − k − q) = δ ¯ − |q| ×δ(k  + q  − q)

(23.87)

reduces the six-dimensional final state measure d 3 k  d 3 q  to the two-dimensional measure d(qˆ  ) ≡ d over direction of the scattered photon after integration over d 3 k  and d|q  |. We include already the factor |q  |E(k  ) in the denominator in Eq. (23.83) in the calculation: 

d 3k 



∞ 0

d|q  |

|q  | f (|q  |)δ(k  + q  − k − q) E(k  )

∞

|q  |f (|q  |) d|q  |  0 c h¯ 2 |q  |2 + h¯ 2 |q|2 − 2h¯ 2 |q  ||q| cos θ + m2 c2    2  2 2  2 2 ×δ |q | + |q| − 2|q ||q| cos θ + (m c /h¯ ) + |q | − (mc/h) ¯ − |q| =

  |q  |f (|q  |) 1 c mc + h|q|(1 − cos θ ) |q  |=mc|q|/[mc+h¯ |q|(1−cos θ)] ¯   m|q| mc|q| . = f 2 mc + h|q| [mc + h|q| ¯ (1 − cos θ ) ¯ (1 − cos θ )] =

(23.88)

This yields the Klein–Nishina cross section dσ0;q,α→q−q  ;q  ,α  = d

αS2 h¯ 2

4mc [mc + h¯ |q| (1 − cos θ )]2  2 2 |q|2 h − cos θ (1 ) ¯ × 4mc( α (q) ·  α  (q  ))2 + . mc + h|q| ¯ (1 − cos θ ) 

(23.89)

Averaging over the initial photon polarization and summing over the final polarization (18.196) yields the unpolarized differential cross section

664

23 Applications of Spinor QED

dσ0;q→q−q  ;q  =

αS2 h¯ 2 1 dσ0;q,α→q−q  ;q  ,α  = d 2  2mc [mc + h¯ |q| (1 − cos θ )]2 α,α



 h¯ 2 |q|2 (1 − cos θ )2 × mc(1 + cos θ ) + . mc + h|q| ¯ (1 − cos θ ) 2

(23.90)

The resulting total cross section is σ0;q→q−q  ;q 

 2 + (h|q|)3 π αS2 2(mc)3 + 8(mc)2 h|q| + 9mc(h|q|) ¯ ¯ ¯ 2 h|q| = ¯ 3 2 mch|q| (mc + 2h|q|) ¯ ¯     2h|q| ¯ 2 2 . (23.91) ln 1 + − 2(mc) + 2mch|q| − (h|q|) ¯ ¯ mc

Photons below the hard X-ray regime satisfy h¯ |q| % mc. This limit is also often denoted as the non-relativistic limit of Compton scattering because the kinetic energy imparted on the recoiling electron is small in this case, h¯ 2 (q − q  )2  2h¯ 2 q 2 (1 − cos θ ) % m2 c2 .

(23.92)

The cross section in the non-relativistic limit yields the Thomson cross section (18.197) and (18.198) for photon scattering, dσ0;q,α→q−q  ;q  ,α  = d



αS h¯ mc

dσ0;q→q−q  ;q  = d σ0;q→q−q  ;q 

8π = 3



αS h¯ mc

2

2



( α (q) ·  α  (q  ))2 ,

αS h¯ mc

2

1 + cos2 θ , 2

≡ σT = 6.652×10−9 Å2 = 0.6652 barn.

(23.93)

(23.94)

(23.95)

The unpolarized differential scattering cross section (23.90) for Compton scattering is displayed for various photon energies in Fig. 23.3. Forward scattering is energy independent, but scattering in other directions is suppressed with energy. The energy dependence of the total Compton scattering cross section (23.91) is displayed in Fig. 23.4.

23.3 Photon Scattering by Free Electrons

665

Fig. 23.3 The differential scattering cross section (23.90) for scattering angle 0 ≤ θ ≤ π . The energy of the incident photon is Eγ = 0 (top black curve), Eγ = 0.2mc2 (center blue curve) and Eγ = 2mc2 (lower red curve)

Fig. 23.4 The total Compton scattering cross section (23.91) in units of the Thomson cross section (23.95) for incident photon energy 0 < Eγ < 3mc2

666

23 Applications of Spinor QED

23.4 Møller Scattering The leading order scattering cross section for electron-electron scattering was calculated in the framework of quantum electrodynamics by C. Møller [118]. The Hamiltonian (22.155) in Coulomb gauge ∇ · A = 0 for the photon field is H = H0 + HI + HC ,

(23.96)

with the electron-photon interaction term  HI ≡ Heγ = ec

d 3 x (x, t)γ · A(x, t)(x, t)

(23.97)

and the Coulomb interaction term   e2 1 3 HC = d x d 3 x  s+ (x, t)s+ (x  , t) s  (x  , t)s (x, t). 8π 0  |x − x  | ss

Note that the summation is over Dirac indices, which are related to spin projections through the corresponding u or v spinors. The corresponding Hamiltonian on the states in the interaction picture is HD (t) = ec



d 3 x  s (x, t)γ · A(x, t)s (x, t)

s

+

e2 8π 0



 d 3x

ss 

d 3 x  s+ (x, t)s+ (x  , t)

1 s  (x  , t)s (x, t) |x − x  |

with the freely evolving field operators A(x, t) (18.32) and (x, t) (22.94) of the interaction picture. The scattering matrix element for electron-electron scattering Sf i ≡ Sk 1 ,s  ;k 2 ,s  |k1 ,s1 ;k 2 ,s2 1

=

2

k 1 , s1 ; k 2 , s2 |T exp

   i ∞ − dt HD (t) |k 1 , s1 ; k 2 , s2  (23.98) h¯ −∞

becomes in leading order O(e2 ) (γ )

(C)

Sf i = Sf i + Sf i , with the photon contribution

(23.99)

23.4 Møller Scattering

(γ )

Sf i = −

667





e2

d 4 x  (t − t  )k 1 , s1 ; k 2 , s2 |(x)γ · A(x)(x)

d 4x

h¯ 2

×(x  )γ · A(x  )(x  )|k 1 , s1 ; k 2 , s2 

(23.100)

and the Coulomb term μ0 e2     k , s ; k , s | 8π ih¯ 1 1 2 2

(C)

Sf i =

×



 d 4x

d 4 x   + (x) + (x  )

δ(ct − ct  ) (x  )(x)|k 1 , s1 ; k 2 , s2 . |x − x  |

(23.101)

(γ )

We evaluate Sf i first. Substitution of the relevant parts of the mode expansions yields (here we also use summation convention for the helicity and spin polarization indices) (γ ) Sf i

 ×

e2 h¯ μ0 c =− 2 h¯ 8(2π )9 d 3q  1 E(q 1 )





 4

d x

d 3q  2 E(q 2 )



4 



d x (t − t )



d 3q   1 E(q 1 )



d 3q   2 E(q 2 )



d 3q   |q  |

d 3q √ exp[i(q  + q2 − q1 ) · x − i(q + q1 − q2 ) · x  ] |q|

× u(q 1 , σ )γ u(q 2 , σ  ) ·  β (q  ) α (q) · u(q 1 , s)γ u(q 2 , s  )0|b(k 1 , s1 )b(k 2 , s2 ) × b+ (q 1 , σ )b(q 2 , σ  )aβ (q  )aα+ (q)b+ (q 1 , s)b(q 2 , s  )b+ (k 2 , s2 )b+ (k 1 , s1 )|0. Elimination of the photon operators yields (γ )

Sf i =  ×

μ0 e2 c 8(2π )9 h¯

d 3q  1 E(q 1 )





 d 4x

d 4 x  (t − t  )



d 3q   1 E(q 1 )



d 3q   2 E(q 2 )



d 3q |q|

d 3q  2 exp[i(q + q2 − q1 ) · x − i(q + q1 − q2 ) · x  ] E(q 2 )

× u(q 1 , σ )γ u(q 2 , σ  ) ·  α (q) α (q) · u(q 1 , s)γ u(q 2 , s  )0|b(k 1 , s1 )b(k 2 , s2 ) × b+ (q 1 , σ )b+ (q 1 , s)b(q 2 , σ  )b(q 2 , s  )b+ (k 2 , s2 )b+ (k 1 , s1 )|0,

(23.102)

where fermionic operators were also re-ordered such that only the connected amplitude contributes. Evaluation of the fermionic operators yields (γ )

Sf i =

e2 μ0 c 8h¯ (2π )9



 d 4x

d 4x



d 3 q (t − t  ) exp[iq · (x − x  )]  |q| E(k 1 )E(k 2 )E(k 1 )E(k 2 )

668

23 Applications of Spinor QED

 × u(k 2 , s2 )γ u(k 1 , s1 ) ·  α (q) α (q) · u(k 1 , s1 )γ u(k 2 , s2 )

 × exp[i(k1 − k2 ) · x − i(k1 − k2 ) · x  ] + exp[i(k2 − k1 ) · x − i(k2 − k1 ) · x  ] − u(k 1 , s1 )γ u(k 1 , s1 ) ·  α (q) α (q) · u(k 2 , s2 )γ u(k 2 , s2 )

× exp[i(k1 − k1 ) · x − i(k2 − k2 ) · x  ]  + exp[i(k2 − k2 ) · x − i(k1 − k1 ) · x  ] .

(23.103)

This yields after changing the integration variables x ↔ x  in the second and fourth term (γ )

Sf i =

  3  1 e2 μ0 c d q 4 4   x d x d 8h¯ (2π )9 E(k  )E(k  )E(k )E(k ) |q| 1 2 1 2

× u(k 2 , s2 )γ u(k 1 , s1 ) ·  α (q) α (q) · u(k 1 , s1 )γ u(k 2 , s2 ) × exp[i(k1 − k2 ) · x + i(k2 − k1 ) · x  ] − u(k 1 , s1 )γ u(k 1 , s1 ) ·  α (q) α (q) · u(k 2 , s2 )γ u(k 2 , s2 )  × exp[i(k1 − k1 ) · x + i(k2 − k2 ) · x  ] 

× (t − t  ) exp[iq · (x − x  )] + (t  − t) exp[iq · (x  − x)] .

We can use the followingequation for parity invariant functions f (q), f (− q) = f (q), and with ω(q) = c q 2 + (m2 c2 /h¯ 2 ), 

to find

f (q) exp(iq · x)





cf (q) exp(iq · x) exp(− iωt) dω [ω − ω(q) + i][ω + ω(q) − i] q 2 + (m2 c2 /h¯ 2 ) − i  exp(− iω(q)t) = 2π ci(t) d 3 qf (q) exp(iq · x) 2ω(q)  exp(iω(q)t) − 2π ci(− t) d 3 qf (q) exp(iq · x) −2ω(q)   3  d q f (q) exp(iq · x) = iπ c (t) ω(q) ω=ω(q)   3  d q f (q) exp(− iq · x) + iπ c (− t) (23.104) ω(q) ω=ω(q) d 4q

=−

d 3q

23.4 Møller Scattering

(γ )

Sf i =

669

   1 μ0 e2 c d 4q 4 4   x d x d 4ih(2π )10 E(k  )E(k  )E(k )E(k ) q 2 − i ¯ 1 2 1 2

× u(k 2 , s2 )γ u(k 1 , s1 ) ·  α (q) α (q) · u(k 1 , s1 )γ u(k 2 , s2 ) × exp[i(k1 − k2 + q) · x + i(k2 − k1 − q) · x  ] − u(k 1 , s1 )γ u(k 1 , s1 ) ·  α (q) α (q) · u(k 2 , s2 )γ u(k 2 , s2 )  × exp[i(k1 − k1 + q) · x + i(k2 − k2 − q) · x  ] .

(23.105)

The integrations then yield (γ )

Sf i =

δ(k1 + k2 − k1 − k2 ) μ0 e2 c  2 16iπ h¯ E(k  )E(k  )E(k )E(k ) 1 2 1 2



u(k 2 , s2 )γ u(k 1 , s1 ) ·  α (k 2 − k 1 ) α (k 2 − k 1 ) · u(k 1 , s1 )γ u(k 2 , s2 ) (k2 − k1 )2 − i  u(k 1 , s1 )γ u(k 1 , s1 ) ·  α (k 1 − k 1 ) α (k 1 − k 1 ) · u(k 2 , s2 )γ u(k 2 , s2 ) − . (k1 − k1 )2 − i

×

Taking into account the energy-momentum conserving δ function, the transversal projectors can e.g. be written as  α (k 1 − k 1 ) ⊗  α (k 1 − k 1 ) = 1 +

(k 1 − k 1 ) ⊗ (k 2 − k 2 ) . (k 1 − k 1 )2

(23.106)

The Dirac equation implies u(k  , s  )γ · (k  − k)u(k, s) =

E(k  ) − E(k) u(k  , s  )γ 0 u(k, s). hc ¯

(23.107)

This yields the photon exchange contribution to the electron-electron scattering matrix element, (γ )

Sf i =  × −

δ(k1 + k2 − k1 − k2 ) μ0 ce2  16iπ 2 h¯ E(k  )E(k  )E(k )E(k ) 1 2 1 2

u(k 2 , s2 )γ u(k 1 , s1 ) · u(k 1 , s1 )γ u(k 2 , s2 ) (k2 − k1 )2 − i

u(k 1 , s1 )γ u(k 1 , s1 ) · u(k 2 , s2 )γ u(k 2 , s2 ) (k1 − k1 )2 − i

670

23 Applications of Spinor QED

−

u(k 2 , s2 )γ 0 u(k 1 , s1 )u(k 1 , s1 )γ 0 u(k 2 , s2 ) [E(k 2 ) − E(k 1 )]2 (k2 − k1 )2 − i h¯ 2 c2 (k 2 − k 1 )2

 u(k 1 , s1 )γ 0 u(k 1 , s1 )u(k 2 , s2 )γ 0 u(k 2 , s2 ) [E(k 1 ) − E(k 1 )]2 + . (k1 − k1 )2 − i h¯ 2 c2 (k 1 − k 1 )2

(23.108)

For the evaluation of the Coulomb term, substitution of the mode expansions and (C) evaluation of the operators in Sf i yields (C) Sf i

2 μ0 ce2  = 8π ih¯ 4(2π )6 E(k  )E(k  )E(k )E(k ) 1 2 1 2



 4

d x

d 4x

δ(ct − ct  ) |x − x  |

! " × u+ (k 1 , s1 )u(k 1 , s1 )u+ (k 2 , s2 )u(k 2 , s2 ) exp i(k2 − k2 ) · x  + i(k1 − k1 ) · x ! " − u+ (k 2 , s2 )u(k 1 , s1 )u+ (k 1 , s1 )u(k 2 , s2 ) exp i(k1 − k2 ) · x  + i(k2 − k1 ) · x 1 μ0 ce2  16π ih¯ (2π )6 E(k  )E(k  )E(k )E(k ) 1 2 1 2 ! "     4 3  exp i(k1 + k2 − k1 − k2 ) · x × d x d x |x − x  |

! " × u(k 1 , s1 )γ 0 u(k 1 , s1 )u(k 2 , s2 )γ 0 u(k 2 , s2 ) exp i(k 2 − k 2 ) · (x  − x) ! " − u(k 2 , s2 )γ 0 u(k 1 , s1 )u(k 1 , s1 )γ 0 u(k 2 , s2 ) exp i(k 1 − k 2 ) · (x  − x) . =

In the next step we use the Fourier decomposition (23.34) of the Coulomb potential to find (C)

Sf i =

δ(k1 + k2 − k1 − k2 ) μ0 ce2  16π 2 ih¯ E(k  )E(k  )E(k )E(k ) 1 2 1 2  u(k 1 , s1 )γ 0 u(k 1 , s1 )u(k 2 , s2 )γ 0 u(k 2 , s2 ) × (k 2 − k 2 )2  u(k 2 , s2 )γ 0 u(k 1 , s1 )u(k 1 , s1 )γ 0 u(k 2 , s2 ) − . (k 2 − k 1 )2

(γ )

(23.109)

(C)

For the addition of Sf i and Sf i , we observe 1  (k − k)2



 [ω(k  ) − ω(k)]2 1 +1 =  2  2 c (k − k) (k − k)2

(23.110)

23.4 Møller Scattering

671

Fig. 23.5 Contributions to the Møller scattering amplitude (23.111)

k’1

k’2

k’2

k1−k’1 k1

k’1

k1−k’2 k2

k1

k2

to find δ(k1 + k2 − k1 − k2 ) μ0 ce2  16π 2 h¯ E(k  )E(k  )E(k )E(k ) 1 2 1 2  u(k 1 , s1 )γ μ u(k 1 , s1 )u(k 2 , s2 )γμ u(k 2 , s2 ) × (k1 − k1 )2  u(k 2 , s2 )γ μ u(k 1 , s1 )u(k 1 , s1 )γμ u(k 2 , s2 ) − (k2 − k1 )2

Sf i = i

= − iMf i δ(k1 + k2 − k1 − k2 ),

(23.111)

where the last equation defines the scattering amplitude Mf i ≡ Mk 1 ,s  ;k 2 ,s  |k 1 ,s1 ;k 2 ,s2 1

2

(23.112)

for Møller scattering. The two contributions to the scattering amplitude can be interpreted as virtual photon exchange with virtual photon 4-momentum k1 − k1 or k1 − k2 , respectively. This is shown in Fig. 23.5. The scattering amplitude (23.111) yields the spin polarized differential cross section (23.22) 2    vdσk 1 ,s  ;k 2 ,s  |k1 ,s1 ;k 2 ,s2 = 4π 2 c Mk 1 ,s  ;k 2 ,s  |k 1 ,s1 ;k 2 ,s2  1

2

1

2

1 ×δ(k1 + k2 − k1 − k2 ) d 3 k 1 d 3 k 2 , (23.113) 2 where v=

 c3 (h¯ 2 k1 · k2 )2 − m4 c4 E(k 1 )E(k 2 )

(23.114)

is the relative speed (23.8) between the two electrons with momentum 4-vectors hk ¯ 1 and hk ¯ 2. The differential cross section is often averaged over initial spin states and summed over final spin states,

672

23 Applications of Spinor QED

dσk 1 ;k 2 |k1 ;k 2 =

1 4

s1 ,s2 ,s1 ,s2

dσk 1 ,s  ;k 2 ,s  |k1 ,s1 ;k 2 ,s2 . 1

2

(23.115)

The property (22.101)

 μ  u(k, s)u(k, s) = mc2 − hcγ k ¯ μ

s

k 0 =ω(k)/c

(23.116)

yields  2 δ(k1 + k2 − k1 − k2 ) c 3  3  μ0 ce2 vdσ = d k 1 d k 2 2 16π h¯ E(k 1 )E(k 2 )E(k 1 )E(k 2 )  

  

1 2  μ 2 γ mc γν tr mc − hcγ · k − h cγ · k × ¯ ¯ 1 1 (k1 − k1 )4 



  ×tr mc2 − h¯ cγ · k2 γμ mc2 − hcγ ¯ · k 2 γν 

  

1 2  μ 2 γ mc γν tr mc − hcγ · k − h cγ · k ¯ ¯ 1 2 (k2 − k1 )4 



  ×tr mc2 − h¯ cγ · k1 γμ mc2 − hcγ ¯ · k 2 γν

+



 

2 2 μ 2  γ mc tr mc − hcγ · k − hcγ · k ¯ ¯ 2 2 (k1 − k1 )2 (k2 − k1 )2



  ν 2 2  (23.117) × γ mc − h¯ cγ · k1 γμ mc − hcγ ¯ · k 1 γν ,

−

where it is understood that all 4-momenta of electrons are on shell. The 4-momenta of the intermediate photons are then automatically off shell with dominant spacelike components, (k  − k)2 > 0 (except in the zero momentum transfer limit k  = k). The traces in Eq. (23.117) are readily evaluated using the contraction and trace theorems for γ matrices from Appendix G. This yields together with 4-momentum conservation k1 + k2 = k1 + k2 the result vdσ =

cd 3 k 1 d 3 k 2  × +



e2 c 4π 0 h¯

2

δ(k1 + k2 − k1 − k2 ) E(k 1 )E(k 2 )E(k 1 )E(k 2 )

h¯ 4 (k1 · k2 )2 + h¯ 4 (k1 · k2 )2 + 2m2 c2 h¯ 2 k1 · k1 + 2m4 c4 (k1 − k1 )4

h¯ 4 (k1 · k2 )2 + h¯ 4 (k1 · k1 )2 + 2m2 c2 h¯ 2 k1 · k2 + 2m4 c4 (k2 − k1 )4

23.4 Møller Scattering

673

 h¯ 4 (k1 · k2 )2 + 2m2 c2 h¯ 2 k1 · k2 +2 . (k1 − k1 )2 (k2 − k1 )2

(23.118)

We further evaluate the cross section through integration over d 3 k 2 and d|k 1 | in the center of mass frame k 1 + k 2 = 0 of the colliding electrons. If we integrate over the final states d 3 k 2 of one of the electrons to get a single-electron differential cross section dσ/d, we have to include a factor of 2 because we could just as well observe the electron with momentum k 2 being scattered into the direction d, see Eq. (23.25). It is convenient to define k = k 1 , k  = k 1 . The integration with the energymomentum δ function then yields 

d 3 k 2

 0

∞

d|k  | |k  |2 f (k, k  , k 2 )δ(k 2 + k  )

     2 2 2 2 × δ 2 |k | + (mc/h) ¯ − 2 |k| + (mc/h) ¯ =

 |k|   |k|2 + (mc/h) ¯ 2 f (k, |k|kˆ , −|k|kˆ ). 2

(23.119)

The scalar products in the center of mass frame are 2 k1 · k2 = − 2|k|2 − (mc/h) ¯ ,

(23.120)

2 k1 · k1 = − |k|2 (1 − cos θ ) − (mc/h) ¯ ,

(23.121)

2 k1 · k2 = − |k|2 (1 + cos θ ) − (mc/h) ¯ ,

(23.122)

where θ is the angle between k and k  . The relative speed (23.114) of the electrons in the center of mass frame is v=

2ch|k| ¯ h¯ 2 |k|2 + m2 c2

.

(23.123)

The differential scattering cross section is then with the factor of 2 from Eq. (23.25), and using the fine structure constant αS = e2 /(4π 0 hc), ¯ dσ h¯ 4 k 4 (3 + cos2 θ )2 + m2 c2 (4h¯ 2 k 2 + m2 c2 )(1 + 3 cos2 θ ) = αS2 . d 4h¯ 2 k 4 (h¯ 2 k 2 + m2 c2 ) sin4 θ

(23.124)

This is symmetric under θ → (π/2) − θ with a minimum for scattering angle θ = π/2 and divergences in forward and backward direction. This divergence in the zero momentum transfer limit is due to the vanishing photon mass, or in other words

674

23 Applications of Spinor QED

due to the infinite range of electromagnetic interactions. It is the same divergence which rendered the Rutherford cross section non-integrable. Equation (23.124) looks fairly complicated, but in terms of energy it essentially entails that Møller scattering is suppressed with kinetic energy K like K −2 : The low energy result for K  h¯ 2 k 2 /2m % mc2 is dσ = d



αS hc ¯ 4K sin2 θ

2 (1 + 3 cos2 θ ),

(23.125)

and the high energy result for K  hck ¯  mc2 is dσ = d



αS hc ¯ 3 + cos2 θ 2K sin2 θ

2 .

(23.126)

With respect to low-energy electron-electron scattering, we also note that the scattering matrix element (23.111) is dominated by the Coulomb contribution (23.109) if the electrons are non-relativistic, h|k| % mc, h¯ |k  | % mc. The estimate ¯ (23.52) for the ratio of scattering amplitudes in the low energy limit applies here too, and this confirms again the domination of Coulomb interactions between nonrelativistic charged particles.

23.5 Problems 23.1 Derive the relation (23.17) between the differential scattering cross section and the scattering amplitude in box normalization. 23.2 Calculate the differential scattering cross section for scattering of a relativistic fermion off a heavy nucleus. Assume that the heavy nucleus is non-relativistic and that the scalar particle cannot resolve its substructure, such that you can describe the nucleus with Schrödinger field operators. 23.3 Calculate the differential scattering cross section dσ0;q,α→k ;q  /d for electron-photon scattering with polarized initial photons, i.e. sum over the polarizations of the scattered photons but do not average over the initial polarization. 23.4 Show that the differential cross sections (23.83) and (23.89) for Compton scattering can also be written in the form dσ0;q,α→k  ;q  ,α  = d 3 k  d 3 q 

αS2 h¯ 2 δ(k  + q  − k − q) 4m|q||q  |E(k  )

  |q| |q  | +  − 2 , (23.127) × 4( α (q) ·  α  (q  ))2 + |q| |q |

23.5 Problems

dσ0;q,α→q−q  ;q  ,α  = d

675



αS h¯ |q  | 2mc|q|

2   |q| |q  | 4( α (q) ·  α  (q  ))2 + +  −2 . |q| |q |

23.5 Calculate the kinetic energy imparted on the recoiling electron in Compton scattering as a function of |q| and θ . 23.6 Derive the scattering amplitude for electron-nucleus scattering using covariant quantization for the photon. 23.7 Derive the scattering amplitude for Compton scattering using covariant quantization for the photons. 23.8 Derive the scattering amplitude for Møller scattering using covariant quantization for the photon.

Appendix A

Lagrangian Mechanics

Lagrangian mechanics is not only a very beautiful and powerful formulation of mechanics, but it is also needed as a preparation for a deeper understanding of all fundamental interactions in physics. All fundamental equations of motion in physics are encoded in Lagrangian field theory, which is a generalization of Lagrangian mechanics for fields. Furthermore, the connection between symmetries and conservation laws of physical systems is best explored in the framework of the Lagrangian formulation of dynamics, and we also need Lagrangian field theory as a basis for field quantization. Suppose we consider a particle with coordinates x(t) moving in a potential V (x). Then Newton’s equation of motion mx¨ = −∇V (x)

(A.1)

is equivalent to the following statement (Hamilton’s principle, 1834): The action integral  S[x] =

t1



t1

˙ = dt L(x, x)

dt

t0

t0

m 2

x˙ 2 − V (x)

 (A.2)

is in first order stationary under arbitrary perturbations x(t) → x(t) + δx(t) of the path of the particle between fixed endpoints x(t0 ) and x(t1 ) (i.e. the perturbation is only restricted by the requirement of fixed endpoints: δx(t0 ) = 0 and δx(t1 ) = 0). This is demonstrated by straightforward calculation of the first order variation of S,  δS[x] = S[x + δx] − S[x] =  =−

t1

dt [mx˙ · δ x˙ − δx · ∇V (x)]

t0 t1

dt δx · (mx¨ + ∇V (x)).

(A.3)

t0

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 R. Dick, Advanced Quantum Mechanics, Graduate Texts in Physics, https://doi.org/10.1007/978-3-030-57870-1

677

678

A Lagrangian Mechanics

Partial integration and δx(t0 ) = 0, δx(t1 ) = 0 were used in the last step. Equation (A.3) tells us that δS[x] = 0 holds for arbitrary path variation with fixed endpoints if and only if the path x(t) satisfies Newton’s equations, mx¨ + ∇V (x) = 0.

(A.4)

This generalizes to arbitrary numbers of particles (x(t) → x I (t), 1 ≤ I ≤ N ). Application of Hamilton’s principle to the N-particle action  S[x 1 , · · · , x N ] =

t1

˙ dt L(x, x)

(A.5)

t0

with the N -particle Lagrange function ˙ = L(x, x)

N mI I =1

2

x˙ 2I − V (x 1 , · · · , x N )

(A.6)

yields the Newton equations of motion for the N-particle system mI x¨ I = −

∂ V (x 1 , · · · , x N ). ∂x I

(A.7)

On the level of unconstrained particle motion in a potential V , Hamilton’s principle does not seem to offer any obvious advantage over the study of the corresponding Newton equations. However, Hamilton’s principle becomes a particularly powerful tool when we are dealing with constrained mechanical systems, where the constraints can be expressed in the form ζc (x 1 , · · · , x N , t) = 0,

1 ≤ c ≤ C.

(A.8)

Here C is the number of constraints, and the explicit time dependence indicates that the constraints themselves can change with time. For example, we could have a two-particle system with the constraint that the two particles must maintain a certain distance, where the distance may be a given function of time R(t), ζ (x 1 , x 2 , t) = |x 1 (t) − x 2 (t)| − R(t) = 0,

(A.9)

whereas the constraint to maintain constant distance R would be written as ζ (x 1 , x 2 ) = |x 1 (t) − x 2 (t)| − R = 0. Hamilton’s principle becomes very powerful because it still applies if we solve the C constraints (A.8) through introduction of 3N − C generalized coordinates qa which parametrize all the remaining allowed motions of the system under the constraints, i.e. we express the coordinates x I of the system through the allowed motions qa (t) under the constraints (A.9),

A Lagrangian Mechanics

679

xI j = xI j (q, t).

(A.10)

The explicit time-dependence in xI j (q, t) will arise if the constraints are timedependent, and the full time-dependence xI j (t) = xI j (q(t), t) of the particle coordinates during the motion of the system will also involve an implicit timedependence from the motions qa (t) of the generalized coordinates. The velocity components are therefore     ∂ ∂ j j  x˙I (t) = q˙a (t) xI (q, t) + xI (q, t) . ∂qa ∂t q=q(t) q=q(t) j

(A.11)

Here the substitutions qa = qa (t) are performed after evaluating the partial derivatives. For example, for the 2-particle system with the constraint (A.9) we could use the five generalized coordinates qa = X1a , 1 ≤ a ≤ 3, q4 = ϑ, q5 = ϕ, where ⎛

m1 x 1 + m2 x 2 X= , m1 + m2

⎞ sin ϑ cos ϕ x 1 − x 2 = R(t)⎝ sin ϑ sin ϕ ⎠. cos ϑ

(A.12)

The mappings (A.10) are then ⎛ ⎞ sin ϑ cos ϕ m2 R(t)⎝ sin ϑ sin ϕ ⎠, x1 = X + m1 + m2 cos ϑ ⎛ ⎞ sin ϑ cos ϕ m1 x2 = X − R(t)⎝ sin ϑ sin ϕ ⎠. m1 + m2 cos ϑ

(A.13)

(A.14)

Performing the transformation (A.10) and (A.11) to the actual degrees of freedom qa in the Lagrange function (A.6) of the unconstrained N-particle system yields the Lagrange function L(q, q, ˙ t) of the constrained N-particle system and the corresponding action  S[q] =

t1

dt L(q, q, ˙ t).

(A.15)

t0

The nontrivial and very useful observation is that Hamilton’s principle also applies to the constrained system: The constrained mechanical system will still evolve in such a way that the action S[q] is stationary under arbitrary first order perturbations qa (t) → qa (t) + δqa (t) subject only to the conditions that the initial and final state are not perturbed: δqa (t) = 0, δqa (t1 ) = 0.

680

A Lagrangian Mechanics

First order variation of the action with fixed endpoints (i.e. δq(t0 ) = 0, δq(t1 ) = 0) yields after partial integration   ∂L ∂L δqa + δ q˙a ∂qa ∂ q˙a t0 a    t1 ∂L d ∂L , dt δqa − = ∂qa dt ∂ q˙a t0 a 

δS[q] = S[q + δq] − S[q] =

t1

dt

(A.16)

where again the fixation of the endpoints eliminated the boundary terms. δS[q] = 0 for arbitrary path variation qa (t) → qa (t) + δqa (t) with fixed endpoints then immediately tells us the equations of motion in terms of the generalized coordinates, d ∂L ∂L − = 0. ∂qa dt ∂ q˙a

(A.17)

These equations of motion are called Lagrange equations of the second kind or Euler–Lagrange equations or simply Lagrange equations. The quantity pa =

∂L ∂ q˙a

(A.18)

is denoted as the conjugate momentum to the coordinate qa . The conjugate momentum is conserved if the Lagrange function depends only on the generalized velocity component q˙a but not on qa , dpa /dt = 0. Furthermore, if the Lagrange function does not explicitly depend on time, we have   dL ∂L pa q¨a + = q˙a . dt ∂qa a

(A.19)

The Euler–Lagrange equation then implies that the Hamilton function1

1 The Hamilton function is always conserved if there is no explicit time-dependence in the Lagrange

function, and generically it corresponds to the energy of the system. However, if there are explicitly time-dependent constraints such that L = L(q, q) ˙ is not explicitly time-dependent, then the conserved Hamilton function may not be the energy of the system. An example for this is provided e.g. by a particle which is tied to an upright circular wire while the wire is rotating around the z-axis: x(t) = R[ex cos ϑ(t) · cos(ωt) + ey cos ϑ(t) · sin(ωt) + ez sin ϑ(t)], ˙ = mR 2 (ϑ˙ 2 + ω2 cos2 ϑ)/2, H = mR 2 (ϑ˙ 2 − ω2 cos2 ϑ)/2. H is conserved, but the L(ϑ, ϑ) energy of the particle is the non-conserved kinetic energy T = L. One can show in this system that the constraint force which forces the particle to rotate with angular velocity ω around the z-axis performs work.

A Lagrangian Mechanics

681

H =



pa q˙a − L

(A.20)

a

is conserved, dH /dt = 0. For a simple example, consider a particle of mass m in a gravitational field g = −gez . The particle is constrained so that it can only move on a sphere of radius r. An example of generalized coordinates are angles ϑ, ϕ on the sphere, and the Cartesian coordinates {X, Y, Z} of the particle are related to the generalized coordinates through X(t) = r sin ϑ(t) · cos ϕ(t),

(A.21)

Y (t) = r sin ϑ(t) · sin ϕ(t),

(A.22)

Z(t) = r cos ϑ(t).

(A.23)

The kinetic energy of the particle can be expressed in terms of the generalized coordinates,  m

 m m ˙2 K = r˙ 2 = X + Y˙ 2 + Z˙ 2 = r 2 ϑ˙ 2 + ϕ˙ 2 sin2 ϑ , (A.24) 2 2 2 and the potential energy is V = mgZ = mgr cos ϑ.

(A.25)

This yields the Lagrange function in the generalized coordinates, L=

 m 2 m

r˙ − mgZ = r 2 ϑ˙ 2 + ϕ˙ 2 sin2 ϑ − mgr cos ϑ, 2 2

(A.26)

and the Euler–Lagrange equations yield the equations of motion of the particle, ϑ¨ = ϕ˙ 2 sin ϑ cos ϑ +

g sin ϑ, r

(A.27)

 d

ϕ˙ sin2 ϑ = 0. dt

(A.28)

∂L = mr 2 ϑ˙ ∂ ϑ˙

(A.29)

∂L = mr 2 ϕ˙ sin2 ϑ ∂ ϕ˙

(A.30)

The conjugate momenta pϑ = and pϕ =

682

A Lagrangian Mechanics

are just the angular momenta for rotation in ϑ or ϕ direction. The Hamilton function is the conserved energy H = pϑ ϑ˙ + pϕ ϕ˙ − L =

pϕ2 pϑ2 + + mgr cos ϑ = K + U. 2mr 2 2mr 2 sin2 ϑ

(A.31)

The immediately apparent advantage of this formalism is that it directly yields the correct equations of motion (A.27) and (A.28) for the system without ever having to worry about finding the force that keeps the particle on the sphere. Beyond that the formalism also provides a systematic way to identify conservation laws in mechanical systems, and if one actually wants to know the force that keeps the particle on the sphere (which is actually trivial here, but more complicated e.g. for a system of two particles which have to maintain constant distance), a simple extension of the formalism to the Lagrange equations of the first kind can yield that, too. The Lagrange function is not simply the difference between kinetic and potential energy if the forces are velocity dependent. This is the case for the Lorentz force. The Lagrange function for a non-relativistic charged particle in electromagnetic fields is L=

m 2 x˙ + q x˙ · A − q. 2

(A.32)

This yields the correct Lorentz force law mx¨ = q(E + v × B) for the particle, cf. Sect. 15.1. The relativistic versions of the Lagrange function for the particle can be found in Eqs (B.91) and (B.93).

Derivation of the Lagrange Equations for the Generalized Coordinates qa from d’Alembert’s Principle A derivation of the Lagrange equations for the generalized coordinates of a constrained N -particle system from Newton’s equations works in the following way: On a microscopic level, the equations of motion of the constrained N-particle system with coordinates xI j , 1 ≤ i ≤ N, 1 ≤ j ≤ 3, should be ∂ d (mI x˙I j ) + V (x 1...N ) = CI j . dt ∂xI j

(A.33)

Here FI j = − ∂V (x 1...N )/∂xI j are the forces which the particles could respond to through virtual (i.e. instantaneous) displacements δxI j without violating the constraints, whereas CI j are the forces which enforce the constraints. Constraints prevent particles or systems of particles to move in certain directions, i.e. the constraint forces must act opposite to the directions of motion which are prevented

A Lagrangian Mechanics

683

by the constraints. This implies that all the allowed instantaneous displacements δx I of the particles must be orthogonal to the constraint forces:2

δx I · C I = 0.

(A.34)

I

However, all the allowed movements under the constraints where encoded in the generalized coordinates qa in the form (A.10), and the condition on the allowed virtual displacements of the system therefore amounts to I

δqa

∂x I · C I = 0. ∂qa

(A.35)

We can also understand this equation in a complementary way: The constraint forces C I restrict the motions of the N-particle system to the allowed motions δx I = (∂x I /∂qa )δqa , but they cannot interfere in any way with those allowed motions. Both arguments lead to Eq. (A.35): The constraint forces must be orthogonal to the allowed motions. This is d’Alembert’s principle. Furthermore, since δqa is an otherwise unconstrained permissible shift of the system, d’Alembert’s principle can be expressed without it, ∂x I I

∂qa

· C I = 0.

(A.36)

Due to the Newton equations (A.33), d’Alembert’s principle (A.36) implies  ∂x I  d ∂ j (mI x˙I ) + · V (x 1...N ) = 0. ∂qa dt ∂xI j

(A.37)

I

Note that this is not a trivial consequence of the Newton equations for the N particles, because the term in the bracket is not the Newton equation for the particle with mass mI . The Newton equation for that particle is Eq. (A.33). With the definition (A.6) of the Lagrange function of the N-particle system, we can write (A.37) in the form  ∂xI j  d ∂L ∂L = 0. − ∂qa dt ∂ x˙I j ∂xI j

(A.38)

I,j

2 Equation

(A.34) only limits the allowed virtual displacements of the system. It does not prevent the constraint forces from performing work on the system during the actual time evolution x I (t) = x I (q(t), t) of the system.

684

A Lagrangian Mechanics

To analyze the implications of (A.38), we repeat Eq. (A.11) for the velocity components in shorthand notation, x˙I j =

∂xI j dxI j ∂xI j = . q˙a + dt ∂qa ∂t a

(A.39)

This implies in particular the two equations ∂ x˙I j ∂xI j = ∂ q˙a ∂qa

(A.40)

∂ 2 xI j ∂ x˙I j ∂ 2 xI j = q˙b + . ∂qa ∂qa ∂qb ∂t∂qa

(A.41)

and

b

Substitution of (A.40) into (A.41) yields ∂ x˙I j ∂ 2 x˙I j ∂ 2 x˙I j = q˙b + . ∂qa ∂qb ∂ q˙a ∂t∂ q˙a

(A.42)

b

Equation (A.40) also yields ∂ 2 x˙I j = 0, ∂ q˙a ∂ q˙b

(A.43)

∂ 2 x˙I j ∂ 2 x˙I j d ∂ x˙I j = q˙b + , dt ∂ q˙a ∂qb ∂ q˙a ∂t∂ q˙a

(A.44)

and therefore

b

and this implies with (A.42) d ∂ x˙I j ∂ x˙I j = . dt ∂ q˙a ∂qa

(A.45)

With these preliminaries we can now revisit Eqs. (A.38). Substitution of Eqs. (A.40) and (A.45) into (A.38) yields

d ∂ x˙ j  ∂ x˙I j d ∂L ∂xI j ∂L ∂ x˙I j  ∂L  I = 0, − + − ∂ q˙a dt ∂ x˙I j ∂qa ∂xI j dt ∂ q˙a ∂qa ∂ x˙I j I,j

or after combining terms:

(A.46)

A Lagrangian Mechanics

685

d ∂ x˙I j ∂L  ∂L − = 0. dt ∂ q˙a ∂ x˙I j ∂qa

(A.47)

I,j

However, the coordinates xI j are independent of the generalized velocities q˙a , and therefore Eq. (A.47) is just the Euler–Lagrange equation (A.17), d ∂L(q, q) ˙ ∂L(q, q) ˙ − = 0. dt ∂ q˙a ∂qa

(A.48)

The Euler–Lagrange equations for a constrained particle system therefore follow from the Newton equations if we take into account d’Alembert’s principle. On the other hand, Newton’s equation for a particle is a special case of the Euler– Lagrange equations in the case of unconstrained motion. Classical mechanics as a theory of both constrained and unconstrained mechanical systems can therefore be based either on Hamilton’s principle with action (A.15), which only incorporates the actual mechanical degrees of freedom qa , or on the Newton equations (A.33) combined with d’Alembert’s principle. However, once this is realized, Hamilton’s principle provides the most efficient way to derive the equations of motion and the conservation laws of a mechanical system.

Symmetries and Conservation Laws in Classical Mechanics We call a set of first order transformations t → t  = t − (t),

qa (t) → qa (t  ) = qa (t) + δqa (t)

(A.49)

a symmetry of a mechanical system with action  S[q] =

t1

dt L(q(t), q(t), ˙ t)

(A.50)

to

if it changes the form dt L(q(t), q(t), ˙ t) in first order of (t) and δqa (t) at most by a term of the form dB = dt (dB/dt): ˙ t) δ(dt L(q, q, ˙ t)) ≡ dt  L(q  (t  ), q˙  (t  ), t  ) − dt L(q(t), q(t), = dt

d Bδq, (q(t), t). dt

(A.51)

This is equivalent to the statement that the new coordinates qa (t  ) satisfy the same differential equations with respect to t  as the old coordinates qa (t) satisfy with respect to t.

686

A Lagrangian Mechanics

To see how this implies conservation laws in the mechanical system, we have to evaluate δ(dt L(q, q, ˙ t)) for the transformations (A.49). We have to take into account that (A.49) implies dt  = dt(1 − ˙ (t)),

d d , = (1 + (t)) ˙  dt dt

(A.52)

and therefore also δ q˙a (t) =

d   d d d q (t ) − qa (t) = ˙ (t) qa (t) + δqa (t). dt  a dt dt dt

(A.53)

The first order change in dt L is therefore (with summation convention for the index a) ˙ t) δ(dt L) = dt  L(q  (t  ), q˙ (t  ), t  ) − dt L(q(t), q(t),     ∂L ∂L d d ∂L . (A.54) + ˙ qa + δqa − ˙ L −  = dt δqa ∂qa dt dt ∂ q˙a ∂t Now we substitute δ q˙a

  ∂L ∂L d ∂L d δqa − δqa = ∂ q˙a dt ∂ q˙a dt ∂ q˙a

and     ∂L ∂L ∂L d ∂L d dL ˙ q˙a  q˙a −L = − L −  q¨a −  q˙a + ∂ q˙a dt ∂ q˙a ∂ q˙a dt ∂ q˙a dt     ∂L ∂L d ∂L d ∂L +  q˙a . = − L +  q˙a − dt ∂ q˙a ∂qa dt ∂ q˙a ∂t

(A.55)

(A.56)

This yields 

 ∂L d ∂L δ(dt L) = dt(δqa +  q˙a ) − ∂qa dt ∂ q˙a   d ∂L + dt − L . (δqa +  q˙a ) dt ∂ q˙a

(A.57)

Comparison of Eqs. (A.51) and (A.57) implies an on-shell conservation law d Qδq, = 0 dt with the conserved charge

(A.58)

A Lagrangian Mechanics

Qδq,

687

  ∂L ∂L =  L − q˙a + Bδq, . − δqa ∂ q˙a ∂ q˙a

(A.59)

Bδq, is the one-dimensional version of the current K μ in Lagrangian field theory and Qδq, is the one-dimensional version of the conserved current J μ , see the paragraph after Eq. (16.27). Bδq, = 0 in most cases. However, a noticeable exception are Galilei boosts in nonrelativistic N-particle mechanics (where I enumerates the particles). The Lagrange function L=

1 mI x˙ 2I − VI J (|x I − x J |) 2 I

(A.60)

I 0. Multiplication of Eq. (C.1) with ψ(x) and integration yields  H [ψ] ≡

b

 b

 dx g(x)ψ 2 (x) + V (x)ψ 2 (x) = E dx (x)ψ 2 (x) ≡ Eψ|ψ,

a

a

where the last equation defines the scalar product 

b

φ|ψ =

dx (x)φ(x)ψ(x).

(C.4)

a

It is easy to prove that (C.4) defines a scalar product since ψ|ψ ≥ 0 and ψ|ψ = 0 ⇔ ψ(x) = 0, and 0 ≤ ψ + λφ|ψ + λφ = ψ|ψ + 2λψ|φ + λ2 φ|φ

(C.5)

has a minimum for λ=−

ψ|φ , φ|φ

(C.6)

which after substitution in (C.5) yields the Schwarz inequality ψ|φ2 ≤ ψ|ψφ|φ.

(C.7)

The Sturm–Liouville equation (C.1) arises as an Euler–Lagrange equation from variation of the action S[ψ] = Eψ|ψ − H [ψ]  b

 = dx E (x)ψ 2 (x) − g(x)ψ 2 (x) − V (x)ψ 2 (x) a

with fixed endpoints ψ(a) and ψ(b).

(C.8)

C Completeness of Sturm–Liouville Eigenfunctions

713

The stationary values of S[ψ] for arbitrary fixed endpoints ψ(a) and ψ(b) are   S[ψ]

on−shell

= g(a)ψ(a)ψ  (a) − g(b)ψ(b)ψ  (b),

(C.9)

where the designation “on-shell” means that ψ(x) satisfies the Euler–Lagrange equation (C.1) of S[ψ]. If we think of the Sturm–Liouville problem as a one-dimensional scalar field theory, G(x) = 1/4g 2 (x) would play the role of a metric in a ≤ x ≤ b and H [ψ] would be the energy of the field ψ(x) if ψ(x) is normalized, ψ|ψ = 1. Suppose ψi (x) and ψj (x) are solutions of the Sturm–Liouville problem (C.1) and (C.2) with eigenvalues Ei and Ej , respectively. Use of the Sturm–Liouville equation (C.1) and partial integration yields    d d g(ξ ) ψi (ξ ) Ei dξ (ξ )ψj (ξ )ψi (ξ ) = dξ ψj (ξ ) V (ξ )ψi (ξ ) − dξ dξ a a  x

 d dξ V (ξ )ψj (ξ )ψi (ξ ) + g(ξ )ψj (ξ )ψi (ξ ) − g(x)ψj (x) ψi (x), = dx a (C.10) 



x

x

and after another integration by parts we find  (Ei − Ej ) 

x

dξ (ξ )ψi (ξ )ψj (ξ ) a

 d d = g(x) ψi (x) ψj (x) − ψj (x) ψi (x) . dx dx

(C.11)

This equation implies for Ei = Ej d d ln ψi (x) = ln ψj (x), dx dx

(C.12)

i.e. ψi (x) has to be proportional to ψj (x): There is no degeneracy of eigenvalues in the one-dimensional Sturm–Liouville problem. For x = b, Eq. (C.11) implies the orthogonality property (Ei − Ej )ψi |ψj  = 0

(C.13)

and taking into account the absence of degeneracy yields ψi |ψj  ∝ δij .

(C.14)

714

C Completeness of Sturm–Liouville Eigenfunctions

Liouville’s Normal Form of Sturm’s Equation We can gauge the functions g(x) and (x) away through a transformation of variables 

x

x→X=

(ξ ) , g(ξ )

dξ a

ψ(x) → (X) = ( (x)g(x))1/4 ψ(x).

(C.15)

(x) , g(x)

(C.16)

This yields  0≤X≤B=

b

dx a

(0) = 0,

(B) = 0,

and the Sturm–Liouville equation (C.1) assumes the form of a one-dimensional Schrödinger equation, d2 (X) − V (X)(X) + E(X) = 0 dX2

(C.17)

with V (X) =

g 2 (x) V (x) g(x)  (x) + (x)g  (x) 5g(x) 2 (x) + − − (x) 16g(x) (x) 4 2 (x) 16 3 (x) +

g  (x)  (x) . 8 2 (x)

(C.18)

Second order differentiability of (x) and g(x) is usually assumed. However, we only have to require continuity of the positive functions (x) and g(x) since we can deal with δ-function singularities in one-dimensional potentials. Equation (C.17) is Liouville’s normal form of the Sturm–Liouville equation.

Nodes of Sturm–Liouville Eigenfunctions For the following reasoning we assume that we have smoothly continued the functions V (x), (x) > 0 and g(x) > 0 for all values of x ∈ R. It does not matter how we do that. To learn more about the nodes of the eigenfunctions ψi (x) of the Sturm– Liouville boundary value problem, let us now assume that ψ(x, λ) and ψ(x, μ) are solutions of the incomplete initial value problems

C Completeness of Sturm–Liouville Eigenfunctions

715

  dψ(x, λ) g(x) , ψ(a, λ) = 0, (C.19) dx   dψ(x, μ) d g(x) , ψ(a, μ) = 0, (C.20) μ (x)ψ(x, μ) = V (x)ψ(x, μ) − dx dx d λ (x)ψ(x, λ) = V (x)ψ(x, λ) − dx

with λ > μ, but contrary to the boundary value problem (C.1) and (C.2) we do not impose any conditions at x = b. In that case there exist solutions to the Sturm– Liouville equations for arbitrary values of the parameters λ, μ, and we can again require  dψ(x, λ)  > 0, dx x=a

 dψ(x, μ)  > 0.  dx x=a

(C.21)

We recall the following facts from the theory of differential equations: The solution ψ(x, λ) to the initial value problem (C.19) is unique up to a multiplicative constant, and ψ(x, λ) depends continuously on the parameter λ. The last fact is important, because it implies that the nodes y(λ) of ψ(x, λ), ψ(y(λ), λ) = 0, depend continuously on λ. Continuity of y(λ) is used in the demonstration below that the boundary value problem (C.1) and (C.2) has a solution for every value of b. Multiplication of Eq. (C.19) with ψ(x, μ) and Eq. (C.20) with ψ(x, λ), integration from a to x > a, and subtraction of the equations yields 

x

(λ − μ)

dξ (ξ )ψ(ξ, λ)ψ(ξ, μ) a

     dψ(ξ, μ) d dψ(ξ, λ) d g(ξ ) − ψ(ξ, μ) g(ξ ) dξ ψ(ξ, λ) dξ dξ dξ dξ a   dψ(x, λ) dψ(x, μ) − ψ(x, μ) . (C.22) = g(x) ψ(x, λ) dx dx 

=

x

Now assume that y(μ) is the first node of ψ(x, μ) larger than a: ψ(y(μ), μ) = 0,

y(μ) > a.

(C.23)

Substituting x = y(μ) in (C.22) yields 

y(μ)

(λ − μ)

dx (x)ψ(x, λ)ψ(x, μ) = g(y(μ))ψ(y(μ), λ)

a

 dψ(x, μ)  .  dx x=y(μ)

We know that (λ − μ) (x)ψ(x, μ) > 0

(C.24)

716

C Completeness of Sturm–Liouville Eigenfunctions

for a < x < y(μ) and that g(y(μ))

 dψ(x, μ)  < 0.  dx x=y(μ)

(C.25)

This implies ψ(x, λ) must change its sign at least once for a < x < y(μ), and in particular y(λ) < y(μ): The location of the leftmost node y(λ) > a of the function ψ(x, λ) moves closer to a if λ increases. We are not really concerned with differentiability properties of the leftmost node y(λ), but we can express the previous observation also as y(λ) > a,

dy(λ) < 0. dλ

(C.26)

Now assume that λ is small enough2 so that even y(λ) > b. Then we can increase the parameter λ until we reach a value λ = E1 such that y(E1 ) = b. This is then the lowest eigenvalue of our original Sturm–Liouville boundary value problem (C.1), and the corresponding eigenfunction is ψ1 (x) = ψ(x, λ = E1 ).

(C.27)

The eigenfunction ψ1 (x) for the lowest eigenvalue E1 has no nodes in a < x < b. Now we consider the first and the second node of ψ(x, μ) for x > a, a < y(μ) ≡ y1 (μ) < y2 (μ),

ψ(y1 (μ), μ) = 0,

ψ(y2 (μ), μ) = 0,

(C.28)

and we integrate from y1 (μ) to y2 (μ), 

y2 (μ)

(λ − μ)

dx (x)ψ(x, λ)ψ(x, μ) y1 (μ)

     dψ(x, μ) d dψ(x, λ) d g(x) − ψ(x, μ) g(x) dx ψ(x, λ) dx dx dx dx y1 (μ)  dψ(x, μ)  = g(y2 (μ))ψ(y2 (μ), λ)  dx x=y2 (μ)  dψ(x, μ)  − g(y1 (μ))ψ(y1 (μ), λ) . (C.29)  dx x=y1 (μ) 

=

2 The

y2 (μ)

alert reader might worry that all y(λ) might be smaller than b, so that there is no finite small value λ with y(λ) > b, or otherwise that all y(λ) might be larger than b, so that no finite value E1 with y(E1 ) = b would exist. These cases can be excluded through Sturm’s comparison theorem, to be discussed later.

C Completeness of Sturm–Liouville Eigenfunctions

717

We know (λ − μ) (x)ψ(x, μ) < 0

(C.30)

for y1 (μ) < x < y2 (μ), and g(y1 (μ))

 dψ(x, μ)  < 0,  dx x=y1 (μ)

g(y2 (μ))

 dψ(x, μ)  > 0.  dx x=y2 (μ)

(C.31)

This tells us that ψ(x, λ) has to change sign in the interval y1 (μ) < x < y2 (μ), i.e. it must have at least one node there. We know that the first node y1 (λ) < y1 (μ) is outside of this interval. Therefore we can infer that at least the second node y2 (λ) of ψ(x, λ) must be smaller than y2 (μ): y2 (λ) < y2 (μ). We can repeat this reasoning for the pair of adjacent nodes yn−1 (μ), yn (μ) of ψ(x, μ), and we always find for λ > μ that yn (λ) < yn (μ), a < yn (λ),

ψ(yn (λ), λ) = 0,

dyn (λ) < 0. dλ

(C.32)

All nodes of the function ψ(x, λ) on the right-hand side of x = a move closer to a if λ increases. Therefore we can repeat the reasoning above which had let us to the first solution ψ1 (x) with eigenvalue E1 of our Sturm–Liouville problem. To find the second eigenfunction, we increase λ > E1 until we hit a value λ = E2 such that y2 (E2 ) = b, and the corresponding eigenfunction ψ2 (x) = ψ(x, E2 )

(C.33)

will have exactly one node y1 (E2 ) in the interval, a < y1 (E2 ) < b. The corresponding result for yn (λ) tells us that in the n-th step we will find a parameter λ = En with yn (En ) = b and eigenfunction ψn (x) = ψ(x, En ),

(C.34)

and this function will have n − 1 nodes a < y1 (En ) < y2 (En ) < . . . < yn−1 (En ) < yn (En ) = b inside the interval.

Sturm’s Comparison Theorem and Estimates for the Locations of the Nodes yn (λ) Sturm’s comparison theorem makes a statement about the change of the nodes yn > a of the solution ψ(x, λ) of

718

d dx

C Completeness of Sturm–Liouville Eigenfunctions

  dψ(x, λ) g(x) + (λ (x) − V (x)) ψ(x, λ) = 0, dx

ψ(a, λ) = 0,

(C.35)

if the functions g(x), (x) and V (x) change. To prove the comparison theorem, we do not use Liouville’s normal form, but perform the following simple transformation of variables,  X= a

x

dx  , g(x  )

(X, λ) = ψ(x, λ).

(C.36)

This transforms (C.35) into the following form, d 2 (X, λ) + (λR(X) − V (X)) (X, λ) = 0, dX2 R(X) = g(x) (x) > 0,

(0, λ) = 0,

V (X) = g(x)V (x),

(C.37) (C.38)

and the nodes Yn > 0 of (X, λ) are related to the nodes yn > a of ψ(x, λ) through 

yn

Yn = a

dx . g(x)

(C.39)

Now we consider another Sturm–Liouville problem of the form (C.37), but with different functions λS(X) − W (X) > λR(X) − V (X), d 2 (X, λ) + (λS(X) − W (X)) (X, λ) = 0, dX2

(C.40)

(0, λ) = 0,

(C.41)

and we denote the positive nodes of (X, λ) with Zn . We also require again   (0) > 0,  (0) > 0. Equations (C.37) and (C.41) imply 

Yn

dX [V (X) − W (X) − λ (R(X) − S(X))] (X, λ)(X, λ)

Yn−1

= (Yn , λ)

  d(X, λ)  d(X, λ)  − (Y , λ) . n−1 dX X=Yn dX X=Yn−1

(C.42)

The following terms in (C.42) have all the same sign,   [V (X) − W (X) − λ (R(X) − S(X))] (X, λ)

Yn−1 0 which arise from the nodes of the solutions of  min (X, λ) + (λRmax − Vmin ) min (X, λ)  = min (X, λ) + gmax (λ max − Umin ) min (X, λ) = 0,

min (0, λ) = 0,

(C.46) (C.47)

and  (X, λ) + (λRmin − Vmax ) max (X, λ) max  = max (X, λ) + gmin (λ min − Umax ) max (X, λ) = 0,

max (0, λ) = 0.

(C.48) (C.49)

Here we use the bounds of the continuous functions g(x), V (x), (x) on a ≤ x ≤ b, 0 < gmin ≤ g(x) ≤ gmax ,

Umin ≤ V (x) ≤ Umax ,

0 < min ≤ (x) ≤ max .

(C.50) (C.51)

The solutions of both Eqs. (C.46) and (C.48) have nodes if (recall that both g(x) > 0 and (x) > 0) λ > Umin / max ,

(C.52)

  min (X, λ) ∝ sin gmax (λ max − Umin )X ,

(C.53)

  max (X, λ) ∝ sin gmin (λ min − Umax )X .

(C.54)

and the two solutions are

This yields bounds for the nodes Yn > 0 of (X, λ),

720

C Completeness of Sturm–Liouville Eigenfunctions

√

nπ nπ ≤ Yn ≤ √ . gmax (λ max − Umin ) gmin (λ min − Umax )

(C.55)

However, we also know from Eq. (C.39) that gmin Yn ≤ yn − a ≤ gmax Yn , and therefore3 gmin nπ gmax nπ a+√ ≤ yn ≤ a + √ . gmax (λ max − Umin ) gmin (λ min − Umax )

(C.56)

This implies in particular that there is no accumulation point for the nodes yn of ψ(x, λ), and yn must grow like n for large n. For our previous proof that ψ1 (x) (C.27) has its first node at y1 = b, we needed the assumption that there are small enough values of λ such that the first node y1 (λ) of ψ(x, λ) satisfies y1 (λ) > b. We can now confirm that from the lower bound in (C.56). It will suffice to choose 2 π2 gmin Umin Umin a would be smaller than b. This is easily confirmed from the upper bound in (C.56). It is sufficient to choose λ>

2 π2 Umax gmax + . min min gmin (b − a)2

(C.58)

Eigenvalue Estimates for the Sturm–Liouville Problem We have found that the Sturm–Liouville boundary value problem (C.1) and (C.2) has an increasing, non-degenerate set of eigenvalues E1 < E2 < . . .

(C.59)

and arises as an Euler–Lagrange equation for the action S[ψ] = Eψ|ψ − H [ψ]  b

 dx E (x)ψ 2 (x) − g(x)ψ 2 (x) − V (x)ψ 2 (x) . =

(C.60)

a

For every continuous function ψ(x) in a ≤ x ≤ b we define the normalized function 3 These bounds can be strengthened by a longer proof, but the present result is completely sufficient

for our purposes.

C Completeness of Sturm–Liouville Eigenfunctions

ψ(x) ˆ ψ(x) =√ . ψ|ψ

721

(C.61)

Since S[ψ] is homogeneous in ψ, ψ(x) is a stationary point of S[ψ] if and only if ˆ ψ(x) is a stationary point of ˆ = E − H [ψ], ˆ S[ψ]

(C.62)

ˆ which implies also that ψ(x) is a stationary point of the functional H [ψ] ˆ = H [ψ] = ψ|ψ

b a

! " dx g(x)ψ 2 (x) + V (x)ψ 2 (x) . b 2 a dx (x)ψ (x)

(C.63)

We have already found that there is a discrete subset ψˆ n (x), n ∈ N, of stationary ˆ which satisfy the boundary conditions ψˆ n (a) = 0, ψˆ n (b) = 0, and points of H [ψ] are mutually orthogonal, ψˆ m |ψˆ n  = δmn .

(C.64)

Use of the Sturm–Liouville equation and the boundary conditions yields the values ˆ at the stationary points ψˆ n (x), of the functional H [ψ] H [ψˆ n ] = En .

(C.65)

We already know E1 < E2 < . . ., and therefore we have found that the functional ˆ has a minimum H [ψ] H [ψˆ 1 ] = E1

(C.66)

on the space of functions Fa,b = {ψ(x), a ≤ x ≤ b|ψ(a) = 0, ψ(b) = 0, ψ|ψ = 1} ,

(C.67)

and in general we have a minimum H [ψˆ n ] = En

(C.68)

on the space of functions (n) Fa,b = {ψ(x), a ≤ x ≤ b|ψ(a) = 0, ψ(b) = 0, ψ|ψ = 1, ψi |ψ = 0,

1 ≤ i ≤ n − 1} .

(C.69)

722

C Completeness of Sturm–Liouville Eigenfunctions

ˆ in Eq. (C.63) shows that all the eigenvalues En increase The explicit form of H [ψ] if g(x) increases or V (x) increases or (x) decreases. However, those continuous functions must be bounded on the finite interval a ≤ x ≤ b, 0 < gmin ≤ g(x) ≤ gmax ,

Umin ≤ V (x) ≤ Umax ,

0 < min ≤ (x) ≤ max .

(C.70) (C.71)

Therefore we can replace those functions with their extremal values to derive estimates for the eigenvalues En . The Sturm–Liouville problems for the extremal values are % & gmin/max ψn (x) + En,min/max max/min − Umin/max ψn (x) = 0, ψn (a) = 0,

(C.72)

ψn (b) = 0,

(C.73)

  x−a ψn (x) ∝ sin nπ b−a

(C.74)

with solutions

and corresponding eigenvalues En,min/max =

1 max/min

  n2 π 2 Umin/max + gmin/max . (b − a)2

(C.75)

This implies the bounds 1 max

    n2 π 2 n2 π 2 1 Umin + gmin ≤ E U . ≤ + g n max max min (b − a)2 (b − a)2

In particular, at most a finite number of the lowest eigenvalues En can be negative, and the eigenvalues for large n must grow like n2 . Both of these observations are crucial for the proof that the set ψn (x) of eigenfunctions of the Sturm–Liouville problem (C.1) and (C.2) provide a complete basis for the expansion of piecewise continuous functions in a ≤ x ≤ b.

Completeness of Sturm–Liouville Eigenstates We now assume that the Sturm–Liouville eigenstates are normalized, ψi |ψj  = δij .

(C.76)

C Completeness of Sturm–Liouville Eigenfunctions

723

Let φ(x) be an arbitrary smooth function on a ≤ x ≤ b with φ(a) = 0 and φ(b) = 0, and define ϕn (x) = φ(x) −

n

ψi (x)ψi |φ.

(C.77)

i=1

Then we have n 0 ≤ ϕn |ϕn  = φ|φ − ψi |φ2 ,

(C.78)

i=1

i.e. for all n we have a Bessel inequality φ|φ ≥

n ψi |φ2 .

(C.79)

i=1

We also have ϕn |ψi  = 0, 1 ≤ i ≤ n, and ϕn (a) = 0, ϕn (b) = 0, i.e. (n+1)

ϕn (x) ∈ Fa,b

(C.80)

.

Therefore the minimum property of the eigenvalue En+1 implies En+1 ≤

H [ϕn ] . ϕn |ϕn 

(C.81)

We have H [ϕn ] = H [φ] − 2

n



i=1

+

n i,j =1

b

ψi |φ a



% & dx g(x)φ  (x)ψi (x) + V (x)φ(x)ψi (x)

b

ψi |φψj |φ a

 dx g(x)ψi (x)ψj (x) + V (x)ψi ψj (x) . (C.82)

In the first sum, partial integration and use of the Sturm–Liouville equation yields 

b

%

dx g(x)φ a



(x)ψi (x) + V (x)φ(x)ψi (x)

&



b

= Ei

dx (x)φ(x)ψi (x) a

= Ei ψi |φ.

(C.83)

In the double sum, partial integration and use of the Sturm–Liouville equation yields

724

C Completeness of Sturm–Liouville Eigenfunctions



b a

 b

 dx g(x)ψi (x)ψj (x) + V (x)ψi ψj (x) = Ei dx (x)ψi (x)ψj (x) a

= Ei δij .

(C.84)

Ei ψi |φ2 .

(C.85)

This implies H [ϕn ] = H [φ] −

n i=1

Since at most finitely many of the eigenvalues Ei can be negative, Eq. (C.85) tells us that the functional H [ϕn ] must remain bounded from above for n → ∞, e.g. for E1 < E2 < · · · < EN < 0 ≤ EN +1 < . . .

(C.86)

we have the bound H [ϕn ] ≤ H [φ] +

N

|Ei |ψi |φ2 .

(C.87)

i=1

On the other hand, Eq. (C.81) yields for n > N (to ensure En+1 > 0), ϕn |ϕn  = φ|φ −

n

ψi |φ2 ≤

i=1

H [ϕn ] En+1

(C.88)

and since En+1 grows like n2 for large n while H [ϕn ] must remain bounded, we find the completeness relation  lim ϕn |ϕn  = lim

n→∞

n→∞ a

b

 dx (x) φ(x) −

n

2 ψi (x)ψi |φ

=0

(C.89)

i=1

or equivalently, φ|φ = lim

n→∞

n φ|ψi ψi |φ.

(C.90)

i=1

Completeness of the series ∞ i=1

ψi (x)ψi |φ ∼ φ(x)

(C.91)

C Completeness of Sturm–Liouville Eigenfunctions

725

in the sense of Eq. (C.89) is denoted as completeness in the mean, and is sometimes also expressed as l.i.m.n→∞

n

ψi (x)ψi |φ = φ(x),

(C.92)

i=1

where  l.i.m. stands for “limit in the mean”. Completeness in the mean says that the series ∞ i=1 ψi (x)ψi |φ approximates φ(x) in the least squares sense. Completeness in the mean also implies for the two piecewise continuous functions f and g f (x) ± g(x) ∼

∞

ψi (x)ψi |f  ±

i=1

∞

ψi (x)ψi |g

(C.93)

i=1

and therefore 1 f |ψi ψi |g. (f + g|f + g − f − g|f − g) = lim n→∞ 4 i=1 (C.94) Completeness in the sense of (C.94) is enough for quantum mechanics, because it says that we can use the completeness relation n

f |g =

1 = lim

n→∞

n

|ψi ψi |

(C.95)

i=1

in the calculation of matrix elements between sufficiently smooth functions (where “sufficiently smooth = continuously differentiable to a required order” depends on the operators we use). This is all that is really needed in quantum mechanics. However, for piecewise smooth functions, the relation also holds pointwise almost everywhere (see remark 3 below). I would like to add a few remarks: 1. The completeness property (C.89) also applies to piecewise continuous functions in a ≤ x ≤ b and functions which do not vanish at the boundary points, because every piecewise continuous function can be approximated in the mean by a smooth function which vanishes at the boundaries. 2. If φ(x) is a smooth function satisfying the Sturm–Liouville boundary conditions, as we have assumed in the derivation of (C.89), the series under the integral sign will even converge uniformly to φ(x), lim

n→∞

n i=1

ψi (x)ψi |φ = φ(x),

(C.96)

726

C Completeness of Sturm–Liouville Eigenfunctions

i.e. for all a ≤ x ≤ b and all values  > 0, there exists an n() such that   n     (C.97) ψi (x)ψi |φ <  if n ≥ n(). φ(x) −   i=1

Uniformity of the convergence refers to the fact that the same n() ensures (C.97) for all a ≤ x ≤ b. 3. If φ(x) is piecewise smooth in a ≤ x ≤ b, it can still be expanded pointwise in Sturm–Liouville eigenstates. Except for points of discontinuity of φ(x), and except for the boundary points if φ(x) does not satisfy the same boundary conditions as the eigenfunctions ψi (x), the expansion φ(x) = lim

n→∞

n

ψi (x)ψi |φ

(C.98)

i=1

holds pointwise, and the series converges uniformly to φ(x) in every closed interval which excludes discontinuities of φ(x) (and the series converges to the arithmetic mean in the points of discontinuity). The boundary points must also be excluded if φ(x) does not satisfy the Sturm–Liouville boundary conditions. For example, the functions  x|sn  = sn (x) =

2 nπ x  , sin a a

n ∈ N,

(C.99)

provide a complete set of eigenfunctions on 0 ≤ x ≤ a for the Sturm–Liouville problem 

 d2 2 + k n sn (x) = 0, dx 2

sn (0) = sn (a) = 0.

(C.100)

On the other hand, the functions  x|cn  = cn (x) =

nπ x  2 − δn,0 cos , a a

n ∈ N0 ,

(C.101)

provide a complete set of eigenfunctions on 0 ≤ x ≤ a for the Sturm–Liouville problem 

 d2 2 + kn cn (x) = 0, dx 2

The expansion |cm  = (C.99) follows from



n |sn sn |cm 

cn (0) = cn (a) = 0.

(C.102)

of the basis (C.101) in terms of the basis

C Completeness of Sturm–Liouville Eigenfunctions

1 − (−)n−m n 2 . sn |cm  =  1 + δm,0 n2 − m2 π

727

(C.103)

On the level of the unrenormalized cosine functions, this yields ∞

πx   4 2n + 1 πx  cos 2m = , sin (2n + 1) a π a (2n + 1)2 − 4m2

(C.104)

n=0



∞

πx  πx  8 n cos (2m + 1) = , sin 2n a π a 4n2 − (2m + 1)2 n=1

pointwise in 0 < x < a, and in the mean for 0 ≤ x ≤ a.

(C.105)

Appendix D

Properties of Hermite Polynomials

We use the following equation as a definition of Hermite polynomials,  Hn (x) = exp

1 2 x 2

    1 d n exp − x 2 , x− dx 2

(D.1)

because we initially encountered them in this form in the solution of the harmonic oscillator in Chap. 6. We can use the identity         d d 1 1 x+ f (x) = exp − x 2 exp x 2 f (x) dx 2 dx 2

(D.2)

to rewrite Eq. (D.1) in the form        1 2 d 1 2 n 1 2 1 2 Hn (x) = exp x 2x − exp − x exp x exp − x 2 2 dx 2 2          1 2 d 1 2 n 1 2 1 2 2x − exp − x exp − x exp x = exp x 2 2 dx 2 2 n  d = 2x − 1, (D.3) dx 

or we can use the identity         d d 1 1 x− f (x) = − exp x 2 exp − x 2 f (x) dx 2 dx 2

(D.4)

to rewrite Eq. (D.1) in the Rodrigues form

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 R. Dick, Advanced Quantum Mechanics, Graduate Texts in Physics, https://doi.org/10.1007/978-3-030-57870-1

729

730

D Properties of Hermite Polynomials



  d n

Hn (x) = exp x 2 − exp −x 2 . dx

(D.5)

The Rodrigues formula implies ∞

Hn (x)

n=0

∞ 

 ∂n 

zn zn 2 exp x 2 exp −(x − z) = n n! ∂z z=0 n! n=0





 = exp x 2 exp −(x − z)2 = exp 2xz − z2 .

(D.6)

The residue theorem then also yields the representation n! Hn (x) = 2π i

*

& % exp 2xz − z2 , dz zn+1

(D.7)

where the integration contour encloses z = 0 in the positive sense of direction, i.e. counter clockwise. Another useful integral representation for the Hermite polynomials follows from (D.3) and the equation 

∞ −∞

  

∂ n du − 2v − exp −(u + v)2 ∂v −∞  n √ ∂ = − 2v − π. (D.8) ∂v

 

du (2u)n exp −(u + v)2 =

∞

This yields in particular for v = −ix, 

∞

−∞



√ du (2u)n exp −(u − ix)2 = in π Hn (x).

(D.9)

Combination of Eqs. (D.6) and (D.9) yields Mehler’s formula [111], ∞ n=0

Hn (x)Hn (x  )

 ∞ ∞ 

zn 1 = Hn (x) √ du (−2iuz)n exp −(u − ix  )2 n! π n! −∞ n=0  ∞ 



1 = √ du exp −4ixuz + 4u2 z2 exp −(u − ix  )2 π −∞   & % z x 2 + x 2 − xx  1 exp −4z = √ . (D.10) 1 − 4z2 1 − 4z2

This requires |z| < 1/2 for convergence. In Sects. 6.3 and 13.1 we need this in the form for |z| < 1,

D Properties of Hermite Polynomials

731

  x 2 + x 2 zn exp − 2n n! 2

(D.11)

 %  &% & 1 + z2 x 2 + x 2 − 4zxx  & % =√ exp − . 2 1 − z2 1 − z2

(D.12)

∞ n=0

1

Hn (x)Hn (x  )

Indeed, applications of this equation for the harmonic oscillator are usually in the framework of distributions and require the limit |z| → 1. In principle we should therefore replace the corresponding phase factors z in Sects. 6.3 and 13.1 with z exp(−), and take the limit  → +0 after applying any distributions which are derived from (D.12).

Appendix E

The Baker–Campbell–Hausdorff Formula

The Baker–Campbell–Hausdorff formula explains how to combine the product of operator exponentials exp(A) · exp(B) into a single operator exponential exp[(A, B)], if the series expansion for (A, B) provided by the Baker– Campbell–Hausdorff formula converges. We try to determine (A, B) as a power series in a parameter λ, exp[λA] · exp[λB] = exp[(λA, λB)],

(λA, λB) =

∞

λn cn (A, B).

n=1

We also use the notation of the adjoint action of an operator A on an operator B, A(ad) ◦ B = −[A, B].

(E.1)

exp[αA] · exp[βB] = exp[(αA, βB)].

(E.2)

We start with

This implies with Lemma (6.46) the equations ∞

B = exp[−(αA, βB)]

(−)n n ∂ exp[(αA, βB)] = [(αA, βB), ∂β ] ∂β n! n=1

=−

∞ (−)n n=1

=

n!

n−1

[ (αA, βB), ∂β (αA, βB)]

∞ n−1 1

(αA, βB)(ad) ◦ ∂β (αA, βB) n! n=1

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 R. Dick, Advanced Quantum Mechanics, Graduate Texts in Physics, https://doi.org/10.1007/978-3-030-57870-1

733

734

E The Baker–Campbell–Hausdorff Formula

=

exp[(αA, βB)(ad) ] − 1 ◦ ∂β (αA, βB) (αA, βB)(ad)

(E.3)

and ∞

A = − exp[(αA, βB)]

1 n ∂ exp[−(αA, βB)] = − [(αA, βB), ∂α ] ∂α n! n=1

=

∞ n=1

=

1 n−1 [ (αA, βB), ∂α (αA, βB)] n!

∞ n−1 1

−(αA, βB)(ad) ◦ ∂α (αA, βB) n! n=1

=

1 − exp[−(αA, βB)(ad) ] ◦ ∂α (αA, βB). (αA, βB)(ad)

(E.4)

For the inversion of these equations, we note 

exp(z) − 1 z

−1

=

exp(−z/2) z =z exp(z) − 1 exp(z/2) − exp(−z/2)

=

z exp(z/2) + exp(−z/2) z − 2 exp(z/2) − exp(−z/2) 2

=

(−)n+1 z z z z coth − = 1 + Bn z2n − , 2 2 2 (2n)! 2

∞



1 − exp(−z) z

−1

(E.5)

n=1

=

exp(z/2) z =z 1 − exp(−z) exp(z/2) − exp(−z/2)

=

z exp(z/2) + exp(−z/2) z + 2 exp(z/2) − exp(−z/2) 2

=

(−)n+1 z z z z coth + = 1 + Bn z2n + , 2 2 2 (2n)! 2

∞

(E.6)

n=1

where the coefficients Bn are Bernoulli numbers. The previous equations yield (with (αA, βB)(ad) ◦ A = −[(αA, βB), A]) ∂α (αA, βB) =

(αA, βB)(ad) (αA, βB)(ad) coth ◦A 2 2 1 − [(αA, βB), A], 2

(E.7)

E The Baker–Campbell–Hausdorff Formula

∂β (αA, βB) =

735

(αA, βB)(ad) (αA, βB)(ad) coth ◦B 2 2 1 + [(αA, βB), B], 2

  ∂λ (λA, λB) = ∂α (αA, βB) + ∂β (αA, βB) =

(λA, λB)(ad) 2

coth

α=β=λ

(λA, λB)(ad) 2

(E.8)

◦ (A + B)

1 + [A − B, (λA, λB)], 2

(E.9)

i.e. ∂λ (λA, λB) = A + B +

∞ (−)n+1 n=1

(2n)!

Bn · [(λA, λB)(ad) ]2n ◦ (A + B)

1 + [A − B, (λA, λB)]. 2

(E.10)

Equation (E.10) provides us with a recursion relation for the n-th order coefficient functions cn (A, B), -n/2. (−)m+1 1 Bm (n + 1)cn+1 (A, B) = [A − B, cn (A, B)] + 2 (2m)! m=1 × [ck2m (A, B), [. . . , [ck2 (A, B), [ck1 (A, B), A + B]] . . .]],

(E.11)

1≤k1 ,k2 ,...k2m , k1 +...+k2m =n

with c0 (A, B) = 0,

c1 (A, B) = A + B.

(E.12)

The floor function -x. maps to the next lowest integer smaller or equal to x, i.e. -n/2. = n/2 if n is even, -n/2. = (n − 1)/2 if n is odd. Equation (E.11) yields for the next three terms c2 (A, B) =

1 [A, B], 2

(E.13)

1 1 [A − B, [A, B]] + B1 [A + B, [A + B, A + B]] 12 6 1 1 = [A, [A, B]] + [B, [B, A]], (E.14) 12 12

c3 (A, B) =

736

E The Baker–Campbell–Hausdorff Formula

1 [A − B, [A, [A, B]] + [B, [B, A]]] 96 1 + B1 [A + B, [[A, B], A + B]] 16 1

[A, [A, [A, B]]] − [B, [B, [B, A]]] + [A, [B, [B, A]]] = 96 −[B, [A, [A, B]]] − [A, [A, [A, B]]] + [B, [B, [B, A]]]  +[A, [B, [B, A]]] − [B, [A, [A, B]]]

c4 (A, B) =

=

1 1 1 [A, [B, [B, A]]] − [B, [A, [A, B]]] = [A, [B, [B, A]]]. 48 48 24

The Jacobi identity [A, [B, C]] + [B, [C, A]] + [C, [A, B]] = 0 was used in the last step for c4 .

(E.15)

Appendix F

The Logarithm of a Matrix

∞ n Exponentials of square matrices G, M = exp G = n=0 G /n!, are frequently used for the representation of symmetry transformations. Indeed, the properties of continuous symmetry transformations are often discussed in terms of their first order approximations 1 + G, where it is assumed that continuity of the symmetries allows for parameter choices such that max |Gij | % 1. It is therefore of interest that the logarithm G = ln M of invertible square matrices can also be defined. Suppose M is an invertible square matrix which is related to its Jordan canonical form through M = T −1 · ⊕n J n · T .

(F.1)

Each of the smaller square matrices J n has the form J = λ1

(F.2)

or the form ⎛

λ ⎜0 ⎜ ⎜. ⎜. . J =⎜ ⎜ ⎜0 ⎜ ⎝0 0

⎞ 0 0⎟ ⎟ .. ⎟ ⎟ . ⎟, ⎟ 0 0 0 ... λ 1 0⎟ ⎟ 0 0 0 ... 0 λ 1⎠ 0 0 0 ... 0 0 λ

10 λ1 .. .. . .

0 ... 0 ... .. .. . .

0 0 .. .

0 0 .. .

(F.3)

and det(M) = 0 implies that none of the eigenvalues λ can vanish. We do not presume whether the matrix M is real or complex. However, the Jordan canonical form may require that we allow for complex eigenvalues λn and complex

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 R. Dick, Advanced Quantum Mechanics, Graduate Texts in Physics, https://doi.org/10.1007/978-3-030-57870-1

737

738

F The Logarithm of a Matrix

transformation matrices T to ensure that the characteristic equation det(M−λ1) = 0 for the eigenvalues and the corresponding eigenvector conditions can be solved. In the case (F.2) we have & % J = exp ln λ1 ,

ln J = ln λ1.

(F.4)

However, it is also possible to construct the logarithm of a Jordan block matrix (F.3). The direct sum of the logarithms of all the matrices J n then yields the logarithm of the matrix M, M = exp(T −1 · ⊕n ln J n · T ),

ln M = T −1 · ⊕n ln J n · T .

(F.5)

Suppose the Jordan matrix (F.3) is a (ν + 1) × (ν + 1) matrix. We define (ν + 1) × (ν + 1) matrices N n , 0 ≤ n ≤ ν, according to (N n )ij = δi+n,j , i.e. N 0 is the (ν + 1) × (ν + 1) unit matrix and N 1≤n≤ν has non-vanishing entries 1 only in the n-th diagonal above the main diagonal. These matrices satisfy the multiplication law N m · N n = (ν − m − n + )N m+n , which also implies N n = (N 1 )n . Each (ν + 1) × (ν + 1) Jordan block can be written as J = λN 0 + N 1 , and its logarithm can be defined through ⎛

ln λ λ−1 −λ−2 /2 λ−3 /3 ⎜ 0 ln λ λ−1 −λ−2 /2 ⎜ ⎜ . .. .. .. ⎜ . . . . . X = ln J = ⎜ ⎜ ⎜ 0 0 0 0 ⎜ ⎝ 0 0 0 0 0 0 0 0 = N 0 ln λ −

ν (−λ)−n

n

n=1

⎞ ... (−)ν−1 λ−ν /ν . . . (−)ν−2 λ−(ν−1) /(ν − 1) ⎟ ⎟ ⎟ .. .. ⎟ . . ⎟ ⎟ −2 ⎟ ... −λ /2 ⎟ ⎠ ... λ−1 ... ln λ (F.6)

N n.

We can prove exp(X) = J in the following way. The N -th power of X is X

N

= N 0 (ln λ) + (−) N

N

ν+1−N

  ν+−

1≤n1 ,n2 ...nN

N

 ni

i=1

(−λ)−n1 −n2 −...−nN N n1 +n2 +...+nN . n1 · n2 · . . . · nN   ν+2−N N −1 N − (−) N ln λ  ν+− ni ×

1≤n1 ,n2 ...nN−1

×

i=1

(−λ)−n1 −n2 −...−nN−1 N n1 · n2 · . . . · nN −1 n1 +n2 +...+nN−1

F The Logarithm of a Matrix

739

 + (−) ×

N

N 2

 (ln λ)



ν+3−N

2

 ν+−

1≤n1 ,n2 ...nN−2

N −2

 ni

i=1

(−λ)−n1 −n2 −...−nN−2 N + ... n1 · n2 · . . . · nN −2 n1 +n2 +...+nN−2

− N(ln λ)N −1

ν (−λ)−n

n

n=1

(F.7)

N n.

The parameter 0 <  < 1 is introduced in this equation to avoid the ambiguity of (x) at x = 0. We can combine terms in Eq. (F.7) in the form N

XN = N 0 (ln λ)N +

 (−)m

m=1

×

ν+1−m



 ν+−

1≤n1 ,n2 ...nm

N m



m

 ni

i=1 ν

= N 0 (ln λ)N + M+1−m

1≤n1 ,n2 ...nm n1 +n2 +...+nm =M

(−λ)−n1 −n2 −...−nm N n1 +n2 +...+nm n1 · n2 · . . . · nm

min(N,M)

(−λ)−M N M

M=1

×

(ln λ)N −m



(−)m

m=1

N m



(ln λ)N −m

1 . n1 · n2 · . . . · nm

(F.8)

In the next step we isolate the term with M = 1 in the sum, (ln λ)N −1 N 1 + (ν − 2 + ) λ  min(N,M)   ν −M m N (−λ) N M (−) × (ln λ)N −m m

X N = N 0 (ln λ)N + N

M=2

×

m=1

M+1−m

1≤n1 ,n2 ...nm n1 +n2 +...+nm =M

 1 . n1 · n2 · . . . · nm

(F.9)

If we sum these expressions to calculate exp(X), only the first two terms survive in exp(X) = 1 +

∞ XN = λN 0 + N 1 = J , N!

N =1

(F.10)

740

F The Logarithm of a Matrix

because the sum over N in the term of order M reduces to ∞ (ln λ)N −m = λ, (N − m)!

(F.11)

N =m

and the remaining sums yield for M ≥ 1 the result M (−)m m!

m=1

=

=

1 2π i 1 2π i

1 = 2π i = =

1 2π i 1 2π i

M+1−m 1≤n1 ,n2 ...nm n1 +n2 +...+nm =M

* |z| 0, into the complex plane is such that this reproduces the retarded non-relativistic Green’s function (J.30) in the non-relativistic limit ω⇒

mc2 + E , h¯

(J.86)

when terms of order O(E 2 ) are neglected. However, in the relativistic case this yields both retarded and advanced contributions in the time domain. This convention for the poles in the relativistic theory was introduced by Feynman [52] and yields the Green’s functions of Stückelberg and Feynman. The solution in x = (ct, x) space is then Gd (x, t; x  , t  ) = Gd (x − x  , t − t  ),  1 Gd (x, t) = dω Gd (x, ω) exp(− iωt), 2π  1 Gd (x, ω) = d d k Gd (k, ω) exp(ik · x). (2π )d

(J.87) (J.88)

(J.89)

The integral is the same as in (J.63) with the substitution 2m h¯ 2

E→

ω2 m2 c 2 − , c2 h¯ 2

(J.90)

i.e.  d−2  2 m2 c4 − h¯ 2 ω2 (mc2 − h¯ |ω|) Gd (x, ω) = √ d h¯ cr 2π   r ×K d−2 m2 c4 − h¯ 2 ω2 2 hc ¯  d−2  2 − mc2 ) h¯ 2 ω2 − m2 c4 π (h|ω| ¯ +i √ d h¯ cr 2 2π   r (1) . ×H d−2 h¯ 2 ω2 − m2 c4 hc 2 ¯

(J.91)

788

J Green’s Functions in d Dimensions

The ω = 0 Green’s functions  −

m2 c 2 h¯ 2

 Gd (x) = − δ(x),

1

Gd (x) = √ d 2π



mc hr ¯



 d−2 2

K d−2 2

 mc r , h¯

yield again the results (J.51) in the limit m → 0, albeit with diverging integration constants in low dimensions, ad=1 =

h¯ , mc

ad=2 =

2h¯ exp(−γ ). mc

(J.92)

In terms of poles in the complex k plane, the complex shift in (J.85) implies c2 (hω ¯ − mc2 ) 

 Gd (k, ω) =

  ck − ω2 − (mc2 /h) ¯ 2 − i ck + ω2 − (mc2 /h) ¯ 2 + i c2 (mc2 − hω) ¯ 

. +

  2 2 2 ck − i (mc /h) ck − i (mc2 /h) ¯ −ω ¯ 2 − ω2

(J.93)

However, in terms of poles in the complex ω plane, Eq. (J.85) implies c2 

. Gd (k, ω) = −

  ω − c k 2 + (mc/h) ¯ 2 + i ω + c k 2 + (mc/h) ¯ 2 − i Fourier transformation to the time domain therefore yields a representation of the relativistic free Green’s function which explicitly shows the combination of retarded positive frequency and advanced negative frequency components, Gd (k, t) =

1 2π

 dω Gd (k, ω) exp(− iωt)

   2 2 exp − i k + (mc/h) ¯ ct  = ic(t) 2 k 2 + (mc/h) ¯ 2   2 2 exp i k + (mc/h) ¯ ct  + ic(− t) 2 k 2 + (mc/h) ¯ 2    2 2 exp − i k + (mc/h) ¯ c|t|  = ic . 2 2 k 2 + (mc/h) ¯

(J.94)

J Green’s Functions in d Dimensions

789

On the other hand, shifting both poles into the lower complex ω plane, (r)

Gd (k, ω) = −

c2 

,  ω − c k 2 + (mc/h) ¯ 2 + i ω + c k 2 + (mc/h) ¯ 2 + i 

yields the retarded relativistic Green’s function (r)

Gd (k, t) =

1 2π



(r)

dω Gd (k, ω) exp(− iωt)

  2 ct sin k 2 + (mc/h) ¯  = c(t) = c2 (t)Kd (k, t), 2 2 k + (mc/h) ¯

(J.95)

cf. Eq. (22.15). If Kd (x, t) exists, then one can easily verify that the properties   1 ∂2 m2 c 2 − 2 2 − 2 Kd (x, t) = 0, c ∂t h¯   ∂ Kd (x, 0) = 0, Kd (x, t) = δ(x) ∂t t=0

(J.96)

(J.97)

(r)

imply that Gd (x, t) = c2 (t)Kd (x, t) is a retarded Green’s function. Furthermore, shifting both poles into the upper complex ω plane, (a)

Gd (k, ω) = −

c2 

,  ω − c k 2 + (mc/h) ¯ 2 − i ω + c k 2 + (mc/h) ¯ 2 − i 

yields the advanced relativistic free Green’s function (a)

Gd (k, t) =

1 2π



(a)

dω Gd (k, ω) exp(− iωt)

  2 2 sin k + (mc/h) ¯ ct  = − c(− t) k 2 + (mc/h) ¯ 2 (r)

= − c2 (− t)Kd (k, t) = Gd (k, − t).

(J.98)

790

J Green’s Functions in d Dimensions

We have been cautious in our terminology to separate the propagators, which are time evolution kernels and propagate initial conditions, from the Green’s functions, which propagate perturbations or driving terms in a differential equation. However, denoting the Green’s functions with Feynman conventions for the poles as “Feynman Green’s functions” is awkward. At the expense of precision, we will therefore adopt the convention to denote those Green’s functions as Feynman propagators.

Retarded Relativistic Green’s Functions in (x, t) Representation (r)

Evaluation of the Green’s functions Gd (x, t) and Gd (x, t) for the massive Klein– Gordon equation is very cumbersome if one uses standard Fourier transformation between time and frequency. It is much more convenient to use Fourier transformation with imaginary frequency, which is known as Laplace transformation. We will demonstrate this for the retarded Green’s function. We try a Laplace transform of 2 G(r) d (x, t) in the form 

∞

gd (x, w) = 0

(r)

dt exp(−wt)Gd (x, t),

w ≥ 0.

(J.99)

dw exp(wt)

(J.100)

The completeness relation for Fourier monomials, 1 δ(t) = 2π



∞

1 dω exp(− iωt) = 2π i −∞



i∞

−i∞

then yields the inversion of (J.99), (r) Gd (x, t)

1 = 2π i



i∞

−i∞

dw exp(wt)gd (x, w).

(J.101)

The condition (J.83) on the d dimensional scalar Green’s functions then implies

only w ≥ 0 assumes that the retarded Green’s functions are integrable along the time axis. This makes physical sense since the impact of a perturbation which occured at time t  = 0 at the point x  = 0 that is felt at the point x should decrease with time. The assumption can also be justified a posteriori from the explicit results (J.113)–(J.115), which show that the Green’s functions for d ≤ 3 oscillate and decay for t → ∞. For Laplace transforms of less well behaved functions G(x, t) one can require w > ν if exp(−νt)G(x, t) is bounded for t → ∞. The vertical integration contour for the inverse transformation (J.101) must then be to the right of (r) ν −i∞ → ν +i∞. However, this would not save the day for the non-existent functions Gd≥4 (x, t), although we can find functions gd (x, w) (J.109) and (J.110) for every number d of dimensions. 2 Assuming

J Green’s Functions in d Dimensions



791

w2 m2 c 2 − 2 − 2 c h¯

 gd (x, w) = − δ(x)

(J.102)

with solution gd (x, w) =

1 (2π )d

 dd k

exp(ik · x) k + (w/c)2 + (mc/h) ¯ 2 2

.

(J.103)

In one dimension this yields

  c exp − w 2 + (mc2 /h) ¯ 2 |x|/c  g1 (x, w) = . 2 w 2 + (mc2 /h) ¯ 2

(J.104)

In higher dimensions, we need to calculate gd (x, w) =

Sd−2 (2π )d

= √

1 2π

d





∞

π

dk 0



0 ∞

dk 0

dϑ k d−1 sind−2 ϑ

k2

exp(ikr cos ϑ) + (w/c)2 + (mc/h) ¯ 2

1 k d−1 J d−2 (kr). (J.105) √ 2 2 2 d−2 2 k + (w/c) + (mc/h) ¯ kr

We can formally reduce (J.105) for d ≥ 3 to the corresponding integrals in lower dimensions by using the relation   1 d n Jν (x) Jν+n (x) − = , ν x dx x x ν+n

(J.106)

see number 9.1.30, p. 361 in [1]. This yields for d = 2n + 1   1 ∂ n√ −2n − J (kr) = k krJ− 1 (kr) d−2 √ d−2 2 2 r ∂r kr   n 1 ∂ 2 −2n − k cos(kr), = π r ∂r 1

(J.107)

and for d = 2n + 2,   1 ∂ n −2n − J (kr) = k J0 (kr). √ d−2 d−2 2 r ∂r kr 1

(J.108)

The resulting relations for the Green’s functions in the (x, w) representations are then

792

J Green’s Functions in d Dimensions



1 ∂ g2n+1 (x, w) = − 2π r ∂r

n

1 π



∞

cos(kr) + (w/c)2 + (mc/h) ¯ 2 0

    2 2 ¯ 2 r/c 1 ∂ n c exp − w + (mc /h)  = − , (J.109) 2π r ∂r 2 w 2 + (mc2 /h) ¯ 2 dk

k2

and  1 g2n+2 (x, w) = − 2π r  1 = − 2π r

∂ ∂r

n

1 π



∞

kJ0 (kr) + (w/c)2 + (mc/h) ¯ 2 0    ∂ n 1 r K0 (J.110) w 2 + (mc2 /h) ¯ 2 . ∂r 2π c dk

k2

(r)

Inverse Laplace transformation yields the retarded Green’s functions Gd (x, t),  1 G(r) (x, t) = − 2n+1 2π r  1 (r) G2n+2 (x, t) = − 2π r

n

  c (J.111) (ct − r)J0 mc c2 t 2 − r 2 /h¯ , 2 

  ∂ n c (ct − r) cos mc c2 t 2 − r 2 /h¯ . (J.112) √ ∂r 2π c2 t 2 − r 2 ∂ ∂r

Except for m = 0 and d = 1, the terms proportional to (ct − r), i.e. the components inside the light cone, to the retarded Green’s functions decrease like t −d/2 for t → ∞ and r fixed. However, the δ function singularities on the light cone become unacceptable for d ≥ 4. The retarded relativistic Green’s functions in one, two and three dimensions are therefore

  c 2 t 2 − x 2 /h , (ct − |x|)J mc G(r) (x, t) = c (J.113) ¯ 0 1 2 (r)

  c (ct − r) cos mc c2 t 2 − r 2 /h¯ , √ 2π c2 t 2 − r 2

(J.114)

 mc2 (ct − r)  2 2 c δ(r − ct) − J1 mc c t − r 2 /h¯ . √ 4π r 4π h¯ c2 t 2 − r 2

(J.115)

G2 (x, t) = and G(r) 3 (x, t) =

The (x, t) representations of the corresponding advanced Green’s functions then follow from (J.98) as (r) G(a) d (x, t) = Gd (x, − t).

(J.116)

J Green’s Functions in d Dimensions

793

The propagator function Kd (x, t) for the free Klein–Gordon fields follows from (J.95) and (J.98) as (a) c2 Kd (x, t) = G(r) d (x, t) − Gd (x, t).

(J.117)

(r)

The functions Gd≥4 (x, t) and Kd≥4 (x, t) do not exist, but the corresponding (r)

(r)

functions Gd (k, t) = c2 (t)Kd (k, t) (J.95) and Gd (k, ω) exist in any number of dimensions.

Green’s Functions for Dirac Operators in d Dimensions We now restore summation convention. The Green’s functions for the free Dirac operator must satisfy   mc μ iγ ∂μ − Sd (x, t) = − δ(x)δ(t). h¯

(J.118)

Since the Dirac operator is a factor of the Klein–Gordon operator, the solutions of Eqs (J.118) and (J.83) are related by   mc Sd (x, t) = iγ μ ∂μ + Gd (x, t) h¯

(J.119)

and  Gd (x, t) =  =

dd x d 

 

d x

dt  Sd (x  − x, t  − t) · Sd (x  , t  ) dt  Sd (x  , t  ) · Sd (x  + x, t  + t).

(J.120)

The free Dirac Green’s function in wave number representation is (here k 2 ≡ k μ kμ ) Sd (k) =

(mc/h) ¯ −γ ·k , k 2 + (mc/h) ¯ 2 − i

(J.121)

where the pole shifts again correspond to the Feynman propagator with retarded and advanced components. The corresponding retarded Green’s function (both poles in the lower complex ω plane) and the advanced Green’s function (both poles in the upper half-plane) are related to the time evolution kernel of the Dirac field (cf. Eq. (22.108)),

794

J Green’s Functions in d Dimensions

W(k, t) =

p0 =±E(k)/c

 mc − γ · p exp − icp 0 t/h¯ · γ 0 , 0 2p

(J.122)

through Sd(r) (k, t) = ic(t)W(k, t)·γ 0 ,

Sd(a) (k, t) = − ic(− t)W(k, t)·γ 0 .

(J.123)

Green’s Functions in Covariant Notation The relativistic free scalar Green’s function satisfies p 2 + m2 c 2 h¯ 2

Gd = 1,

(J.124)

i.e. in the k = (ω/c, k) domain k|Gd |k   = Gd (k)δ(k − k  ),

(J.125)

where the factor Gd (k) is Gd (k) =

1 k2

+ (m2 c2 /h2 ) − i ¯

.

(J.126)

This yields after transformation into x = (ct, x) space (D = d + 1), 1 x|Gd |x  = (2π )D 

=

1 (2π )D



 D

d k 

d D k  k|Gd |k   exp[i(k · x − k  · x  )],

d D k Gd (k) exp[ik · (x − x  )] = Gd (x − x  ), (J.127)

which satisfies     m2 c 2 m2 c 2 ∂2 − 2 x|Gd |x   = ∂ 2 − 2 x|Gd |x   = − δ(x − x  ). h¯ h¯

(J.128)

The relation to (J.83)–(J.115) is x|Gd |x   =

1 Gd (x, t; x  , t  ), c

k|Gd |k   = cGd (k, ω; k  , ω ),

Gd (k) = Gd (k, ω).

(J.129) (J.130)

J Green’s Functions in d Dimensions

795

Translation invariance (J.125) and (J.127) implies that the Green’s function in mixed representation is proportional to plane waves, x|Gd |k = Gd (k)x|k. The fermionic Green’s function satisfies γ · p + mc Sd = 1, h¯

Sd =

mc − γ · p Gd , h¯

(J.131)

or in various representations,   mc iγ · ∂ − x|Sd |x   = − δ(x − x  ), h¯ k|Sd |k   = Sd (k)δ(k − k  ),  Sd (k) =

x|Sd |k = Sd (k)x|k,

 mc (mc/h) ¯ −γ ·k , − γ · k Gd (k) = 2 2 h¯ k + (m c2 /h¯ 2 ) − i

1 x|Sd |x   = Sd (x − x  ) = Sd (x − x  , t − t  ) c  1 d D k Sd (k) exp[ik · (x − x  )] = (2π )D   mc = iγ · ∂ + Gd (x − x  ). h¯

(J.132) (J.133) (J.134)

(J.135)

The pole shifts in (J.126) and (J.134) correspond to the Feynman conventions. This yields 

" dd k ! (t) exp(ik · x) + (− t) exp(− ik · x) ,(J.136) 2ω(k)     mc ic dd k Sd (x) = (t) − γ · k exp(ik · x) (J.137) (2π )d 2ω(k) h¯    mc + γ · k exp(− ik · x) , (J.138) + (− t) h¯

Gd (x) =

ic (2π )d

where k 0 = ω(k)/c. These Green’s functions are related to matrix elements of the Dirac picture operators (or free Heisenberg picture operators) for the Klein–Gordon or Dirac fields through i0|Tc φ(x)φ + (x  )|0 =

1 x|Gd |x  , c

(J.139)

796

J Green’s Functions in d Dimensions

i0|Tc ψ(x)ψ(x  )|0 = x|Sd |x  ,

(J.140)

where Tc is the chronological time ordering operator (17.205). (r) For the retarded Green’s functions G(r) d and Sd both poles have to be shifted into the lower k 0 plane. Note that as a consequence of (J.135) the free fermionic Green’s functions also satisfy i∂μ Sd (x − x  )γ μ +

mc Sd (x − x  ) = δ(x − x  ). h¯

(J.141)

Green’s Functions as Reproducing Kernels Suppose that V is a (d + 1)-dimensional spacetime volume with boundary ∂V. Equation (J.128) and the Klein–Gordon equation imply for a free field φ(x) and for x in V the relation   2 2  m c D   2 Gd (x − x  ) φ(x) = d x φ(x ) −∂ h¯ 2 V    ↔ = d D x  ∂ μ Gd (x − x  ) ∂ μ φ(x  ) . (J.142) V

The Gauss theorem in D spacetime dimensions then yields a representation for all values of φ(x) inside of V in terms of the values of the Klein–Gordon field on the boundary ∂V,  φ(x) =

∂V

  ↔ d d x  nμ Gd (x − x  ) ∂ μ φ(x  ) ,

(J.143)

where nμ is an outward bound normal vector with n0 = 1 on spacelike boundaries t  = constant, t  > t, and n0 = −1 on t  = constant, t  < t. If Gd (x − x  ) is in particular the retarded Green’s function, Gd (x − x  ) = c(t − t  )Kd (x − x  , t − t  ), (r)

(J.144)

or the advanced Green’s function,     G(a) d (x − x ) = c(t − t)Kd (x − x , t − t),

(J.145)

J Green’s Functions in d Dimensions

797

and ∂V contains only the spacelike surface t  < t, or only the spacelike surface t  > t, then (J.143) is the solution (22.14) of the initial value problem (t  < t) or future value problem3 (t  > t) for the Klein–Gordon field. For free Dirac fields the Dirac equation and (J.141) implies for x ∈ V the equation  ψ(x) = i

∂V

d d x  nμ Sd (x − x  )γ μ ψ(x  ).

(J.146)

This yields again the initial/final value solution (22.107) if ∂V contains only the spacelike surface t  < t or only the spacelike surface t  > t, since the retarded and advanced Green’s functions are related to the time evolution kernel (22.108) through Sd(r) (x − x  ) = i(t − t  )Wd (x − x  , t − t  ) · γ 0 ,

(J.147)

Sd (x − x  ) = − i(t  − t)Wd (x − x  , t − t  ) · γ 0 ,

(J.148)

(a)

see also Eqs. (J.135), (J.144), (J.145), and (22.110).

Liénard–Wiechert Potentials in Low Dimensions The massless retarded Green’s functions solve the basic electromagnetic wave equation for the electromagnetic potentials in Lorentz gauge, ∂μ ∂ μ Aν (x) = − μ0 j ν (x),  A (x) = μ0 μ

∂μ Aμ (x) = 0,

  (r) d d+1 x  Gd (x − x  )

m=0

j μ (x  ).

(J.149) (J.150)

In three dimensions this yields the familiar Liénard–Wiechert potentials from the contributions of the currents on the backward light cone of the spacetime point x, μ

Ad=3 (x, t) =

μ0 4π



d 3x

  1 1 μ   x j |x − x , t − | . |x − x  | c

(J.151)

However, in one and two dimensions, the Liénard–Wiechert potentials sample charges and currents from the complete region inside the backward light cone,

3 The

future value problem or backwards evolution problem asks the question: Which field configuration φ(x) at time t yields the prescribed field configuration φ(x  ) at time t  > t through time evolution with the equations of motion?

798

J Green’s Functions in d Dimensions

μ

Ad=1 (x, t) = μ

Ad=2 (x, t) =

μ0 c 2π



μ0 c 2 d 2x



∞

−∞



dx 



t−(|x−x  |/c) −∞

t−(|x−x  |/c) −∞

dt  

dt  j μ (x  , t  ), j μ (x  , t  )

c2 (t − t  )2 − (x − x  )2

(J.152)

.

(J.153)

Stated differently, a δ function type charge-current fluctuation in the spacetime point x  generates an outwards traveling spherical electromagnetic perturbation on the forward light cone starting in x  if we are in three spatial dimensions. However, in one dimension the same kind of perturbation fills the whole forward light cone uniformly with electromagnetic fields, and in two dimensions the forward light cone is filled with a weight factor [c2 (t − t  )2 − (x − x  )2 ]−1/2 . How can that be? The electrostatic potentials (J.51) for d = 1 and d = 2 hold the answer to this. Those potentials indicate linear or logarithmic confinement of electric charges in low dimensions. Therefore a positive charge fluctuation in a point x  must be compensated by a corresponding negative charge fluctuation nearby. Both fluctuations fill their overlapping forward light cones with opposite electromagnetic fields, but those fields will exactly compensate in the overlapping parts in one dimension, and largely compensate in two dimensions. The net effect of these opposite charge fluctuations at a distance a is then electromagnetic fields along a forward light cone of thickness a, i.e. electromagnetic confinement in low dimensions effectively ensures again that electromagnetic fields propagate along light cones. This is illustrated in Fig. J.1. ct

+

−

x

Fig. J.1 The contributions of nearby opposite charge fluctuations at time t = 0 in one spatial dimension generate net electromagnetic fields in the hatched “thick” light cone region

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Index

Symbols 2-spinor, 178 A Absorption coefficient, 465 Absorption cross section, 292 for photons, 459 Active transformation, 83 Adjoint operator, 78 Amplitude vector, 539 Angular momentum, 134 in relative motion, 130 Angular momentum density for electromagnetic fields, 482 in relativistic field theory, 629 for scalar Schrödinger field, 362 for spin 1/2 Schrödinger field, 363 Angular momentum operator, 134 addition, 181 commutation relations, 136 in polar coordinates, 134 Annihilation operator, 107 for non-relativistic particles, 375 for photons, 439 for relativistic fermions, 604 for relativistic scalar particles, 587 Anti-commutator, 369 Attenuation coefficient, 465 Auger process, 288 B Baker–Campbell–Hausdorff formula, 113, 733 γ matrices, 743 Berry phase, 308

Bethe sum rule, 351 Bloch factor, 206 Bloch function, 206 Bloch operators, 534 Bloch state, 206 Bohmian mechanics, 172 Bohr magneton, 611 Bohr radius, 154 for nuclear charge Ze, 160 Boost parameter, 757 Born approximation, 232 Born-Oppenheimer approximation, 516 Boson number operator, 628 Bra-ket notation, 63, 73 in linear algebra, 73 in quantum mechanics, 74 Bremsstrahlung, 489 Brillouin zone, 206, 530 C Canonical quantization, 367 Capture cross section, 291 due to spontaneous photon emission, 488 Ceiling function, 257 Center of mass frame relativistic, 708 Center of mass motion, 129 Charge conjugation, 584 Dirac field, 605 for general representations of Dirac matrices, 636 Klein-Gordon field, 584 Lorentz covariance, 765 on products of fermion fields, 636

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 R. Dick, Advanced Quantum Mechanics, Graduate Texts in Physics, https://doi.org/10.1007/978-3-030-57870-1

805

806 Charge operator for the Dirac field, 603 for the Klein-Gordon field, 588 for the Schrödinger field, 370 in terms of Schrödinger picture operators, 374 Chirality, 770 Christoffel symbols, 98 in terms of metric, 104 Classical electron radius, 476 Clebsch-Gordan coefficients, 183 Coherent states, 115 for the electromagnetic field, 441 overcompleteness, 119 Collapse of the wave function, 507 Color center, 59 spherical model, 167 Commutator, 88 Commutators involving operator products, 371 Completeness in the mean, 725 Completeness of eigenstates, 30, 711 Completeness relations, 34 in cubic quantum wire, 92 example with continuous and discrete states, 50 for Fourier monomials, 204 for free spherical waves, 149 for hydrogen eigenstates, 165 in linear algebra, 69 for spherical harmonics, 144 for Sturm–Liouville eigenfunctions, 724 for time-dependent Wannier states, 211 for transformation matrices for photon wave functions, 437 for Wannier states, 209, 534 Compton scattering, 654 non-relativistic limit, 664 scattering cross section, 664 Compton wavelength, 16 Confluent hypergeometric function, 154 Conjugate momentum, 680 Conservation laws, 356 Conserved charge, 358 and local phase invariance, 633 as symmetry generator, 428 Conserved current, 358 and local phase invariance, 633 Contravariant transformation behavior, 66 for non-orthogonal transformations, 82 Copenhagen interpretation, 493 Correlation function, 313 Coulomb gauge, 432 Coulomb potential, 448

Index domination for low-energy interactions, 653 Coulomb waves, 162 Covalent bonding, 520 Covariant derivative, 334 Covariant transformation behavior, 66 for non-orthogonal transformations, 82 Creation operator, 107 for non-relativistic particles, 375 for photons, 439 for relativistic fermions, 604 for relativistic scalar particles, 587

D Darwin term, 618 Decay rate, 281 Decoherence induced by environment, 492 Degeneracy pressure, 426 δ function, 25 Density of states, 249 in d dimensions, 257 in the energy scale, 255 inter-dimensional, 574 non-relativistic particles, 257 in radiation, 257 relation to non-relativistic Green’s function, 568 relation to relativistic Green’s function, 580 Density of states operator, 259 Derivative operator alternating, 446 Differential scattering cross section, 293 for photons, 474 Dihydrogen cation, 516 Dimensions of states, 95 Dipole approximation, 338 Dipole line strength, 348 Dipole moment induced, 342 intrinsic, 340 Dipole selection rules, 338 Dirac equation, 595 free solution, 600 with minimal coupling, 597 non-relativistic limit, 610 relativistic covariance, 758 Dirac γ matrices, see γ matrices Dirac picture, 266, 272 in quantum field theory, 384 Dirac’s δ function, 25 Dual bases, 67

Index E Eddington tensor, 134 Edelstein effect, 641 Effective mass, 216 Ehrenfest theorem, 24 for N particles, 402 in second quantization, 410 Einstein-Podolsky-Rosen paradox, 497 Electric dipole line strength, 348 Electromagnetic coupling, 331 Electron-electron scattering, 666 Electron interference biprism experiments, 23 Electron-nucleus scattering, 649 Electron-photon scattering, 654 Energy-momentum tensor, 359 for classical charged particle in electromagnetic fields, 706 for the Maxwell field, 437 in quantum optics, 446 for scalar quantum electrodynamics, 625 for spinor quantum electrodynamics, 605 Energy-time uncertainty, 57, 91, 300, 459 Environmentally induced decoherence, 492 Epistemic collapse, 493 Epistemic reduction, 493  tensor, 134 Euler–Lagrange equations, 354, 680 for field theory, 355 Ewald construction, 72 Exchange energy, 405 Exchange hole, 425 Exchange integral, 406 Exchange interaction, 406 Exchange term kinetic, 404

F Fermi momentum, 260 Fermion number operator, 628 Fermi’s trick, 287 improved derivation, 302 Feynman propagators, 790 Field operator, 368 Field quantization, 367 Fine structure constant, 154 Floor function, 257 Fluorescence is different from scattering, 476 Fock space, 376 Foldy-Wouthuysen transformation, 617 Form factors, 244 Fourier transformation

807 between frequency and time domain, 92 Fourier transforms, 25 Frequency-time Fourier transformation, 92 f -sum rule, 347

G γ5 matrix, 769 γ matrices, 596 construction in d dimensions, 746 Dirac basis, 596 invariance under Lorentz transformations, 758 Weyl basis, 597 Gauss bracket, 745 Gaussian wave packet, 53 free evolution, 54 width, 53 Generators, 138 from charge operators, 428 Golden Rule, 286 Golden Rule #1, 288 for scattering cross sections, 298 Golden Rule #2, 290 for scattering cross sections, 298 Green’s function, 53, 227 advanced, 247, 777 in d dimensions, 773 for Dirac operator, 793 for harmonic oscillator, 271 relations between scalar and spinor Green’s functions, 793 inter-dimensional, 573 retarded, 53, 228, 568, 573, 777 as vacuum expectation value, 409 Group, 132 Group theory, 133

H Hamiltonian density in Coulomb gauge, 609 for the Dirac field, 603 for the Klein-Gordon field, 588 for the Maxwell field, 438 in scalar QED, 625 for the Schrödinger field, 360 for spinor QED, 607 Hard sphere, 237 Harmonic oscillator, 105 coherent states, 115 eigenstates, 107 in k-representation, 111 in x-representation, 109

808 Harmonic oscillator (cont.) isotropic, 105 in polar coordinates, 171 solution by the operator method, 106 Hartree-Fock equations, 415 orthogonalized, 404, 417 Heisenberg evolution equation, 271 Heisenberg Hamiltonian, 407 for spin-1 bosons, 415 Heisenberg picture, 266, 271 Heisenberg uncertainty relations, 87 Hellmann-Feynman theorem, 85 Hermite polynomials, 110, 729 Hermitian operator, 32 Higher order commutator, 113 Hubbard model, 534 Hydrogen atom, 152 bound states, 152 ionized states, 162 radial expectation values, 159 Waller’s method, 168 Hydrogen molecule ion, 520

I Interaction picture, see Dirac picture Inverse Edelstein effect, 641 Ionization rate, 281

J Junction conditions for wave functions, 48

K Kato cusp condition, 529 Klein-Gordon equation, 584 with minimal coupling, 584 non-relativistic limit, 590 Klein–Nishina cross section, 663 Klein’s paradox, 592 oblique incidence, 627 Kramers-Heisenberg formula, 475 Kramers-Pasternak relation, 160 Kronig-Penney model, 212 density of states, 262 Van Hove singularities, 262 Kummer’s function, 154 Kummer’s identity, 163

L Ladder operators, 107 Lagrange density, 353

Index for the Maxwell field, 432 for the Dirac field, 603, 760 for the Klein-Gordon field, 587 for the Schrödinger field, 355 Lagrange function, 679 for particle in electromagnetic fields, 331 for small oscillations in N particle system, 537 Lagrangian field theory, 353 Larmor frequency, 306 Laue conditions, 71 Length dimension of states, 95 Length form matrix elements, 454 oscillator strength, 347 Length gauge, 486 Liénard–Wiechert potentials, 797 Lippmann-Schwinger equation, 227 in terms of the full Green’s function, 246 Local density of states, 259 Lorentz group, 133, 692 generators, 756 spinor representation, 758 in Weyl and Dirac bases, 760 vector representation, 756 construction from the spinor representation, 762 Lorentzian absorption line, 462 Lowering operator, 107

M Matrix logarithm, 737 Maxwell field, 431 covariant quantization, 619 quantization in Coulomb gauge, 438 quantization in Lorentz gauge, 621 Mehler formula, 111, 271, 730 Minimal coupling, 334 in the Dirac equation, 597 in the Klein-Gordon equation, 584 to relative motion in two-body problems, 443 in the Schrödinger equation, 332 Møller operators, 279, 411 Møller scattering, 666 Mollwo’s law, 59 spherical model, 167 Momentum density in Coulomb gauge, 609 for the Dirac field, 603 for the Klein-Gordon field, 588 for the Maxwell field, 438 in scalar QED, 625

Index for the Schrödinger field, 360 in spinor QED, 607 Mott-Gordon states in parabolic coordinates, 245 in polar coordinates, 166 Mott scattering, 653

N Noether’s theorem, 358 Nonlinear Schrödinger equation, 402 Non-relativistic limit Dirac equation, 610 Klein-Gordon equation, 590 Normal coordinates, 539 Normal modes, 539 Normal ordering, 440 for Dirac fields, 603 for Klein-Gordon fields, 588 for photons, 440 N -point function, 313 Number operator, 108 for the Schrödinger field, 370 in terms of Schrödinger picture operators, 374

O Occupation number operator, 108 Ontological collapse, 493 Optical theorem, 233 from partial wave analysis, 237 Oscillator strength, 344

P Parabolic coordinates, 103, 242 Parity transformation, 768 Passive transformation, 66 Path integrals, 311 Pauli equation, 610 Pauli matrices, 177 Pauli term, 463 Perturbation theory, 189, 265 and effective mass, 217 time-dependent, 265 time-independent, 189 with degeneracy, 195, 200 without degeneracy, 189, 194 Phonons, 548 Photoelectric effect, 15 Photon, 438 Photon absorption, 459 Photon coupling

809 to Dirac field, 597 to Klein-Gordon field, 584 Photon coupling to relative motion, 443 Photon emission, 450 Photon flux, 461 Photon scattering, 469 Planck’s radiation laws, 7 Poincaré group, 133, 692 Polar coordinates in d dimensions, 779 Polarizability tensor dynamic, 343 frequency dependent, 344 static, 341 Potential scattering, 225 Potential well, see Quantum well Probability density, 20 Projection postulate, 493 Propagator, 52 for harmonic oscillator, 271 as vacuum expectation value, 410

Q Quantization of the Dirac field, 603 of the Klein-Gordon field, 585 of the Maxwell field, 438 phonons, 548 of the Schrödinger field, 368 Quantum dot, 44 Quantum field, 368 Quantum state uniqueness, 109 Quantum virial theorem, 82 Quantum well, 44 Quantum wire, 44

R Raising operator, 107 Rapidity, 757 Rashba term, 619 Rayleigh-Jeans law, 6 Rayleigh scattering, 477 Reduced mass, 129 Reflection coefficient for δ function potential, 50 for Klein’s paradox, 594 for a square barrier, 42 Relative motion, 129 Relativistic spinor momentum eigenstates, 603 Relativistic spinor plane wave states, 603 Reproducing kernels, 796

810 Rotation group, 133 defining representation, 136 generators, 138 matrix representations, 136 Rutherford scattering, 241

S Scalar, 179 under translations, 125 Scattering Coulomb potential, 241 hard sphere, 237 Scattering amplitude, 294 in potential scattering, 232 Scattering cross section, 226 for two particle collisions, 643 Scattering matrix, 279, 388 for δ potential in one dimension, 49 in higher order, 298 with vacuum processes divided out, 388 Scattering phase shifts, 152 and cross sections, 235 for the hard sphere, 248 SSchrödinger equation time-dependent, 18 time-independent, 39 Schrödinger picture, 266 Schrödinger’s cat, 505 SSchrödinger’s equation, 17 Second quantization, 367 Self-adjoint operator, 30, 79 Separation of variables, 100 Shift operator, 112 Singlet state, 397 Sokhotsky-Plemelj relations, 29 Sommerfeld’s fine structure constant, 154 Spherical Coulomb waves, 162 Spherical harmonics, 141 Spin, 389 Spinor, 179 left-handed, 770 right-handed, 770 Spin-orbit coupling, 181, 613, 618 vector model, 365 Squeezed states, 121 Stark effect, 339 Stefan-Boltzmann constant, 10 Stimulated emission, 467 Stress-energy tensor, see Energy-momentum tensor, 360 Summation convention, 64

Index Symmetric operator, 32 Symmetries and conservation laws, 356 Symmetry group, 132

T Temporal gauge, 479 Tensor product, 65 Thomas-Reiche-Kuhn sum rule, 347 Thomson cross section, 476, 664 Time evolution kernel for free Dirac fields, 601 for free Klein-Gordon fields, 586 Time evolution operator, 52, 267 composition property, 269 for harmonic oscillator, 270 as solution of initial value problem, 269 on states in the interaction picture, 273 two evolution operators in the interaction picture, 274 unitarity, 270 Time ordering operator, 267 Transition frequency, 278 Transition probability, 278 Transmission coefficient for δ function potential, 50 for Klein’s paradox, 594 for a square barrier, 42 Transverse δ function, 437 Triplet states, 397 Tunnel effect, 42 Two-particle state, 393 Two-particle system, 129

U Uncertainty relations, 87 Unitary operator, 79

V Vector, 179 Vector addition coefficients, 183 Velocity form matrix elements, 454 oscillator strength, 347 Velocity gauge, 486 Virial theorem, 81

W Wannier functions, 208 Wannier operators, 534 Wannier states

Index completeness relations, 209 time-dependent, 210 Wave function collapse, 507 Wave packets, 51 Wave-particle duality, 15 Weyl gauge, 479 Wien’s displacement law, 5

811 Y Yukawa potentials, 565 Z Zeeman term, 176 from minimal coupling, 349