Relativistic and Non-Relativistic Quantum Mechanics: Both at Once (Undergraduate Lecture Notes in Physics) [1st ed. 2023] 3031370724, 9783031370724

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Undergraduate Lecture Notes in Physics

Luis Grave de Peralta · Maricela Fernández Lozada · Hira Farooq · Gage Eichman · Abhishek Singh · Gabrielle Prime

Relativistic and Non-Relativistic Quantum Mechanics Both at Once

Undergraduate Lecture Notes in Physics Series Editors Neil Ashby, University of Colorado, Boulder, CO, USA William Brantley, Department of Physics, Furman University, Greenville, SC, USA Matthew Deady, Physics Program, Bard College, Annandale-on-Hudson, NY, USA Michael Fowler, Department of Physics, University of Virginia, Charlottesville, VA, USA Morten Hjorth-Jensen, Department of Physics, University of Oslo, Oslo, Norway Michael Inglis, Department of Physical Sciences, SUNY Suffolk County Community College, Selden, NY, USA Barry Luokkala , Department of Physics, Carnegie Mellon University, Pittsburgh, PA, USA

Undergraduate Lecture Notes in Physics (ULNP) publishes authoritative texts covering topics throughout pure and applied physics. Each title in the series is suitable as a basis for undergraduate instruction, typically containing practice problems, worked examples, chapter summaries, and suggestions for further reading. ULNP titles must provide at least one of the following: ● An exceptionally clear and concise treatment of a standard undergraduate subject. ● A solid undergraduate-level introduction to a graduate, advanced, or non-standard subject. ● A novel perspective or an unusual approach to teaching a subject. ULNP especially encourages new, original, and idiosyncratic approaches to physics teaching at the undergraduate level. The purpose of ULNP is to provide intriguing, absorbing books that will continue to be the reader’s preferred reference throughout their academic career.

Luis Grave de Peralta · Maricela Fernández Lozada · Hira Farooq · Gage Eichman · Abhishek Singh · Gabrielle Prime

Relativistic and Non-Relativistic Quantum Mechanics Both at Once

Luis Grave de Peralta Department of Physics and Astronomy Texas Tech University Lubbock, TX, USA Hira Farooq Department of Physics and Astronomy Texas Tech University Lubbock, TX, USA Abhishek Singh Department of Physics and Astronomy Texas Tech University Lubbock, TX, USA

Maricela Fernández Lozada Instituto Tecnológico de Hermosillo Hermosillo, Mexico Gage Eichman Department of Physics and Astronomy Texas Tech University Lubbock, TX, USA Gabrielle Prime Department of Physics and Astronomy Texas Tech University Lubbock, TX, USA

ISSN 2192-4791 ISSN 2192-4805 (electronic) Undergraduate Lecture Notes in Physics ISBN 978-3-031-37072-4 ISBN 978-3-031-37073-1 (eBook) https://doi.org/10.1007/978-3-031-37073-1 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 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 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. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

This book is a first attempt at introducing relativistic quantum mechanics to interested learners with no previous knowledge of quantum mechanics. Currently, relativistic quantum mechanics is considered an advanced topic that is only accessible to students that have already received considerable training in non-relativistic quantum mechanics. However, the authors believe to have found an excellent pedagogic approach for simultaneously introducing both topics of non-relativistic and relativistic quantum mechanics. Purposely, we have avoided the utilization of the well-known Lorentz invariant equations. With that avoidance, we only refer to the Klein–Gordon and Dirac equations for justifying the use of the Poveda-Poirier-Grave de Peralta (PPGP) equations. The PPGP equations are the equations this book is solely based on. Nevertheless, to steer clear of unnecessary complications in an introductory book, we sporadically refer to well-known results obtained by using the Klein–Gordon and Dirac equations. There exists a PPGP equation that coincides with the Schrödinger equation in the non-relativistic limit. The solutions of this Schrödinger-like PPGP equation are identical to the solutions of the Klein–Gordon equation associated with positive kinetic energies. Therefore, when this does not affect the comprehension of a topic, we sometimes refer to the well-known results obtained by using the Klein–Gordon equation. We do this instead of solving the Schrödinger-like PPGP equation. Also, there exists a Pauli-like PPGP equation that coincides with the Pauli equation in the non-relativistic limit. The solutions of this Pauli-like PPGP equation are identical to the solutions of the Dirac equation associated with positive kinetic energies. Like the process above, we sometimes refer to the well-known results obtained using the Dirac equation. We utilize this instead of solving the Pauli-like PPGP equation.

v

vi

Preface

In addition, there exist two complementary Schrödinger-like and Pauli-like PPGP equations. The solutions of these equations are identical to the respective solutions of the Klein–Gordon and Dirac equations associated with negative kinetic energies. These equations are not studied at an extensive amount in this introduction to relativistic quantum mechanics. Nevertheless, their relation to the existence of antiparticles is discussed. Lubbock, TX, USA Hermosillo, Mexico Lubbock, TX, USA Lubbock, TX, USA Lubbock, TX, USA Lubbock, TX, USA

Luis Grave de Peralta Maricela Fernández Lozada Hira Farooq Gage Eichman Abhishek Singh Gabrielle Prime

About This Book

Chapters 1 and 2 provide a fast and traditional introduction to non-relativistic quantum mechanics. In Chap. 3, this book diverges from a traditional introduction to quantum mechanics. The first proposed ideas of relativistic quantum mechanics are introduced for a background sense of what this book wants to ultimately pursue. This includes the introduction of the Schrödinger-like PPGP equation and its use to solve simple one-dimensional problems. Chapter 4 contains a novel and ambitious study of the consequences of the special theory of relativity for quantum mechanics. This includes the study of the relativistic harmonic oscillator, and a pedagogical presentation of Klein’s paradox based on solving the Schrödinger-like PPGP equation for a step potential. More realistic three-dimensional problems are solved in Chap. 5. The spin and the Pauli-like PPGP equation are introduced in Chap. 6. The relativistic description of the Hydrogen atom is presented and compared with experimental results. In addition, the relativistic descriptions of a spin-(s = 0) and a spin-(s = 1/2) particle moving in a Coulomb potential are compared. Chapter 7 provides a brief overview of the problem of how to describe systems with more than one quantum particle. This includes the application of Pauli’s exclusion principle for describing atoms, a precise yet brief visit to a relativistic Fermi gas, and the discussion of the importance of including relativity for describing the formation of black holes in Cosmology. Most of the results that are presented in this book are widely known results proposed and discovered by a multitude of physicists. However, we introduced some controversial but exciting topics in the last Chapter. Only time and experiments will judge the validity of the ideas discussed in Chap. 8. These topics were included for celebrating the first century of quantum mechanics, for illustrating that relativistic quantum mechanics remains an open field of research, and for emphasizing that the pleasure of discovery and critical thinking skills should be systematically cultivated.

vii

Contents

1 The Need for Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 The Stability of Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Atoms Absorb and Emit Electromagnetic Radiation and the Absorption and Emission Spectra are Formed by a Discrete Set on Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Particles are Waves Kind of . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 The Spectrum of the Hydrogen Atom . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Hydrogen Spectral Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Fine Structure of the Hydrogen Spectral Lines . . . . . . . . . . . . . . . . . . 1.7 Hyperfine Structure and Lamb Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1

2 The Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 The Schrödinger Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 The Statistical Interpretation of the Wavefunction . . . . . . . . . . . . . . . 2.3 Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 The Time Independent Schrödinger Equation . . . . . . . . . . . . . . . . . . . 2.5 The Infinite One-Dimensional Well . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 The Free Particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Does the Schrödinger Equation Satisfy Expectations? . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7 7 8 10 14 16 22 24 25

3 Introducing Relativity in Quantum Mechanics . . . . . . . . . . . . . . . . . . . . 3.1 The Special Theory of Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 The Klein-Gordon Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 The Dirac’s Sea and the Hole Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 The Poveda-Poirier-Grave De Peralta Equation . . . . . . . . . . . . . . . . . 3.5 The Relationship Between the PPGP and the Klein-Gordon Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 A Relativistic Particle Trapped in an Infinite One-Dimensional Well . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27 28 35 37 40

1 2 3 3 4 5 6

41 44

ix

x

Contents

3.7 A Beam of Relativistic Particles in a Constant Potential . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

46 48

4 Other One-Dimensional Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 The Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 A Beam of Particles Hitting a Sharp Potential Step . . . . . . . . . . . . . . 4.3 Particles, Antiparticles, and Exotic Quantum States with Negative Kinetic Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49 49 53

5 Quantum Mechanics in Three Dimensions . . . . . . . . . . . . . . . . . . . . . . . . 5.1 The Infinite Cubic Well . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 The Coulomb Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 The Angular Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.3 The Radial Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.4 The Spectrum of Hydrogen . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71 71 73 74 76 82 83

6 Angular Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Orbital Angular Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 The Quantum Mechanical Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Electron in a Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 The Pauli-Like PPGP Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 The Hydrogen Atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

85 85 87 89 91 94 99

7 Identical Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Fermi Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Non-relativistic Fermi-Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Relativistic Fermi Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.3 Fermi-Gas Stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

101 101 105 105 107 108 110 114

8 Some Consequences of Relativity for Quantum Mechanics . . . . . . . . . 8.1 Atoms Cannot Be Too Heavy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 A Possible Frontier Between the Classical and the Quantum World . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 About the Observed Asymmetry Between Matter and Antimatter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

115 115

61 69

117 119 125

Annexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

About the Author

Luis Grave de Peralta is currently a full professor in the Department of Physics and Astronomy at Texas Tech University (TTU), USA. He is a Licenciado en Física from the Oriente University in Cuba. He obtained a Ph.D. in Electrical Engineering at TTU. He has published more than 90 papers in archival peer review journals. Maricela Fernández Lozada (Ph.D.) is a Cuban Mexican theoretical physicist with a large experience teaching quantum mechanics. Hira Farooq (Ph.D.) is an instructor in the TTU Department of Physics and Astronomy. Gage Eichman, Abhishek Singh, and Gabrielle Prime are bright undergraduate physics students at the TTU Department of Physics and Astronomy. The classroom interactions between the authors resulted in the novel pedagogical approach to relativistic and non-relativistic quantum mechanics this book is based on.

xi

Chapter 1

The Need for Quantum Mechanics

Abstract Quantum mechanics was discovered in the second decade of the twentieth century. In that time, physicists were trying to explain phenomena involving atoms and molecules, so quantum mechanics was originally developed as the mechanics of the sub-microscopic world. Therefore, the concept of quantum particles referred to molecules, atoms, and the particles forming the atoms. Some of the phenomena that motivated the development of quantum mechanics are mentioned and explained in this chapter.

1.1 The Stability of Atoms A vast number of atoms are stable. An atom with the atomic number Z is formed by a nucleus with a charge eZ, and Z negative-charged electrons with charge -e. The electrons are attracted to the nucleus, so therefore, their acceleration is not null. Classical electromagnetism predicts that any charged particle moving with a nonnull acceleration must radiate, so ultimately it must lose energy. Classical physics predicts that the electrons in an atom must lose energy and will ultimately fall towards the nucleus. Atoms should not be stable in theory, but in reality, many are stable. With that being said, the necessity of a new concentration in physics was essential for explaining the observed stability of many atoms.

1.2 Atoms Absorb and Emit Electromagnetic Radiation and the Absorption and Emission Spectra are Formed by a Discrete Set on Lines Several decades before the discovery of the Schrödinger Equation in 1925, Max Planck and Albert Einstein had a proposition. This proposal stated that the frequency (ν) of the electromagnetic radiation that is emitted or absorbed by an atom, is proportional to the internal energy (∆E) lost or gained by the atom. Respectively, this is represented by: © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L. Grave de Peralta et al., Relativistic and Non-Relativistic Quantum Mechanics, Undergraduate Lecture Notes in Physics, https://doi.org/10.1007/978-3-031-37073-1_1

1

2

1 The Need for Quantum Mechanics

Fig. 1.1 Bohr’s model of the Hydrogen atom. There is a discrete set of possible electron’s orbits. The electron’s energy (E) is larger in most external orbits. A photon is emitted when the electron jumps to an interior orbit. The frequency of the emitted radiation is determined by the relation ∆E = hν

n=3

n=2 n=1

Nucleus

ΔE Electron

ΔE=hν Radiation

∆E = hυ.

(1.1)

In Eq. (1.1), h represents the Planck’s constant. It was experimentally well-known at the beginning of the twentieth century that atoms solely absorb and emit discrete sets of frequencies. As a result, the internal energy of atoms should only have a discrete set of values. In the year 1913, Niels Bohr proposed that the electron in an atom has orbits with a discrete spectrum of energy. This is depicted in Fig. 1.1 above. In the Bohr model, ∆E, is the difference of the electron energy in different orbits. Later (Fig. 1.2), using the ideas proposed in 1924 by Louis De Broglie, the stability of the electron’s orbits was explained by assuming there is a wave associated to any particle with mass. The length of the stable orbits must equal a multiple of the wavelength that corresponds to an electron with a magnitude of its linear momentum (p): p=

h . λ

(1.2)

1.3 Particles are Waves Kind of When these equations are applied to the electrons in atoms, Eqs. (1.1) and (1.2) mix the properties of a subatomic particle with the properties of a wave. Frequency and wavelength are properties of waves. They are properties of something that is not spatially localized but is distributed in a spatial region. However, ∆E and p

1.5 Hydrogen Spectral Series Fig. 1.2 The length of a possible electron’s orbit (2πr) must be equal to a multiple of the De Broglie’s wavelength (λ) associated to the electron’s linear momentum (p)

3

r

2πr = 2λ

are properties of a subatomic particle where they are properties of something that is highly localized in the space or is even a mathematical point with mass. A new physics was in demand to explain the nature of the relationship between a quantum particle and the wave associated with it.

1.4 The Spectrum of the Hydrogen Atom The Hydrogen atom is the simplest atom known to the Universe. It is formed due to the interaction of a single electron with a single proton. The simplicity of the Hydrogen atom helped experimental and theoretical physicists determine the development of quantum mechanics. The spectrum of the Hydrogen atom is summarized below in relation to the principal experimental and theoretical results.

1.5 Hydrogen Spectral Series In 1880, Johannes Rydberg experimentally and empirically discovered the Rydberg’s formula: [ ] 1 1 1 − (1.3) = Z2 R , with R = 1.09678 × 107 m −1 . λ n2 (n ' )2 In Eq. (1.3), n and n' are positive integers, λ is the wavelength of the light emitted or absorbed by the atom, and Z is the atomic number of a Hydrogen-like atom. This is where an atom exists with a single electron. Rydberg’s formula is a recipe for determining a set of λ values corresponding to the experimental lines in the spectra of Hydrogen-like atoms. Later, in 1913, Bohr’s model provided a semiquantitative explanation of Rydberg’s formula. In 1925, Erwin Schrödinger derived Rydberg’s formula from the solution of today’s famous Schrödinger equation. Theoretically, Schrödinger obtained a formula for Rydberg’s constant that is in satisfactory correspondence with its experimental value:

4

1 The Need for Quantum Mechanics

(a)

Balmer Series

(b) Pashen Series

Lyman Series

(c) n=1

2

3 45

Fig. 1.3 a Sketch of the structural functionalism of the Hydrogen’s spectrum, b illustration of the emission lines of the Lyman series, c doublet fine structure of the Lyman-α line

R=

m e e4 . 8εo2 h 3 c

(1.4)

In Eq. (1.4), me symbolizes the electron mass, e represents the charge of a proton, c stands for the speed of the light in the vacuum, and εo represents the dielectric constant of the vacuum. Figure 1.3a depicts a sketch of the structural functionalism of the spectrum of the Hydrogen atom. This is observed and obtained by utilizing a spectrometer with a minimal amount of resolution. Several series of lines can be identified in the spectrum. For instance, the Lyman Series corresponds to n' = 1 and n ≥ 2 in Rydberg’s formula (Fig. 1.3b).

1.6 Fine Structure of the Hydrogen Spectral Lines However, if a spectrometer with better resolution were used to observe the spectrum of the Hydrogen atom, it would be clear that in many cases a single spectrum line is formed by set of several lines close to each other. For instance, Fig. 1.3c displays that two close lines are observed where there should appear a single line corresponding to the Lyman-α line. The Lyman-α doublet corresponding to the electron transition from n = 2 to n' = 1 is exemplified in Fig. 1.3c. As sketched in Fig. 1.4, a theoretical explanation of the Lyman-α doublet includes the existence of the electron’s spin and the introduction of special theory of relativity in quantum mechanics. How this can be accomplished will be discussed later beginning in Chap. 3. The net result is that the energy of the electron in the Hydrogen atom depends on not one, but two quantum numbers, the principal quantum number [n = 1, 2, …], and the total angular momentum quantum number [ j = 1/2, 3/2,…].

1.7 Hyperfine Structure and Lamb Shift

Quantum mechanics En

5

+ Relativity

+ Quantum spin

n=2

En = 2, j = 3/2

n = 2, l= 1, j = 3/2

n = 2, l = 1

En = 2, l = 1

n = 2, l = 0, j = 1/2

En = 2, j = 1/2

n = 2, l = 1, j = 1/2

n = 2, l = 0

En = 2, l = 0 Hydrogen series

Fine structure

Fig. 1.4 Sketch of the theoretical explanation of doublet fine structure of the Lyman-α line

1.7 Hyperfine Structure and Lamb Shift In addition, to explain the so-called hyperfine structure of the Hydrogen’s spectrum, the magnetic interaction between the spin of the electron and the spin of the proton should be considered. Moreover, as sketched in Fig. 1.5, in 1947 Lamb found that the transition from the states n = 2, P1/2 and n = 2, S1/2 (both with quantum numbers n = 2 and j = 1/2) to the state n = 1, S1/2 is not a line, but a doublet. The frequency shift between the two lines in the doublet is ≈ 1000 MHz. This discrepancy, between the predictions of the fully relativistic Dirac’s theory and the experiments, motivated the development of quantum electrodynamics, which is the first quantum field theory ever developed. Quantum mechanics + Relativity + Quantum spin En

+ Quantum electrodynamic

+ Magnetic interaction with proton

n=2

En = 2, j = 3/2

n = 2, l = 1, j = 3/2

2S1/2 n = 2, l = 0, j = 1/2 n = 2, l = 1, j = 1/2

En = 2, j = 1/2 Fine structure

2P1/2 Lamb shift

Hyperfine structure

Fig. 1.5 Sketch of the hyperfine structure the Hydrogen atom spectrum and the Lamb shift

6

1 The Need for Quantum Mechanics

Originally, calculations utilizing quantum electrodynamics were plagued with infinities. The first indication of a possible way out was given by Hans Bethe in 1947. Bethe made the first non-relativistic computation of the shift of the lines of the Hydrogen atom as measured by Lamb. Despite the limitations of the computation, the agreement was excellent in the sense of validity. The sole idea was to simply attach infinities to corrections of mass and charge that were fixed to a finite value by experiments. In such manner, the infinities get absorbed in those constants and yield a finite result in good agreement with experiments. This procedure was named renormalization. Despite the importance of quantum electrodynamics, and due to the mathematical difficulties involved, this topic is not included in this short introduction to relativistic quantum mechanics.

Bibliography 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

R. Harris, Modern Physics, 2nd edn. (Pearson Addison-Wesley, New York, 2008) J.D. Jackson, Classical Electrodynamics, 2nd edn. (J. Wiley & Sons, New York, 1975) Max Planck (Wikipedia), https://en.wikipedia.org/wiki/Max_Planck. Accessed 10 April 2023 Albert Einstein (Wikipedia), https://en.wikipedia.org/wiki/Albert_Einstein. Accessed 10 April 2023 Louis De Broglie (Wikipedia), https://en.wikipedia.org/wiki/Louis_de_Broglie. Accessed 10 April 2023 Niels Bohr (Wikipedia), https://en.wikipedia.org/wiki/Niels_Bohr. Accessed 10 April 2023 Erwin Schrödinger (Wikipedia), https://en.wikipedia.org/wiki/Erwin_Schr%C3%B6dinger. Accessed 10 April 2023 Rydberg’s formula (Wikipedia), https://en.wikipedia.org/wiki/Rydberg_formula. Accessed 10 April 2023 Hydrogen spectral series (Wikipedia), https://en.wikipedia.org/wiki/Hydrogen_spectral_ series. Accessed 10 April 2023 Fine structure (Wikipedia), https://en.wikipedia.org/wiki/Fine_structure. Accessed 10 April 2023 Lamb shift (Wikipedia), https://en.wikipedia.org/wiki/Lamb_shift. Accessed 10 April 2023 Quantum Electrodynamics (Wikipedia), https://en.wikipedia.org/wiki/Quantum_electrodyn amics. Accessed 10 April 2023

Chapter 2

The Wave Equation

Abstract In 1925, quantum mechanics was put on solid ground when Erwin Schrödinger discovered the equation that is named after him today. It was then clear that the analytical expression of the wave associated to a quantum particle, with mass m, could be calculated by solving the Schrödinger equation. This chapter provides a fast and traditional introduction to non-relativistic quantum mechanics.

2.1 The Schrödinger Equation The one-dimensional Schrödinger equation for a quantum particle of mass m in a potential V is given by the following expression: iℏ

∂ ℏ2 ∂ 2 ψ(x, t) + V (x)ψ(x, t). ψ(x, t) = − ∂t 2m ∂x2

(2.1)

The solution of this specific wave equation is the wavefunction ψ that depends on the spatial (x) and temporal (t) variables. For pedagogical reasons, we will first study the one-dimensional Schrödinger equation (Eq. 2.1). However, space is tridimensional, meaning there are more realistic wavefunctions that depend on three spatial variables instead. The tridimensional Schrödinger equation is later introduced and studied in Chap. 5. In Eq. (2.1), V (x) represents a potential that only depends on the spatial variable. It is given that ℏ = h/2π represents the reduced Planck’s constant. The i in this equation represents the imaginary unit. Note that m and V are the only properties of the particle and the medium of where the particle is explicitly included in the Schrödinger equation. The symbol psi, ψ, represents the analytical expression of the wave associated to the quantum particle with mass m, which is moving in a potential V. Further on, we will learn more about the Schrödinger equation by solving and interpreting some simple examples. Now, let us promptly discuss some of the fundamental properties of the Schrödinger equation and its solutions. The explicit presence of the imaginary unit in Eq. (2.1), determines that ψ is considered a complex function, which can be expressed using two real functions. The real part of ψ (Re[ψ]) and the imaginary part of ψ (Im[ψ]): © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L. Grave de Peralta et al., Relativistic and Non-Relativistic Quantum Mechanics, Undergraduate Lecture Notes in Physics, https://doi.org/10.1007/978-3-031-37073-1_2

7

8

2 The Wave Equation

ψ = Re[ψ] + iIm[ψ].

(2.2)

Alternatively, ψ can be expressed in the following way: ψ = |ψ|eiArg[ψ] .

(2.3)

In Eq. (2.3), |ψ| and Arg[ψ] designates the amplitude and phase of the wavefunction ψ, respectively: |ψ| =

/

(Re[ψ])2 + (Im[ψ])2 and Arg[ψ] = ArcTan

Im[ψ] . Re[ψ]

(2.4)

The amplitude of the wavefunction is particularly important due to the statistical interpretation of the wavefunction that will be discussed in the next Section. Another important property of the Schrödinger equation is that it is a linear equation. This means that if ψ 1 , ψ 2,…, ψ N are all solutions of the Schrödinger equation, then the following is also a solution of Eq. (2.1): ψ=

N ⎲

an ψn , an ∈ C.

(2.5)

n=1

2.2 The Statistical Interpretation of the Wavefunction Currently, there is broad consent in the community of physicists regarding the analytical expression that describes the wave associated to a quantum particle with mass m. This could be calculated through solving the Schrödinger equation. However, there are several proposals concerning the physical interpretation of ψ. One of the more popular schemes is the so-called “Statistical Interpretation” of the wavefunction that was proposed in 1926 by Max Born. Born was a member of the Copenhagen group led by Niels Bohr. Where this is also called and known as the “Orthodox Interpretation”, |ψ(x,t)|2 is considered to be a probability density. The symbol (ρ) is noted as the probability per unit of length for finding the particle at the position x with the time of t = τ: ρ(x) = |ψ(x, τ )|2 = ψ ∗ (x, τ )ψ(x, τ ).

(2.6)

In the statistical interpretation, there is a wave of probabilities associated to any particle with mass. Therefore, in this account, ψ does not have a real physical existence. Instead, it is solely a good mathematical tool for calculating the probability (PAB ) of finding the particle in the segment defined by the points A and B at the time t = τ:

2.2 The Statistical Interpretation of the Wavefunction

∫B PAB =

9

∫B |ψ(x, τ )| dx =

ρ(x, τ )dx = A

∫B 2

A

ψ ∗ (x, τ )ψ(x, τ )dx.

(2.7)

A

Other interpretations consider that ψ is a matter wave with a real physical existence. For instance, Erwin Madelung founded the existence of Quantum Hydrodynamics in 1926. In 1952, David Bohn developed earlier ideas from Louis De Broglie given the theory of De Broglie-Bohm’s quantum mechanics. Currently, there is not a set universal consent about this topic in the collective community of physicists. Nevertheless, there is universal consent about the random nature of the measurements given the position of quantum particles. This theory is supported and backed up by a large body of experimental research. If a quantum particle were trapped in the segment defined by the points at A and B, and if each experiment were conveniently prepared so that the initial quantum state of the particle (ψ(x,0)) is always the same, then the position of the particle measured at the same time (t = τ) would be different in each experiment (x n ). The average position ( ) of the particle in a complete set of N experiment would be equal to: ∑ ∑N x=

k=1 xk

N

∫B =

∫B xρ(x, τ )dx =

A

∫B x|ψ(x, t)| dx = 2

A

ψ ∗ (x, t)xψ(x, t)dx. (2.8)

A

The random nature of the measured results is considered by the advocates of the realistic interpretation of the wavefunction as evidence of the incomplete description of reality given by quantum mechanics. Physicists such as Albert Einstein and David Bohm advocated for the existence of hidden variables that are vital for a complete description of the submicroscopic reality. There is another universal consent in which that after the position of the particle is measured at x = x o , the act of measuring the position of the particle puts the particle in the quantum state that ultimately gives ψ(x o , τ). If the position of the particle is measured in several successive experiments without returning the particle to its initial state ψ(x, 0), then all measurements would ultimately end up at the same particle’s position value of x o . This theory is also supported and backed up by a large body of experimental research. There are other supplemental topics with importance related to the physical interpretation of the wavefunction where there is no consent in the modern community of physicists. Some of these topics will be addressed in Sect. 2.6, respectively. The existence of quantum mechanics is almost a century old, but it is still a discipline in development. There exist previous and current novel insights while we are discussing these complex ideas. Let us conclude this Section by discussing another important property of the Schrödinger equation. This property is directly related to the statistical interpretation of the wavefunction. This property guarantees that if the wavefunction is normalized at t = 0, then it will remain normalized forever, thus:

10

2 The Wave Equation

d dt

∫+∞ ∫+∞ ] ∂[ ∗ ∗ ψ (x, t)ψ(x, t) dx = 0. ψ (x, t)ψ(x, t)dx = ∂t

−∞

(2.9)

−∞

But: ∂ ∂( ∗ ) ∂ ψ ψ = ψ ∗ ψ + ψ ψ ∗. ∂t ∂t ∂t

(2.10)

The time derivatives can be evaluated using Eq. (2.1) as: iℏ ∂ 2 ∂ ∂ i iℏ ∂ 2 ∗ i ψ= ψ − V (x)ψ, and ψ ∗ = − ψ + V (x)ψ ∗ . 2 ∂t 2m ∂x ℏ ∂t 2m ∂x2 ℏ

(2.11)

Therefore: ( [ ( )] ) 2 ∂ψ ∗ ∂( ∗ ) iℏ ∂ 2ψ ∗ ∂ iℏ ∗∂ ψ ∗ ∂ψ ψ ψ = ψ ψ − ψ . (2.12) − ψ = ∂t 2m ∂x2 ∂x2 ∂x 2m ∂x ∂x By substituting Eq. (2.12) in Eq. (2.9), we obtain: d dt

( )I+∞ ∫+∞ I ∂ψ ∗ ∂ψ iℏ ψ∗ − ψ II ψ ∗ (x, t)ψ(x, t)dx = = 0. 2m ∂x ∂x −∞

(2.13)

−∞

In Eq. (2.8), we noted that ψ must go to zero as x goes to ± ∞. Consequently, the integral in Eq. (2.13) is independent of time. Therefore, if ψ is normalized at t = 0 and ψ satisfies the Schrödinger equation, then ψ will stay normalized for eternity. Note that the existence of a quantum particle requires that the following normalization condition is fulfilled: ∫+∞ ψ ∗ (x, t)ψ(x, t)dx = 1.

(2.14)

−∞

2.3 Operators The formalism of quantum mechanics is extensively developed. For both practical and fundamental mathematical reasons, it is very common to find the Schrödinger Equation rewritten in the following way: ( ) Hˆ ψ = Kˆ + Vˆ ψ.

(2.15)

2.3 Operators

11

This is a convenient compact notation. In Eq. (2.15), the characters with a hat (^) are the quantum operator’s total energy, potential energy, and kinetic energy, respectively: pˆ 2 ∂ ∂ with pˆ = −iℏ . Hˆ = iℏ , Vˆ = V (x), and Kˆ = ∂t 2m ∂x

(2.16)

In Eq. (2.15), these operators act over ψ and returns a different function: ∂ Hˆ ψ = iℏ ψ = f (x, t), Vˆ ψ = V (x)ψ = g(x, t). ∂t

(2.17)

And: ( ) ∂ 2 1 ℏ2 ∂ 2 1 2 ˆ pˆ ψ = −iℏ ψ =− ψ = h(x, t). Kψ = 2m 2m ∂x 2m ∂x2

(2.18)

The operator of kinetic energy is an example of an operator which has a mathematical expression obtained from another operator. In this case, it is denoted as the linear momentum (p) operator. In quantum mechanics there is an operator for each physical magnitude. Table 2.1 contains a list of some frequently used operators. In the discipline of classical mechanics, the analytical expression corresponding to the physical magnitude O is a function of x and p, given O = f (x, p). For instance, the formula for the kinetic energy of a classical particle is denoted as K = p2 /2 m. Then, the quantum mechanical operator corresponding to O is: ( ) ˆ = f xˆ , pˆ . O

(2.19)

For instance, the analytical expression of the kinetic energy operator given by Eq. (2.16) was obtained by following the first-quantization procedure given by Eq. (2.19). Operators are important in quantum mechanics not only because their use provides for a compact notation. Based on the statistical interpretation of the wavefunction and if the particle is in the quantum state ψ, then the average value of the physical magnitude O is given by the following equation: Table 2.1 Some frequently used operators

Position

xˆ = x

Linear momentum

∂ pˆ = −iℏ ∂x

Total energy

Hˆ = iℏ ∂t∂

Kinetic energy

Kˆ =

Potential energy

pˆ 2 2m

ℏ ∂ = − 2m ∂x2 ( ) Vˆ = V xˆ = V (x) 2

2

12

2 The Wave Equation

∑ ∑N O=

k=1

N

Ok

∫B =

ˆ ψ ∗ (x, t)Oψ(x, t)dx.

(2.20)

A

For instance, Eq. (2.8) takes the form of Eq. (2.20) for the average position of the particle. In addition, equations of the following kind are of particular interest in quantum mechanics: ˆ = OΩ. OΩ

(2.21)

Equations in this form are called “eigenequations”. In general, solving Eq. (2.21) means to find a set of eigenfunctions (Ω n ) and the corresponding set of eigenvalues (On ). Eigenequations are firstly relevant in the case if a quantum particle were in the quantum state Ω n , then a measurement of the physical magnitude O would surely return to the value of On . However, if the particle were in the quantum state of ψ, that is not particularly an eigenfunction of Eq. (2.21), then the result of the measurement could be any one of the eigenvalues of Eq. (2.21). Moreover, if the particle were to exist in a quantum state ψ that is not an eigenfunction of Eq. (2.21), then the result of the measurement could not be a value that is not one of the eigenvalues of Eq. (2.21). Second, if the operators are Hermitic, then eigenequations are relevant from a more mathematical point of view. The operator O es Hermitic if it satisfies the following equation: ∫

ˆ n dx = Ω∗n' OΩ

∫

ˆ n' dx. Ω∗n OΩ

(2.22)

The eigenvalues of Hermitic operators are always real. For this reason, the operators that correspond to physical magnitudes should be Hermitic. In this case, if the set of eigenfunctions were numerable, such as Ω = Ω 1 , Ω 2 … Ω n , then the set of eigenfunctions of Eq. (2.21) is a complete set. This means that any other function ψ can be expressed as the following series: ψ=

⎲

∫ cn Ωn , where cn =

ψ ∗ Ωn (x)dx, and

∫

Ω∗n' (x)Ωn (x)dx = δn' n . (2.23)

n

In Eq. (2.23), δnn' is the delta of Kronecker, which is equal to 1 if n = n', but 0 otherwise. In general, cn is a complex number with an important physical meaning. This is because if the particle were in the quantum state ψ that is not an eigenfunction of Eq. (2.21), then |cn |2 would be equal to the probability of getting the value On . This is when the physical magnitude O is being measured. Although a more formal explanation could be easily found elsewhere in modern physics literature, this Section itself includes a brief overview of the concept of an operator and its sole use in quantum mechanics. Formally, any wavefunction is a vector, and the set of all the wavefunctions form a Hilbert space with very welldefined properties. Using this formalism, Eqs. (2.7), (2.20), (2.22), and (2.23) could

2.3 Operators

13

be rewritten in a more compact form in the following way:

P−∞+∞

∫+∞ = ψ ∗ (x, τ )ψ(x, τ )dx = 〈ψ|ψ〉 = 1.

(2.24)

−∞

O=

∫+∞ < I I > IˆI ˆ ψ ∗ (x, t)Oψ(x, t)dx = ψ IO Iψ . −∞

I I > < I I > IˆI IˆI ' Ωn' IO IΩn = Ωn IO IΩn .

(2.25)


0. 2mL2

(2.62)

The existence of a non-null minimum value of the energy means that the spatially localized particle cannot lose more energy if it is in its ground state (n = 1). This itself explains the stability of atoms. Moreover, from Eqs. (1.1) and (2.54), it follows that a quantum particle with mass m, that is confined in a small spatial region, must have a discrete frequency spectrum: Δυij =

) Ei − Ej ΔEij h (2 = = i − j2 . 2 h h 8mL

(2.63)

It should be noted that Eq. (2.63) does not precisely predict the values of Δν in the spectrum of the Hydrogen atom. This could be expected because the electron in the Hydrogen atom is not moving in the infinite well potential (Eq. (2.48)), but in the Coulomb potential that is produced by the proton. For pedagogical reasons, we will refine the model of a quantum particle confined in a small spatial region until the predicted values of Δν match the spectrum of the Hydrogen atom. Nevertheless, it is astonishing that both the stability of the atoms and the discrete character of their spectra are just consequences of the spatial localization of the quantum particle. Other interesting physical characteristics of the quantum world can be extracted from the mathematical results obtained in this Chapter. From Eq. (2.54), it follows that the probability (P) of a measurement of the position of the particle in the well returns the value x with a precision Δx is: P = ρn (x)Δx =

2Δx 2 ( nπ ) sin x . L L

(2.64)

20

2 The Wave Equation

This probability depends on the quantum state the particle is in. From Fig. 2.1, it follows that if the particle is in the ground state (n = 1), then P is at its maximum at x = L/2. This signifies that the particle is more probable to be found around the center of the well. However, if the particle is in the first excited state (n = 2), the particle is less probable to be found around the walls and the center of the well. This is where P is found to be at a maximum at x = L/4 and x = 3L/4. In both cases, P is different from what we should expect if the particle were to be a classical particle. The potential energy of a particle that lies inside the infinite well is null, which means that all its energy is kinetic. Therefore, if K > 0, the particle must be moving inside of the well. This is correct for both a classical and a quantum particle. Note that a classical particle could exist and be inside the well when K = 0. On the contrary, if a quantum particle is inside of the well then, the relationship will be K > 0. In the stationary states, the kinetic energy of the particle is constant. A classical particle with a constant kinetic energy, K = p2 /2m, contains a constant value of the magnitude of its linear momentum (|p|). This means that if a classical particle with K > 0 were trapped in a one-dimensional infinite well, then the particle would be traveling back and forth with a constant value |p|. Consequently, the probability of finding the particle anywhere inside the well would be the same. If a quantum particle were trapped in a one-dimensional infinite well, so that the particle is in a stationary state, then the particle must be moving with constant values of K and |p|. How does the quantum particle move? A quantum particle does not move like a classical one does because P depends on the nature of the particle. Certainly, if the quantum particle were in the first excited state, then its movement would be extremely unusual. This is because the quantum particle would be highly probable to be found between the points at x = L/2 and x = 3L/4, but very unlikely be found at x = L/2. How can a particle moving with constant value of |p| go from x = L/2 to x = 3L/4 without passing through x = L/2? This is known as the quantum particle propagation paradox. Followers of different interpretations of the mathematical formulism of quantum mechanics give various amounts of answers to this question. In most introductory quantum mechanical courses, the quantum particle propagation paradox is not admittedly mentioned. This is because of historical and pedagogical reasons. Introductory quantum mechanics courses adhere to the orthodox interpretation of quantum mechanics, which is also known as the Copenhagen interpretation. This was the original interpretation proposed and defended by the founders of quantum mechanics. Followers of the orthodox interpretation maintain the belief that particles do not have trajectory. Therefore, they do not welcome or consider the question of how the quantum particle moves. A typical answer to such a simple but challenging question, is that a quantum particle is not in a place until it is measured there. For them, it does not make any sense to talk about the position of a quantum particle before it is measured. The following uncertainty formula is an experimental fact: ΔxΔp > 0.

(2.65)

2.5 The Infinite One-Dimensional Well

21

In Eq. (2.65), Δx and Δp are the uncertainty in the determination of the position and linear momentum of a quantum particle. It is theoretically possible to simultaneously measure the position and the linear momentum of a classical particle with absolute precision. This means that a classical particle will be ΔxΔp ≥ 0. If Δx = Δp = 0, then a classical particle describes a well-defined trajectory. Note that a classical particle could be at rest inside of an infinite well. Thus, it will also reside in a definite position with |p| = 0. In contrast, a quantum particle cannot be at rest in the well because K > 0 for a quantum particle. Followers of the orthodox interpretation of quantum mechanics argue that Eq. (2.65) implies that quantum particles do not move following well-defined trajectories. However, this is not correct as demonstrated by alternative quantum mechanics interpretations that are consistent with Eq. (2.65). Quantum particles move following well-defined trajectories in these theories. However, the trajectories of the quantum particle in these alternative interpretations defies common sense. For pedagogical reasons and for time’s sake, we will limit the discussion about this here. A classical particle trapped in an infinite well can contain any value of K, but a quantum particle can only have the values given by Eq. (2.54). Therefore, it looks like we could possibly distinguish a classical particle from a quantum particle by introducing it in a well. Classical particles can be at rest in the well and can have any value of K. However, quantum particles cannot be at rest and can only occupy a numerable set of K values. Questions arise from these known theories of particles. Is there any way to know if the particle of mass m is classical or quantum without measuring its kinetic energy in the well? So originally, quantum mechanics was created for describing particles of minuscule mass forming part of submicroscopic systems. However, m is the only property of the particle in the Schrödinger equation. This motivates the wonderance of the following questions. Do quantum particles with a large mass exist or could they exist? Is there a mass value that separates the quantum world from the classical one? Currently, there is not a viable consent in the community of physicists about the response to these questions. Nevertheless, let us discuss a possible response to these questions based on Eq. (2.54). From the Eqs. (2.62) and (2.63), it follows that for a given value of L, if m >> 1 then E 1 > 1, then E 1 0 and Δν is large. This insinuates that only particles confined in a very small spatial region should be quantum. Note that the initial focus of quantum mechanics on very small particles with a tiny mass is well-justified. The extension of quantum mechanics for including the study of very massive objects with a very small size is more controversial. Currently, objects like these are of high interest. This includes the mysterious and notorious celestial objects such as black holes and the whole Universe at the first moments of its formation.

22

2 The Wave Equation

2.6 The Free Particle The most basic difference between a classical and a quantum particle is that there is a wave associated to a quantum particle. Different interpretations of quantum mechanics give different answers to the nature of this wave. For instance, as discussed in Sect. 2.2, the orthodox interpretation of quantum mechanics considers that |ψ(x,t)|2 is a probability density. Nevertheless, all interpretations of quantum mechanics coincide in that the wave associated with a free quantum particle is a solution of the simplest Schrödinger equation possible (V = 0): iℏ

ℏ2 ∂ 2 ∂ ψ(x, t) = − ψ(x, t). ∂t 2m ∂x2

(2.66)

Nevertheless, as will be shown below, the simplicity of Eq. (2.66) is only apparent. We’re looking for stationary solutions of Eq. (2.66) with energy E = K > 0, where K is the kinetic energy of the free particle: ψ(x) = ϕ(x)e−iwt , with w =

E . ℏ

(2.67)

We obtain the time independent Schrödinger equation for a free particle: −

ℏ2 d 2 ϕ(x) = Eϕ(x) = 0. 2m dx2

(2.68)

Thus: d2 ϕ(x) + k 2 ϕ(x) = 0, with k = dx2

√ 2mE . ℏ

(2.69)

Therefore: ϕ(x) = A± e±ikx .

(2.70)

Consequently, the stationary solutions of Eq. (2.66) are plane waves traveling from left to right (→) and plane waves traveling from right to left (←): ψ→ (x, t) = A+ ei(kx−wt) , ψ← (x, t) = A− e−i(kx+wt) .

(2.71)

Unfortunately, a plane wave cannot be the wave associated to a free quantum particle because a plane wave cannot be normalized since: ∫+∞ ∫+∞ ∗ 2 ϕ (x)ϕ(x)dx = |A± | dx = ∞. −∞

−∞

(2.72)

2.6 The Free Particle

23

Nevertheless, each plane wave in Eq. (2.71) could be used for describing not just a single quantum particle, but a stationary and homogeneous beam formed by many free quantum particles of mass m that are traveling with K = p2 /2m = E. This is because |A± |2 could be interpreted as the constant average density of particles (particles per unit length) at any point in such a beam. Consequently, Eq. (2.71) could be utilized for describing not just a single particle, but a stationary and homogeneous beam formed by many identical quantum particles. In such a beam, the average number of particles per unit length (average particle density) is: ρ± = |A± |2 .

(2.73)

These probability densities remain constant amid all possibilities. Note that there is a constant stream of particles (current) associated to each one of the plane waves given by Eq. (2.71): j→ = |A+ |2

√ p p , and j← = |A− |2 , with p = ℏk = 2mE. m m

(2.74)

Maintaining these currents requires a constant source of particles. As sketched in Fig. 2.2, and due to the linearity of the Schrödinger equation, we could look for a normalized solution of Eq. (2.66). This would be found for a convenient superposition of plane wave forming the wave-packet: 1 ψ(x, t) = √ 2π

∫+∞ ℏ2 k 2 E . φ(k)ei(kx−wt) dk, with w = , and E = ℏ 2m

(2.75)

−∞

In Eq. (2.75), the linear superposition of plane waves traveling in both directions is expressed as an integral because the values of k can take any real numbered value. The function φ(k) can be determined from the initial condition: 1 ψ(x, t = 0) = √ 2π

∫+∞ φ(k)eikx dk. −∞

Therefore, φ(k) is the Fourier transform of ψ(x, t = 0): Fig. 2.2 The wavefunction of a free quantum particle may be a spatially localized wave packet. However, a quantum particle in wave packet quantum state cannot have well-defined value of K and |p|

(2.76)

24

2 The Wave Equation

1 φ(k) = √ 2π

∫+∞ ψ(x, t = 0)e−ikx dx.

(2.77)

−∞

If φ(k) is narrowly peaked about some value k = k o , then the wave-packet looks like a classical particle because it is spatially localized (Fig. 2.2). Moreover, the group velocity of the spatially localized wave-packet is equal to the speed of a classical particle (vparticle ). This is a classical particle with approximately the same energy as the quantum particle associated to the wave-packet: vg ≈

I ( )I d w II d ℏk 2 II ℏko po = = = = vparticle . dk Ik=ko dk 2m Ik=ko m m

(2.78)

It should be noted that in contrast with a free classical particle, a free quantum particle in the quantum state corresponding to wave-packet cannot have a welldefined value of its linear momentum (p = èk) and kinetic energy (K = E = è2 k 2 /2m). This is because a wave-packet contains numerous plane waves with various values of k. Moreover, in contrast with solid classical particles, the shape of the wave-packet depends on time because different plane waves forming the wave-packet travel with different phase velocities: vph =

ℏ w = k. k 2m

(2.79)

2.7 Does the Schrödinger Equation Satisfy Expectations? The answer to this question should be an unquestionable and an undeniable yes. This is because as discussed in Sect. 2.5, solving the Schrödinger equation for a particle with mass m in an infinite one-dimensional well, allows for the explanation of why atoms are stable and why the atoms’ spectra are formed by a set of bright or dark lines. However, the answer to this question should also be an undoubted no. This is because the experimental values of the frequencies in the Hydrogen spectra are far from the theoretical values. Several things should be improved in the crude model discussed in Sect. 2.5: (a) The infinite well potential is not the real one, therefore, a more realistic potential should be introduced. (b) Real atoms are three dimensional. For that reason, the one-dimensional Schrödinger equation should be substituted by its tridimensional version. (c) The Schrödinger equation is not relativistic. This means that it only describes a particle moving slowly compared to the speed of the light in the vacuum.

Bibliography

25

As a result, it is necessary to substitute the Schrödinger equation with another relativistic wave equation. (d) Electrons have spin-(s = 1/2), but there is no reference to the particle’s spin in the Schrödinger equation (only m is included on it). This is because the Schrödinger equation only describes spinless particles with spin-(s = 0). On that account, it is necessary to substitute the Schrödinger equation with another wave equation that explicitly refers to the spin of the particle. In the rest of the course, each one of these necessary improvements will be studied. We will start these studies in the next Chapter, where we will learn the consequences for quantum mechanics produced by including the special theory of relativity.

Bibliography 1. 2. 3. 4. 5.

6. 7. 8.

D.J. Griffiths, Introduction to Quantum Mechanics (Prentice Hall, USA, 1995) D. Bohm, Quantum Theory, 11th edn. (Prentice -Hall, USA, 1964) A.S. Davydov, Quantum Mechanics (Pergamon Press, USA, 1965) Schrödinger equation (Wikipedia), https://en.wikipedia.org/wiki/Schr%C3%B6dinger_equ ation. Accessed 10 April 2023 Wavefunctions (Wikipedia), https://en.wikipedia.org/wiki/Wave_function#:~:text=A%20w ave%20function%20in%20quantum%20physics%20is%20a,on%20the%20system%20can% 20be%20derived%20from%20it. Accessed 10 April 2023 Interpretations (Wikipedia), https://en.wikipedia.org/wiki/Interpretations_of_quantum_mech anics. Accessed 10 April 2023 Pilot wave theory (Wikipedia), https://en.wikipedia.org/wiki/Pilot_wave_theory. Accessed 10 April 2023 Quantum mechanical operators (Wikipedia), https://en.wikibooks.org/wiki/Quantum_Mech anics/Operators_and_Commutators. Accessed 10 April 2023

Chapter 3

Introducing Relativity in Quantum Mechanics

Abstract In 1905, Albert Einstein discovered the special theory of relativity. In 1925, Schrödinger knew that any foundational law of physics should be Lorentz covariant. Schrödinger knew that the equation that is named after him today, was not Lorentz’s covariant, but Galileo’s invariant. For practical reasons, he chose to publish the equation that carries his name today. He chose to publish this instead of a Lorentz covariant equation that is known as the Klein-Gordon equation in modern times. The reason for Schrödinger’s decision was because of the evidence that the KleinGordon equation for a free particle has solutions with negative kinetic energy. On the contrary, the kinetic energy of classical particles is always positive. In this chapter, this book diverges from a traditional introduction to quantum mechanics. The first proposed ideas of relativistic quantum mechanics are introduced for a background sense of what this book wants to ultimately pursue. This includes the introduction of the Schrödinger-like PPGP equation and its use to solve simple one-dimensional problems.

Almost a century later, as it will be discussed below in Sects. 3.2 and 3.3, it is well known that the quantum states that are solutions of relativistic (Lorentz’s covariant) wave equations can be grouped into two branches. In the first branch, the total relativistic energy of the quantum particle is E T = E + mc2 . In the other branch, the total energy of the quantum particle is E T = E ' − mc2 . Here, we will refer to these states as “exotic” quantum states for reasons that will be clearly explained below. The apostrophe (' ) attached to the variable representing a magnitude will be used to exemplify the magnitude in an exotic quantum state. In addition, it is important to recognize that there is an antiparticle associated with each elemental particle in nature. The associated particle and the antiparticle have the same mass and charges of equal magnitude. On the contrary, the associated particle and antiparticle have charges with opposite signs. For instance, a positron is the antiparticle associated with the electron. As will be discussed below, there is a close relationship between the antiparticle states with E Ta = E a + mc2 and the particle states with E T = E' − mc2 .

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L. Grave de Peralta et al., Relativistic and Non-Relativistic Quantum Mechanics, Undergraduate Lecture Notes in Physics, https://doi.org/10.1007/978-3-031-37073-1_3

27

28

3 Introducing Relativity in Quantum Mechanics

3.1 The Special Theory of Relativity We use reference frames to communicate our observations with other people. If there were to be two observers at rest respect to an object, then observer A could indicate the position of the object using a three-dimensional vector rA = (x A , yA , zA ) originating at a reference point PA and ending at the object. However, observer B could indicate the position of the object using other three-dimensional vector rB = (x B , yB , zB ) originating at a reference point PB and ending at the object. In general, rA /= rB , meaning both observers will each state that the object is in different positions respect to the reference points at PA and PB . However, the observers would agree with the mass of this object. In the world of physics, matters are fundamentally more complicated than the trivial example given above. Nevertheless, for facilitating the communication between observers, we use frames of reference. In general, two observers usually do not agree with their individual observations referred to their different frames of references. However, they often agree on a reduced number of observations. In non-relativistic mechanics and in special theory of relativity, there are specific matters the observers should take on to understand the experiment to their fullest potential. The observers should refer to their observations about the movement of objects to inertial frames that move respectively to each other with constant velocity. For instance, in non-relativistic mechanics, two observers could agree on the time their observations were made. They can also agree about the mass and the acceleration of a moving object, but they could disagree on the position and velocity of the object. In special theory of relativity, contents are a bit more complicated. Events that are simultaneous for an observer could appear to occur at different times for other observers. Nevertheless, there are observations where all the observers agree. As it will be shown below, these observer-independent observations are the ones used for introducing the special theory of relativity in quantum mechanics. These are the most important results of the special theory of relativity in quantum mechanics: (a) The speed of the light in the vacuum (c) and the particle’s mass (m) are relativistic invariant. This means these are observer-independent magnitudes. (b) The magnitude of the linear momentum of a free classical particle (p) moving with velocity (v) is not a relativistic invariant measure and it is given by the following equation: 1 p = γ mv ≥ 0, with γ = / 1−

v2 c2

.

(3.1)

In Eq. (3.1), γ ≥ 1 is the Lorentz factor of special theory of relativity. (c) In the special theory of relativity there are some four-dimensional vectors which magnitudes (modules) are Lorentz (relativistic) invariant. Two notable instances are:

3.1 The Special Theory of Relativity

29

|(ct, i x, i y, i z)| =

√

c2 t 2 − x 2 − y 2 − z 2 = 0.

(3.2)

I( )I / I E T − V , i px c, i p y c, i pz c I = (E T − V )2 − p 2 c2 − p 2 c2 − p 2 c2 = mc2 . x y z (3.3) The energy-linear-momentum four-dimensional vector in Eq. (3.3) corresponds to a particle with mass m that is moving with total energy E T in the potential V. This four-dimensional vector is particularly important for quantum mechanics because the energy operator explicitly appears in the Schrödinger equation (Eq. 2.15). Classical Particles Classical particles have constant mass m > 0 and kinetic energy K > 0. Equation (3.3) can be rewritten as: (E T − V )2 = p 2 c2 + m 2 c4 , with p 2 = px2 + p 2y + pz2 .

(3.4)

√ E T − V = ± p 2 c2 + m 2 c4 .

(3.5)

Or:

For classical particles the + sign in Eq. (3.5) must be taken; therefore: ET =

√

p 2 c2 + m 2 c4 + V .

(3.6)

From Eq. (3.6), it follows when p = 0 and V = 0, then: E T = E m = mc2 .

(3.7)

There is then an energy (E m = mc2 ) associated with the mass of a classical particle. Consequently: E T = E + mc2 , with E = K + V .

(3.8)

Substituting E T with K + V + mc2 in Eq. (3.6), we obtain: K + mc = 2

√

/ p 2 c2

+

m 2 c4

= γ mc , with γ = 2

1+

p2 ≥ 1. m 2 c2

(3.9)

Note that Eq. (3.9) gives an alternative formula for the Lorentz factor. This is because substituting p given by Eq. (3.1) in Eq. (3.9), we obtain the original definition of the Lorentz factor:

30

3 Introducing Relativity in Quantum Mechanics

/ γ =

1 + γ2

v2 1 ⇒γ = / 2 c 1−

v2 c2

.

(3.10)

From Eq. (3.9), we obtain the kinetic energy of a classical particle is given by: K = (γ − 1)mc2 ≥ 0.

(3.11)

Substituting Eq. (3.11) in Eq. (3.8), we obtain the following equations of: E T − V = γ mc2 .

(3.12)

K = E − V = (γ − 1)mc2 .

(3.13)

Along with:

From Eq. (3.13), we obtain another very useful alternative formula for γ: γ =1+

E−V . mc2

(3.14)

Note that γ ≥ 1 in Eq. (3.14) because for a classical particle E = K + V ≥ V. Figure 3.1 depicts an instance corresponding to a classical particle with energy E moving in a parabolic potential. V (x = 0) = 0, thus at x = 0 all the energy of the particle is kinetic (E = K); however, at x = x min and x = x max all the particle’s energy is potential energy (K = 0). A classical particle can only be found in the classical region x min ≤ x ≤ x max . A classical particle cannot be in the inaccessible classical region where V > E. An unfamiliar but particularly useful alternative equation for K can be obtained from Eq. (3.9):

V(x) = x2 Classical Inaccessible classical region

region Inaccessible classical region

E

xmin

xmax

x

Fig. 3.1 Energy considerations for a classical particle moving in a quadratic potential

3.1 The Special Theory of Relativity

31

γ2 = 1 +

p2 . m 2 c2

(3.15)

Therefore: (γ + 1)(γ − 1)mc2 =

p2 . m

(3.16)

Thus: (γ − 1)mc2 = K =

p2 . (γ + 1)m

(3.17)

Note that for classical particles γ ≈ 1 in the non-relativistic limit; thus, K and p are given by the non-relativistic formulas: K =

p2 , with p = mv. 2m

(3.18)

As it would be discussed in Sect. 3.4, it is useful to introduce the concept of the effective mass of a classical particle as: μ=

E−V 1+γ m ≥ m, with γ = 1 + . 2 mc2

(3.19)

Thus: ) ( E−V m ≥ m. μ= 1+ 2mc2

(3.20)

Using Eqs. (3.19) and (3.20), Eq. (3.17) can be rewritten as: K =

p2 ≥ 0. 2μ

(3.21)

Quantum Particles In contrast to classical particles, a quantum particle with mass m and that is moving with total energy E T in a potential V, can be in the classical inaccessible region where E = K + V < V. This will be explained further in Sect. 4.2. Consequently, quantum particles can have negative kinetic energies (K < 0) in the inaccessible classical region. Moreover, relativistic quantum particles in exotic states can also have K ' < 0 in the classical region where E > V. In the classical region where E > V, such as classical particles, quantum particles can exist in quantum states where E T = E + mc2 and the particle has K > 0. For these states, all the formulas discussed above are valid in the classical region.

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3 Introducing Relativity in Quantum Mechanics

In addition to quantum states where the particle has K > 0 in the classical region, a relativistic quantum particle can be in other “exotic” quantum states where E T = E' − mc2 . In contrast with classical particles, quantum particles in these exotic states can have kinetic energy K ' < 0 in the classical region. For these exotic states, the formulas discussed above should be modified. The special theory of relativity was developed by Albert Einstein for classical particles. Therefore, we must be careful in extrapolating the valid results for classical particles to relativistic quantum particles in exotic states where E T = E' − mc2 . In Sect. 3.3 a brief discussion about the Hole Theory and the Dirac’s Sea is presented. Here it is enough to know that there exist particles and antiparticles in Mother Nature. An antiparticle is a particle that has the same mass as the associated particle but contains an opposite charge. For instance, the positron is the antiparticle of the electron. Although they have the same mass, the positron and electron electric charges are e and –e, respectively. According to the Hole Theory, the existence of a hole in Dirac’s Sea means there exists an unoccupied exotic quantum state of a particle moving in the potential V with total energy E T = E ' − mc2 . This hole is perceived as an antiparticle moving in the potential −V with total energy E Ta = E a + mc2 and E a = −E'. Consequently, due to the relation of E a = −E ' , we can start by obtaining the relativistic equations which are valid for the antiparticle in the states with E Ta = E a + mc2 . After they are found and known, the relativistic equations valid for the exotic quantum states of the particle can be deducted from them. If the particle is moving in the external potential V, then the associated antiparticle is a particle moving in the external potential −V. Therefore, for the quantum states of the antiparticle with E Ta = E a + mc2 , Eqs. (3.1)–(3.21) are valid after modifying V to −V. Therefore: (E T a + V )2 = p 2 c2 + m 2 c4 , with p 2 = px2 + p 2y + pz2 .

(3.22)

Or: E T a + V = E a + mc2 + V =

√

p 2 c2 + m 2 c4 .

(3.23)

Using the relation E' = −E a , we obtain: √ −E T a = −E a − mc2 = E ' − mc2 = E T' = − p 2 c2 + m 2 c4 + V .

(3.24)

Therefore, if the particle is in an exotic state with E ' T = E ' − mc2 , from Eqs. (3.23) and (3.24) follows when p = 0 and V = 0, then: E T' = −E T a = −E m = −mc2 .

(3.25)

Therefore, mc2 is the absolute minimum value of E T , if a free quantum particle is in a state where E T = E + mc2 (Eq. (3.7)). However, −mc2 is the absolute maximum value of E T , if a free quantum particle is in an exotic state where E T = E ' − mc2 .

3.1 The Special Theory of Relativity

33

From Eq. (3.23) follows that: E T a = E a + mc2 , with E a = K a − V .

(3.26)

) ( E T' = −E T a = − E a + mc2 = −K a + V − mc2 .

(3.27)

Thus:

We can rewrite Eq. (3.27) as: E T' = E ' − mc2 , with E ' = K ' + V = −E a , and K ' = −K a .

(3.28)

Therefore, E = K + V in a state where E T = E + mc2 (Eq. (3.8)). Also, E' = K ' + V in an exotic state where E T = E ' − mc2 . Substituting E Ta by K a − V + mc2 in (3.23), we obtain: K a + mc = 2

√

/ p 2 c2

+

m 2 c4

= γa mc , with γa = 2

1+

pa2 ≥ 1. m 2 c2

(3.29)

From Eqs. (3.28) and (3.29), we obtain: K a = (γa − 1)mc2 = −K ' .

(3.30)

Substituting K a given by Eq. (3.30) in Eq. (3.26), we obtain the following equation: E T a + V = γa mc2 .

(3.31)

E a + V = (γa − 1)mc2 .

(3.32)

And:

From Eq. (3.32), we obtain an alternative formula for γa : γa = 1 +

E a − (−V ) Ea + V =1+ . 2 mc mc2

(3.33)

Note that γa ≥ 1 in Eq. (3.33) when E a ≥ −V. Also, in general, γa /= γ. This is because γ is determined by V but γa is determined by −V. Instead of the substitutions above, if we substitute K' given by Eq. (3.30) in Eq. (3.27), we obtain the following equations: E T' − V = −γa mc2 = γ ' mc2 . And:

(3.34)

34

3 Introducing Relativity in Quantum Mechanics

( ) E ' − V = γ ' + 1 mc2 .

(3.35)

From Eq. (3.35), we obtain a formula for γ' : E' − V mc2

γ ' = −1 +

(3.36)

Note that in Eq. (3.34) we defined γ' = −γa as the Lorentz factor corresponding to a quantum particle in an exotic state where E T = E ' − mc2 . By substituting γ by γa in Eqs. (3.15)–(3.21), we obtain: p2 Ka = , with μa = 2μa

(

] ) [ E a − (−V ) γa + 1 m. m = 1+ 2 2mc2

(3.37)

Also, due to γ' = −γa , Eq. (3.15) is also valid for γ' , thus: ( ' )( ) p2 . γ − 1 γ ' + 1 mc2 = m

(3.38)

Nothing that Eq. (3.30) can be rewritten as: ( ) ( ) K ' = − −γ ' − 1 mc2 = γ ' + 1 mc2 .

(3.39)

From Eqs. (3.38) and (3.39), we obtain: K' =

p2 p2 = , with μ' = ' 2μ' (γ − 1)m

(

) ) ( E' − V γ' − 1 m. m = −1 + 2 mc2 (3.40)

We have then obtained similar kinetic energy equations, K = p2 /2μ (Eq. 3.21) and K = p2 /2μ' . These equations are valid for both kinds of quantum states. However, the effective masses μ (Eq. 3.20) and μ' are different. The equations relating μ y γ (Eq. 3.19) and μ' y γ' (Eq. 3.36) are also different. Also, due to the relation of E a = −E ' , if follows that −μ' = μa (Eq. 3.37). Finally, as will be studied later, quantum particles can tunnel through a potential barrier and be in spatial regions where a classical particle cannot be. As discussed in Sect. 1.2, the magnitude of the linear momentum of a quantum particle is p = ℏk. When a quantum particle is in a classically forbidden region where E < V, then k can take pure imaginary values; therefore, from Eq. (3.9) follows that: '

/ γ =

1−

| p|2 . m 2 c2

(3.41)

3.2 The Klein-Gordon Equation

35

Table. 3.1 Relevant relativistic formulas for quantum mechanics “Like-classical” quantum particles states

Exotic quantum particles states

E T = E + mc2 ⇒ E m = mc2

E T' = E ' − mc2 ⇒ E m = −mc2

γ = 1 + E−V mc2 ) [ ( γ +1 m = 1+ μ= 2

γ ' = −1 + Emc−V 2 [ ( ' ) γ −1 m = −1 + μ' = 2

'

] E−V m 2mc2

K = E − V = (γ − 1)mc2 =

p2 2μ

E ' −V 2mc2

] m

( ) K ' = E ' − V = γ ' + 1 mc2 =

p2 2μ'

This means that |γ| could be less than 1 in the classical forbidden region where E < V. In general, the potential depends on the position of the particle. Therefore, γ is a local function on the position of the particle: ] [ E − V (x) m. γ (x) = 1 + mc2

(3.42)

Consequently, μ, μ' , K, and K ' also depends on x. While K(x) > 0, and K ' (x) < 0 in the classical region where E > V; this is not true in a classical inaccessible region. In addition, when describing the interaction of a charged quantum particle (with charge e) with an external electromagnetic field, Eq. (3.13) should be modified in the following way: I( ( e ) ( e ) ( e ) )II I I E T − e Ao , −i px − A x c, i p y − A y c, i pz − A z c I c c c / )2 )2 ( ( ( e e )2 e = (E T − e Ao )2 − px − A x c2 − p y − A y c2 − pz − A z c2 = mc2 . c c c (3.43) In Eq. (3.43), Ao and A = (Ax , Ay , Az ) are the scalar and vector potentials of Electrodynamics, respectively. Besides Eqs. (3.3) and (3.43), the more relevant relativistic formulas for quantum mechanics are summarized in Table 3.1. As discussed above, the formulas for the exotic quantum particle states where E T = E ' − mc2 are obtained by defining γ' = − γa , μ' = (γ' − 1)/2 and changing V by −V and E a by −E ' in the relativistic formulas for the antiparticle in states where E Ta = E a + mc2 .

3.2 The Klein-Gordon Equation The Klein-Gordon equation is the correct Lorentz covariant equation for a spin-(s = 0) particle. In 1925, Schrödinger, Klein and Gordon discovered this equation. However, Schrödinger found this equation independently, but decided not to publish it. This equation is named after Klein and Gordon because they were the first to

36

3 Introducing Relativity in Quantum Mechanics

publish it. Schrödinger did not like the fact that this equation, in contrast to all previous knowledge about classical particles, has solutions where the kinetic energy of the quantum particle is negative. In modern days, we know this is a feature of all Lorentz covariant wave equations. This short introduction to quantum mechanics does not focus on the best-known Lorentz invariant wave equations. Nevertheless, we will discuss how the Klein-Gordon equation can be formally obtained using the first quantization procedure. In Sect. 2.3, we show how the Schrödinger equation can be obtained. The kinetic energy of a non-relativistic classical particle is K = p2 /2m. The corresponding operator is then given by Eq. (2.16). Therefore, the Schrödinger equation in operator notation is (Eq. 2.15): ( ) ℏ2 ∂ 2 ∂ ψ + V ψ. Hˆ ψ = Kˆ + Vˆ ψ ⇔ iℏ ψ = − ∂t 2m ∂ x 2

(3.44)

Consequently, the Schrödinger equation is not Lorentz covariant, but Galileo invariant. For obtaining the correct Lorentz covariant equation, we should start from a relativistic formula that is valid in all frames of reference traveling at constant velocity respect to each other. For a free quantum particle such a formula could be Eq. (3.4) with V = 0: E T2 = p 2 c2 + m 2 c4 .

(3.45)

We can then make the following formal substitutions in Eq. (3.45): ∂ ∂2 ∂2 ∂2 E T → Hˆ = iℏ , p 2 → pˆ 2 = −ℏ2 ∇ 2 , with ∇ 2 = ∇.∇ = 2 + 2 + 2 . ∂t ∂x ∂y ∂y (3.46) In Eq. (3.46), we have used the tri-dimensional version of Eq. (2.16) in cartesian coordinates: ) ( ∂ ∂ ∂ , , . (3.47) pˆ = −iℏ∇, with ∇ = ∂x ∂y ∂y After utilizing Eq. (3.46) for the first quantization substitution in Eq. (3.45), and after a few algebraic manipulations, the Klein–Gordon equation for a free particle is obtained: 1 ∂2 m 2 c2 ψK G = ∇ 2ψK G − 2 ψK G . 2 2 c ∂t ℏ

(3.48)

The more notable difference between the Schrödinger and the Klein-Gordon equations is in the order of the time derivatives. This is first-order for the Schrödinger equation but second-order for the Klein-Gordon equation. This determines that the

3.3 The Dirac’s Sea and the Hole Theory

37

Schrödinger equation only has solutions with K > 0, but the Klein-Gordon equation has solutions with both K > 0 and K’ < 0. The latter is related to the fact that Eq. (3.4) with V = 0 has two solutions.

3.3 The Dirac’s Sea and the Hole Theory The existence of solutions of the Klein-Gordon equation, where the kinetic energy of the quantum particle could have negative values, let Dirac propose a physical model that is qualitatively described in this Section. A quantitative justification of this model will be provided later in Chap. 4. The vacuum in relativistic quantum mechanics is not the same vacuum as in classical mechanics. Figure 3.2a sketched the Dirac’s Sea. The vacuum in this model of nature is formed by an infinitude of free quantum particles occupying the infinite exotic states with E ' = K ' = −E < 0. In the vacuum all the free particle quantum states with E = K > 0 are empty. Note that the values of energies E for a free particle are not quantized. There is no explanation in this model of why the particles occupying the exotic states with K ' = −E < 0 are not observable. In the modern quantum field theory, the Dirac’s Sea is substituted by a vacuum where quantum fluctuations continuously occur. Nevertheless, the model is very useful. For instance, as sketched in Fig. 3.2b, the Dirac’s Sea model can explain the creation of a paired particle-antiparticle if the vacuum absorbs a quantum of energy containing E q > 2mc2 . When this happens, a particle jumps from an occupied exotic state with K ' < 0 to a quantum state with K = −K ' > 0. The occupied quantum state with K > 0 is observed as a particle of mass m and kinetic energy K > 0. As sketched in Fig. 3.2c, the empty quantum state with K ' < 0 is observed as an antiparticle of mass m and kinetic energy K a = −K ' > 0. Note that the antiparticle is just another particle with the same mass, but with the opposite charge than the one that we originally called the particle. Due to the conservation of the linear momentum, if the linear momentum of the particle were p, the linear momentum of the antiparticle would be -p. In this model, particles and antiparticles are formed in pairs. Therefore, the number of particles in the universe should be equal to the number of antiparticles. The fact that we seem to live in a Universe where there are many more particles than antiparticles is an unsolved mystery. While in this model, particles and antiparticles are born in pairs, it is possible that in a local spatial region there is a single free particle or a single free antiparticle. These possibilities are sketched in Fig. 3.3a, c, respectively. In Fig. 3.3a all the exotic quantum states with K ' < 0 are occupied. Therefore, there is not an observable antiparticle. However, there is an occupied free quantum particle state with K > 0. Hence, this is perceived as an observable free particle. In contrast, in Fig. 3.3b, all the quantum states with K > 0 are empty. For that reason, there is not an observable particle. However, there is an unoccupied exotic free quantum particle state with K ' < 0. Consequently, as sketched in Fig. 3.3c, this is perceived as an observable free antiparticle with E a = −E ' = E.

38

3 Introducing Relativity in Quantum Mechanics Free antiparticle states

Free particle states

(a)

ET = E + mc2

(b)

ET = E + mc2

(c)

ETa = Ea + mc2

Empty particle states with K > 0

Antiparticle states with Ka > 0

+mc2

+mc2

+mc2

0

0

0

-mc2

-mc2

-mc2

Occupied particle states with Kꞌ < 0

Antiparticle states with Kaꞌ < 0

ETa = Eaꞌ - mc2

ET = Eꞌ- mc2

ET = Eꞌ- mc2

Fig. 3.2 a The vacuum in relativistic quantum mechanics is a Dirac’s Sea full of unobservable free particles with K ' < 0. Occupation of b particle’s states and c antiparticle’s states when a particle-antiparticle pair is created from the quantum vacuum Free antiparticle states

Free particle states

(a)

ET = E + mc2

(b)

ET = E + mc2

(c)

ETa = Ea + mc2 Antiparticle states with Ka > 0

Particle states with K > 0

+mc2

+mc2

+mc2

0

0

0

-mc2

-mc2

-mc2

Occupied free particle states with Kꞌ < 0

Occupied antiparticle states with Kaꞌ < 0

ET = Eꞌ- mc2

ET = Eꞌ- mc2

ETa = Eaꞌ - mc2

Fig. 3.3 a Occupation of free quantum states for a single observable particle. b A hole in the Dirac’s Sea otherwise full of particles with K ' < 0 is perceived as c a single observable antiparticle

The relation of E a = −E ' is always correct, but in general E a /= E because E ' /= −E. As discussed in Sect. 2.5, the quantum states of particles (and antiparticles) can be bound and spatially localized by an external potential. In general, as sketched in Fig. 3.4, an external potential introduces an asymmetry in the energy distribution of quantum states with energies E and E'. As an example, Fig. 3.4 corresponds to a particle moving in an attractive Coulomb potential produced by another particle with a charge of opposite sign. A charged particle moving in a Coulomb potential

3.3 The Dirac’s Sea and the Hole Theory

39

will be studied later in Sect. 5.2, but here it is solely used as an example. The external Coulomb potential changes all the free particle quantum states. As shown in Fig. 3.4a, the lower energy quantum states with K > 0 are transformed in bound states, but all the exotic states with E T = E ' − mc2 remains unbound. In Fig. 3.4a, all the exotic particle states with E T = E ' − mc2 are occupied, so this indicates there is not an antiparticle present. However, all the particle states with E T = E + mc2 are empty except for one. This indicates there is a single observable particle present. Consequently, Fig. 3.4a may correspond to a single electron moving in the attractive Coulomb potential produced by the proton in the Hydrogen atom. In contrast, in Fig. 3.4b, all the particle states with E T = E + mc2 are empty. This indicates there is no observable particle present. However, all the exotic particle states with E T = E ' − mc2 are occupied except one. Here, this indicates that there is a single observable antiparticle. The energy of the antiparticle is E a = −E ' . Therefore, as shown in Fig. 3.4c, the antiparticle is in a free particle state. The energy of the antiparticle, E a , is different than E which is the bound energy of the particle in Fig. 3.4a. This is because the antiparticle is moving in the potential created by the same external particle. Consequently, Fig. 3.4c may correspond to a single positron moving in the repulsive Coulomb potential produced by a proton. A matter world should be undistinguishable from an antimatter world. For instance, Fig. 3.4a may correspond to a single positron moving in the attractive Coulomb potential produced by the antiproton in an anti-Hydrogen atom. Also, Fig. 3.4c may correspond to a single electron moving in the repulsive Coulomb potential produced by an antiproton. Antiparticle states

Particle states

Bound particle states

ET = E +

(b) ET = E +

(c) ETa = Ea + mc2

mc2

+mc2

+mc2

0

0

-mc2

-mc2

Occupied unbound particle states

-mc2 ET = Eꞌ - mc2

ET = Eꞌ - mc2

Unbound antiparticle states

+mc2 0

(a)

mc2

Occupied bound antiparticle states

ETa = Eaꞌ - mc2

Fig. 3.4 Sketch of occupation of quantum states for a the electron in the Hydrogen atom, b a hole in the Dirac’s Sea in the presence of a Coulomb potential, and c a positron interacting with the nucleus of a Hydrogen atom (a proton)

40

3 Introducing Relativity in Quantum Mechanics

3.4 The Poveda-Poirier-Grave De Peralta Equation An interesting alternative to the first quantization procedure discussed in Sect. 3.2 consists of not starting it from Eq. (3.4), but from the following classical relativistic equation: ) ( E−V 1 2 m. (3.49) p , and μ = 1 + E T = K + V + mc , with K = 2μ 2mc2 2

Note that we have applied K and μ given by Eqs. (3.21) and (3.20), respectively. K > 0 and γ > 1 in the classical region where E > V. Making the formal substitutions given by Eq. (3.46), we obtain: ) ( ℏ2 ∂ Hˆ φ = Kˆ + Vˆ + mc2 φ ⇔ iℏ φ = − ∇ 2 φ + V φ + mc2 φ. ∂t 2μ

(3.50)

Equation (3.50) resembles a tridimensional version of the Schrödinger equation in cartesian coordinates. This similarity can be improved by introducing a new wavefunction: Ψ = φei

mc2 ℏ

t

.

(3.51)

Note that both wavefunctions represent the same probability density because |Ψ |2 = |φ|2 . By substituting Eq. (3.51) into Eq. (3.50), we obtain the Poveda-Poirier-Grave de Peralta (PPGP) equation: iℏ

) ( ℏ2 E−V ∂ m. Ψ = − ∇ 2 Ψ + V Ψ, with μ = 1 + ∂t 2μ 2mc2

(3.52)

The PPGP equation corresponds to a spin-(s = 0) quantum particle with mass m which is moving in a potential V. This is a relativistic Schrödinger-like equation that correctly describes the quantum states with E T = E + mc2 . Its one-dimensional version is: ) ( ℏ2 ∂ 2 E−V ∂ m. (3.53) Ψ + V Ψ, with μ = 1 + iℏ Ψ = − ∂t 2μ ∂ x 2 2mc2 The only difference between Eq. (3.53) and the one-dimensional Schrödinger equation is that μ is not the mass of the particle, but its effective mass which is given by Eq. (3.20). Accordingly, if we know how to solve the one-dimensional Schrödinger equation, then we should be able to solve the corresponding PPGP equation. By comparing the solutions of the PPGP and Schrödinger equations for the same problem, we could obtain how the introduction of the special theory of relativity modifies quantum mechanics.

3.5 The Relationship Between the PPGP and the Klein-Gordon Equations

41

As discussed in Sect. 3.2, if a quantum particle moves in a non-constant potential V, then μ is an operator depending on the position of the particle: ] [ E − V (x) m. μ(x) = 1 + 2mc2

(3.54)

Consequently, the order of the operators μ and the spatial derivatives should be the one indicated in Eqs. (3.52) and (3.53). It should not be the opposite because in general: {

[ ]} 1 ∂2 1 ∂2 Ψ /= 0. − μ(x) ∂ x 2 ∂ x 2 μ(x)

(3.55)

It is worth noting that Eq. (3.54) is valid whatever the position of the particle is in. Therefore, μ could be less than m or negative in the classical inaccessible region where E < V. Further discussion about this topic will be introduced and explained later.

3.5 The Relationship Between the PPGP and the Klein-Gordon Equations The time independent PPGP equation corresponding to Eq. (3.52) is: ) ( E−V ℏ2 2 m > m if E > V . (3.56) − ∇ ϕ + V ϕ = Eϕ, with μ = 1 + 2μ 2mc2 By pre-multiplying both sides of the Eq. (3.56) by μ/m, substituting μ with his value in Eq. (3.56) into the resulting equation, and after some algebraic manipulation, we obtain: [ ] ℏ2 2 (E − V )2 − ϕ = Eϕ. (3.57) ∇ +V − 2m 2mc2 In what follows, we will see that we can also arrive at Eq. (3.57), but only from starting from the Klein-Gordon equation. The Klein-Gordon equation, for a spin-(s = 0) particle moving in a time independent potential V, can be easily found using the formal first quantization procedure discussed in Sect. 3.2. Starting from Eq. (3.4) and making the substitutions given by Eq. (3.46), we obtain: )2 ( ∂ (E T − V )2 = p 2 c2 + m 2 c4 → iℏ − V ψ K G = −ℏ2 c2 ∇ 2 ψ K G + m 2 c4 ψ K G . ∂t (3.58)

42

3 Introducing Relativity in Quantum Mechanics

We can obtain the time independent Klein-Gordon equation corresponding to Eq. (3.58) by looking for stationary solutions of the form: ψ K G (r, t) = ϕ(r )e− ℏ E T t . i

(3.59)

Substituting Eq. (3.59) in Eq. (3.58) and considering that ∂V /∂t = 0, thus: ( ) ) ( ∂ ∂ iℏ − V ψ K G = (E T − V )ψ K G ⇒ iℏ − V [(E T − V )ψ K G ] ∂t ∂t = (E T − V )2 ψ K G .

(3.60)

Therefore, the time independent Klein–Gordon equation is: ] [ −ℏ2 c2 ∇ 2 ϕ = (E T − V )2 − m 2 c4 ϕ.

(3.61)

Then, substituting E T by E + mc2 , and after some algebraic manipulations, we obtain: ] [ 2 2 2 2 2 (E − V ) (3.62) + (E − V ) . −ℏ c ∇ ϕ = 2mc 2mc2 Then, after dividing by 2mc2 both sides of Eq. (3.62), we obtain: ] [ ℏ2 2 (E − V )2 ∇ ϕ= − + (E − V ) ϕ. 2m 2mc2

(3.63)

Evidently, Eq. (3.63) is equivalent to Eq. (3.57). We have then demonstrated that by solving either the Klein-Gordon or the PPGP equations, we could obtain the energies (E) and wavefunctions (ϕ) corresponding to a relativistic quantum particle containing a total energy E T = E + mc2 . Note that this demonstration requires substituting E T by E + mc2 in Eq. (3.61). A reader interested in these topics should also consult Annexes A and D. Exotic Quantum States In this short introduction to quantum mechanics, the attention is focused on quantum states where E T = E + mc2 . Nevertheless, we include extra material about the discussion of exotic quantum states introduced in Sect. 3.1. An important consequence of the Klein-Gordon equation is that a quantum particle can also be in exotic states Ω with energy E ' T = E ' − mc2 . For obtaining the quantum wave equation that Ω satisfies, we should modify Eq. (3.52) in the following way: ) ( ℏ2 2 E' − V ∂ ' m. iℏ Ω = − ' ∇ Ω + V Ω, with μ = −1 + ∂t 2μ 2mc2

(3.64)

3.5 The Relationship Between the PPGP and the Klein-Gordon Equations

43

The stationary solutions of Eq. (3.64) are in the form: '

Ω(r, t) = χ (r )e− ℏ E t . i

(3.65)

Therefore, the time independent equation corresponding to Eq. (3.64) is: −

ℏ2 2 ∇ χ + V χ = E 'χ . 2μ'

(3.66)

By pre-multiplying both sides of Eq. (3.66) by μ' /m, substituting μ' with his value in Eq. (3.64) into the resulting equation, and after some algebraic manipulation, we obtain: [ )2 ] ( ' E −V ℏ2 2 ∇ +V + (3.67) χ = E 'χ . 2m 2mc2 In what follows, we will prove that Eq. (3.67) can also be obtained starting from the time independent Klein-Gordon equation. Substituting E T by E ' − mc2 in Eq. (3.61) with ϕ substituted by χ, and after some algebraic manipulations, we obtain: [ ( −ℏ c ∇ χ = −2mc 2 2

2

2

E' − V − 2mc2

)2

(

'

+ E −V

] )

.

(3.68)

Then, after dividing by 2mc2 both sides of Eq. (3.68), we obtain the equation: [ ( ] )2 ( ' ) E' − V ℏ2 2 ∇ χ= − + E − V χ. 2m 2mc2

(3.69)

Transparently, Eq. (3.69) is equivalent to Eq. (3.66). We have then demonstrated that by solving either the Klein-Gordon equation or Eq. (3.64), we could obtain the energies (E' ) and wavefunctions (χ) corresponding to a relativistic quantum particle with E T = E ' − mc2 . Note that this demonstration requires substituting E T by E ' − mc2 in Eq. (3.61). It is worth noting here that by substituting E ' by −E a in Eq. (3.66), we obtain the following equation: ] [ ℏ2 2 E a − (−V ) m. − ∇ ϕa − V ϕa = E a ϕa , with μa = 1 + 2μa 2mc2

(3.70)

Equation (3.70) is the time independent PPGP equation that corresponds to the Schrödinger-like PPGP equation associated to a particle of mass m, that is moving in the potential −V. The time-dependent equation that corresponds to Eq. (3.70) is:

44

3 Introducing Relativity in Quantum Mechanics

iℏ

∂ ℏ2 2 Ψa = − ∇ Ψa − V Ψa . ∂t 2μa

(3.71)

We will discern in Sect. 4.3 that Eq. (3.71) is the PPGP equation corresponding to the antiparticle of the particle associated with Eq. (3.52). Equation (3.71) can be obtained from Eq. (3.64) by the so-called charge conjugation operation, meaning by taking the complex conjugate of both sides of Eq. (3.64): −iℏ

) ( ∂ ∗ ℏ2 E' − V m. Ω = − ' ∇ 2 Ω∗ + V Ω∗ , with μ' = −1 + ∂t 2μ 2mc2

(3.72)

Ω* in Eq. (3.72) depends on μ' and thus on E ' . However, after substituting E ' by −E a in μ' , we obtain: iℏ

[ ] ∂ ∗ ℏ2 2 ∗ E a − (−V ) ∇ Ω − V Ω∗ , with μa = −μ' = 1 + m. Ω =− ∂t 2μa 2mc2 (3.73)

Thus, Ω* in Eq. (3.73) depends on μa and thus on E a . A comparison between Eqs. (3.71) and (3.73) reveals validity that Ψ a = Ω*. This means the quantum states of the antiparticle with E Ta = E a + mc2 (Ψ a ) can be obtained from the exotic quantum states of the particle with E T = E ' − mc2 (Ω) and vice versa.

3.6 A Relativistic Particle Trapped in an Infinite One-Dimensional Well We are now prepared to find out what happens to a quantum particle when it is trapped in a small spatial region, and when the particle moves at speeds comparable to the speed of the light in the vacuum. Solving the one-dimensional PPGP equation for the infinite potential well provides a crude, but simple mathematical model for this riveting physical situation. Let us provisionally assume that relativistic quantum particles can be spatially confined inside of the infinite well, such as non-relativistic quantum particles. Therefore, as discussed above in Sect. 2.5, the spatial part and the energies of the corresponding stationary states are solutions of the following mathematical problem: ∂2 ϕ ∂x2

= −k 2 ϕ, i f 0 ≤ x ≤ L , with k = ϕ(x) ≡ 0, otherwise

√ 2μE . ℏ

(3.74)

Note that V = 0 inside of the well; therefore, the μ given by Eq. (3.56) is constant inside of the well. In the non-relativistic limit, E 0 are constants for a given value of E > 0. For that reason, the stationary solutions of the one-dimensional PPGP equation for the one-dimensional infinite well potential are given by the Eq. (2.56): / Ψn (x, t) =

(r el) 2 nπ i sin(kn x)e− ℏ En t , with kn = , and n = 1, 2, . . . (3.75) L L

For the relativistic quantum particle, the energies depend on k n and μ as E n in Eq. (2.54) depends on k n and m: E n(r el) =

ℏ2 π 2 2 ℏ2 kn2 = n . 2μL 2 2μL 2

(3.76)

Equations (3.56) and (3.76) form the following system of two equations with two variables: ) ( En ℏ2 π 2 n 2 (r el) m. (3.77) , μn = 1 + En = 2μn L 2 2mc2 Therefore: ) (μ ℏ2 π 2 n 2 ℏ2 π 2 n 2 n 2 2 − 1 mc = 2 ⇔ μ − mμ − = 0. n n 2μn L 2 m 4c2 L 2

(3.78)

Due to Eq. (3.56), we should require that μn ≥ m. With that being said, the solutions of Eq. (3.78) are: ⎞ ⎛ / ) ( / ( ) h 1 ℏ2 π 2 n 2 n 2 λC 2 ⎠ 1⎝ . μn = m, with λC = 1+ 1+ 2 2 2 m = 1+ 1+ 2 m c L 2 4 L mc (3.79) Therefore: / 2μn −1= γn = m

1+

( ) n 2 λC 2 > 1. 4 L

(3.80)

In Eqs. (3.79) and (3.80), λC is the Compton wavelength associated with a particle of mass m. So, if the particle is in the ground state (n = 1), then the non-relativistic limit occurs (γ ≈ 1 and μn ≈ m) when L >> λC . The particle travels faster (γ increases) when the spatial confinement increases; that is when the width of the well (L) decreases. Substituting Eq. (3.79) in Eq. (3.77), we obtain:

46

3 Introducing Relativity in Quantum Mechanics

ℏ2 π 2 n 2 ) E n(r el) = ( . / ( ) n 2 λC 2 m L2 1+ 1+ 4 L

(3.81)

In the non-relativistic limit (n small, L >> λC ), Eq. (3.81) coincides with Eq. (2.54). If L ≈ λC /2, then Eq. (3.81) reduces to: E n(r el) = (

ℏ2 π 2 n 2 ) . √ 1 + 1 + n2 m L 2

(3.82)

The ratio between the energies given by Eqs. (3.82) and 2.54) is: E n(r el) 2 ). =( √ En 1 + 1 + n2

(3.83)

Consequently, when the particle moves faster (n increases), the energy of the highly confined particle decreases in comparison with the non-relativistic energy value. In contrast, the spatial part of the stationary states (ϕ) does not depend on the speed of the particle. A more notable difference exists in the energy difference between consecutive energy levels (ΔE = E n+1 − E n ) at the non-relativistic and ultra-relativistic limits. At the non-relativistic limit: ΔE n =

] ℏ2 π 2 ℏ2 π 2 [ 2 2 + 1) = − n (n (2n + 1). 2m L 2 2m L 2

(3.84)

Therefore, ΔE increases as n increases at the non-relativistic limit. However, using Eq. (3.82) for estimating ΔE at the ultra-relativistic limit (n >> 1), we obtain: ΔE n =

ℏ2 π 2 ℏ2 π 2 . + 1) − n] = [(n m L2 m L2

(3.85)

Therefore, ΔE is constant at the ultra-relativistic limit. It is worth recalling that we initially began this Section assuming that, like non-relativistic quantum particles, relativistic quantum particles can be spatially confined inside of the infinite well. In Sects. 4.2 and 4.3 this assumption will be challenged furthermore.

3.7 A Beam of Relativistic Particles in a Constant Potential A quantum particle with mass m, that is traveling through of a constant potential V with energy E > V, moves with a constant kinetic energy K = E − V > 0. As a resulted factor, the wavefunction associated to the particle can be obtained from

3.7 A Beam of Relativistic Particles in a Constant Potential

47

solving the following PPGP equation: ) ( E−V ∂ ℏ2 d 2 m > 0. Ψ + V Ψ, with μ = 1 + iℏ Ψ = − ∂t 2μ d x 2 2mc2

(3.86)

Looking for stationary solutions of Eq. (3.86) of the form: Ψ(x, t) = ϕ(x)e−wt , with w =

E−V . ℏ

(3.87)

We obtain that ϕ satisfies the following time independent PPGP equation: −

ℏ2 d 2 ϕ = K ϕ, with K = E − V . 2μ d x 2

(3.88)

Formally, Eq. (3.86) is like the Schrödinger equation of a free particle moving with constant kinetic energy (K = E − V ). The effective mass of the relativistic particle is constant (μ = 1 + K/2mc2 ) for a given K. Therefore, the solution of Eq. (3.86) is given by Eq. (2.71): Ψ(x, t) = Aei(kx−wt) + Be−i(kx+wt) , with k =

√ p 2μ(E − V ) = . ℏ ℏ

(3.89)

Hence, a stationary beam of relativistic quantum particles that is traveling from left to right can be described by the plane wave: √ p 2μ(E − V ) = , Ψ(x, t) = Aei(kx−wt) , with k = ℏ ℏ ) ( E−V E−V m, and w = . μ= 1+ 2 2mc ℏ

(3.90)

On that account, μ ≈ m when E − V 1. mc2

/

E−V ) = ( m 2 1 + E−V 2mc2

/

E−V , with (1 + γ )m (3.91)

Firstly, note that in the non-relativistic limit, Eq. (3.91) coincides with Eq. (2.79), but with kinetic energy E − V. Second, the formula given the speed (v) of a classical particle with mass m, that is traveling with kinetic energy K = E − V >> mc2 , can be obtained in the following way:

48

3 Introducing Relativity in Quantum Mechanics

p2 γ 2 m 2 v2 E−V = = ⇒v= 2μ (1 + γ )m

/

E−V [ ] . 2 γ /(1 + γ ) m

(3.92)

In the non-relativistic limit, v given by Eq. (3.92) is different than vph given by Eq. (2.79), but with kinetic energy E-V; however, in the ultra-relativistic limit (γ >> 1, thus E − V ≈ γmc2 ): / v ph ≈

γ mc2 ≈ c. γm

(3.93)

Bibliography 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

W. Greiner, Relativistic quantum mechanics: wave equations (Spring-Verlag, New York, 1990) C. Christodeulides, The special theory of relativity (Springer, New York, 2016) R. Harris, Modern physics, 2nd edn. (Pearson Addison-Wesley, New York, 2008) J.D. Jackson, Classical electrodynamics, 2nd edn. (J. Wiley & Sons, New York, 1975) L. Grave de Peralta, K.C. Webb, H. Farooq, A pedagogical approach to relativity effects in quantum mechanics. Eur. J. Phys. 43, 045402 (2022) L. Grave de Peralta, L.A. Poveda, B. Poirier, Making relativistic quantum mechanics simple. Eur. J. Phys. 42, 055404 (2021) L. Grave de Peralta, Did Schrödinger have other options? Eur. J. Phys. 41, 065404 (2020) Klein Gordon equation (Wikipedia). https://en.wikipedia.org/wiki/Klein%E2%80%93G ordon_equation. Accessed 10 Apr 2023 Dirac’s Sea (Wikipedia). https://en.wikipedia.org/wiki/Dirac_sea. Accessed 10 Apr 2023 Hole Theory (Wikipedia). https://en.wikipedia.org/wiki/Dirac_equation. Accessed 10 Apr 2023 PPGP equations and antiparticles (YouTube). https://www.youtube.com/@luisgravedeperalta. Accessed 10 Apr 2023

Chapter 4

Other One-Dimensional Problems

Abstract In this Chapter, we continue to solve unrealistic one-dimensional problems, but we improve our models of submicroscopic phenomena by exploring the use of more realistic potentials. This chapter contains a novel and ambitious study of the consequences of special theory of relativity for quantum mechanics. This includes the study of the relativistic harmonic oscillator, and a pedagogical presentation of Klein’s paradox based on solving the Schrödinger-like PPGP equation for a step potential.

4.1 The Harmonic Oscillator The classical non-relativistic harmonic oscillator is a system formed by a mass m attached to a spring of elastic constant k e . The potential energy of the system is: 1 V (x) = mw 2 x 2 , with w = 2

'

ke . m

(4.1)

In Eq. (4.1), w is the angular frequency of the system oscillations. In a classical harmonic oscillator, the total energy of the system of E = K + V, can have any positive value and K ≥ 0. The speed of the mass is zero at the points of maximum deformation of the spring and it is at its maximum at x = 0; therefore, the mass is near the points of maximum deformation of the spring most of the time. Let us find out which of these properties are conserved in a quantum harmonic oscillator. The quantum harmonic oscillator is described at any speed of the quantum particle with mass m by the one-dimensional PPGP equation (Eq. 3.53) with V given by Eq. (4.1). At the non-relativistic limit, the spatial part of the stationary solutions of the quantum harmonic oscillator can be obtained by solving the following time independent Schrödinger equation: −

1 ℏ2 d 2 ϕ + mw 2 x 2 ϕ = Eϕ. 2 2m d x 2

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L. Grave de Peralta et al., Relativistic and Non-Relativistic Quantum Mechanics, Undergraduate Lecture Notes in Physics, https://doi.org/10.1007/978-3-031-37073-1_4

(4.2)

49

50

4 Other One-Dimensional Problems

Equation (4.2) can be solved in different ways. One of them is the algebraic method that is based on the use of the following ladder operators: ( ) ℏ d 1 ± imwx . a± = √ 2m i d x Δ

(4.3)

Using the ladder operators, Eq. (4.3) can be rewritten in the following alternative ways: ) ( ) ( 1 1 a − a + − ℏw ϕ = Eϕ, and a + a − + ℏw ϕ = Eϕ. 2 2 Δ

Δ

Δ

Δ

(4.4)

Note that the order of applications of the ladder operators is important. This is because the ladder operators do not commute, like most operators in quantum mechanics: ] [ aˆ − , aˆ + = aˆ − aˆ + − aˆ + aˆ − = ℏw /= 0 ⇒ aˆ − aˆ + /= aˆ + aˆ − .

(4.5)

The ladder operators are useful for solving Eq. (4.2) because if ϕ satisfies Eq. (4.2) with energy E, then the wavefunction â± ϕ also satisfies Eq. (4.2), but with energy E ± èw. This means that if we find a solution from Eq. (4.2), then we can obtain other solutions by applying the ladder operators to the known solution successively. Therefore, by applying the lowering operator â- to a known solution successively, we will ultimately obtain the ground state ϕo containing the minimum value of E = E o . If we then apply the lowering operator to ϕo one more time, we obtain: ( ) ℏ d 1 − imwx ϕo = 0. a − ϕo = √ 2m i d x Δ

(4.6)

Before solving Eq. (4.6), we can obtain the energy of the ground state by utilizing Eqs. (4.4) and (4.6) in the following manner: 1 1 1 a + a − ϕo + ℏwϕo = ℏwϕo = E o ϕ ⇒ ℏw = E o . 2 2 2 Δ

Δ

(4.7)

From Eq. (4.6) follows: d mw dϕo mw = ϕo = xϕo ⇒ xd x. dx ℏ ϕo ℏ

(4.8)

By integrating both sides of Eq. (4.8), we obtain: ϕo (x) = Ao e− 2ℏ x . mw

As a result, the solutions of Eq. (4.2) are:

2

(4.9)

4.1 The Harmonic Oscillator Fig. 4.1 Energies (in ℏw units) and probability densities corresponding to the first three quantum states (Eq. 4.10) of a particle in a harmonic oscillator

51

|φn (x)|2

En /ℏw kx2/2

0

n=2

0

n=1

0

n=0 x

( ) 2 ( )n − mw 1 2ℏ x ϕn (x) = An a + e , and E n = n + ℏw, with n = 0, 1, 2, . . . 2 Δ

(4.10)

In Fig. 4.1, the probability densities (|ϕn |2 ) are plotted corresponding to the ground state (n = 0) and the first excited states of the quantum harmonic oscillator. A comparison between Figs. 2.5, 3.1, and 4.1 depict that quantum particles can penetrate through the inaccessible classical regions of the harmonic potential where E n < V (x). Nevertheless, there is an evident similitude between Figs. 2.5 and 4.1 Such as the infinite well potential, the quadratic potential spatially localizes the particle’s wavefunction around the classical region. This results in a discrete and numerable set of energy values. Eventually, E n > mc2 when n >> 1; therefore, we should expect that at high enough energies, the results acquired from solving the non-relativistic Schrödinger equation should not be correct. In order to discover how the non-relativistic results changes when the particle moves at speeds close to the speed of the light in the vacuum, we should solve the following time independent PPGP equation: [ ( )] 1 1 1 ℏ2 d 2 2 2 2 2 ϕ + mw x ϕ = Eϕ, with μ(x) = 1 + E − mw x m. − 2μ(x) d x 2 2 2mc2 2 (4.11) Writing Eq. (4.11) as Eq. (3.57), we obtain that Eq. (4.11) is equivalent to the following equation: )2 ] ( E − 21 mw 2 x 2 ℏ2 2 1 2 2 ∇ + mw x − ϕ = Eϕ. − 2m 2 2mc2

[

(4.12)

Or: [

] ℏ2 2 ∇ + Ve f f (x) ϕ = E e f f ϕ − 2m

(4.13)

52

4 Other One-Dimensional Problems

In Eq. (4.13): Ee f f

( = E 1+

) ) ( mw 4 4 E 1 E 2 2 , and V w x − x . (4.14) = m 1 + (x) e f f 2mc2 2 mc2 8c2

Equation (4.13) looks like a time independent Schrödinger equation with a nonharmonic effective potential. Therefore, we should expect that the solutions of Eq. (4.13) should notably differ from the solution of Eq. (4.2). In Sect. 4.3, we will discuss the nature of the solutions of Eq. (4.13). Here, we note that in the nonrelativistic limit Eq. (4.13) does not reduce to Eq. (4.2) because V eff /= ½mw2 x 2 when E 0) decreases faster when V − E increases. However, if E + mc2 < V < E + 2mc2 , the opposite occurs where N(x > 0) decreases slower when V − E increases. We will discuss the physical meaning of this unintuitive result later in this Section. From the continuity of Ψ and ∂Ψ/∂x at x = 0 follows: B = A + C, and

iB√ (A − C) √ 2μ1 E. 2μ2 (V − E) = h h

Therefore: / / ( ) ( ) μ2 (V − E) μ2 (V − E) B B A= 1+i , and C = 1−i . 2 μ1 E 2 μ1 E

(4.44)

(4.45)

The reflection coefficient is then: R=

|C|2 = 1. |A|2

(4.46)

Therefore, if E < V < E + 2mc2 , a beam of quantum particles is totally reflected from a sharp change in potential. The penetration depth of the quantum particles existing in the inaccessible classical region (x > 0, E < V ) depends on V − E. If V − E < mc2 , the penetration length decreases when V − E increases. Evidently, these

4.2 A Beam of Particles Hitting a Sharp Potential Step

59

results are also valid in the non-relativistic limit that occurs when E E + 2mc2 If V > E > 0 and the potential step is very large, that is if V > E + 2mc2 , then μ2 < 0 in Eq. (4.30). In this case, Eq. (4.30) can be rewritten in the following way: ) ( E−V ℏ2 d 2 m < 0. ϕt = (E − V )ϕt , with μ2 = 1 + − 2μ2 d x 2 2mc2

(4.47)

Clearly, quantum particles associated to Eq. (4.47) are not like classical particles because both μ2 and K = E − V are negative for x ≥ 0. Equation (4.47) can be rewritten as: ) ( d2 ℏ2 E−V m > 0. (4.48) ϕt = (V − E)ϕt , with (−μ2 ) = − 1 + − 2(−μ2 ) d x 2 2mc2 The solution of Eq. (4.48) is then given by Eq. (4.31), but with k 2 changed by k 22 : Ψt (x, t) = Be

i(k22 x−wt)

, with k22

p2 = = ℏ

√

2(−μ2 )(V − E) > 0. ℏ

(4.49)

Note that k 22 and k 2 in Eqs. (4.31) and (4.49) can both be considered as depending on the variable ξ = E − V > 0. In particular, if V ≥ E + 4mc2 : ( ) k22 −ξ − 4mc2 = k2 (ξ ).

(4.50)

Consequently, if V ≥ E + 4mc2 , Eq. (4.49) can be rewritten as: Ψt (x, t) = Bei(k2 x−wt) .

(4.51)

As sketched in Fig. 4.6 for V >> E + 4mc2 , Eq. (4.51) also corresponds to a common plane wave traveling from left to right, which is in the same direction of p = +ℏk 2 . More importantly, if V > E + 2mc2 , the average particle density in the transmitted beam is the same everywhere: ρ(x > 0) = ρ(x = 0) = |B|2 .

(4.52)

Correspondingly, we could repeat what was done before for the case E > V and obtain, in the limit V − E >> 4mc2 , that the transmission coefficient is given by Eq. (4.36). Ultimately, we obtain: T ≈ 1, and R ≈ 0.

(4.53)

60

4 Other One-Dimensional Problems

V(x) Isolator

V >> E + 4mc2

E>0

- +

Incident beam of electrons

Transmitted beam of electrons

x

Faraday cage with an excess of electrons

Faraday cage at V = 0

x=0 Fig. 4.6 Klein’s paradox

As discussed above, Eq. (4.46) indicates that a beam of quantum particles cannot pass through an infinitely broad potential barrier if E < V and V − E < 2mc2 . However, Eq. (4.53) indicates that the beam passes through the barrier without reflection if V − E >> 4mc2 . This unintuitive result is known as Klein’s paradox. It is purely a certain relativistic quantum mechanics’ result. Note that this unintuitive result is valid even when the incident beam is non-relativistic (E > 4mc2 the relativistic quantum particle certainly leaves the well. Consequently, the results obtained in Sect. 3.6 need further consideration. It is worth noting that the width of the potential step is infinite; therefore, we should expect that quantum particles should be capable of passing through a finite potential barrier. This is called tunneling and this occurs when quantum particles are moving fast or slowly. Quantum particles have the capacity to be in spatial regions that are prohibited for classical particles. We can now refer to Sect. 4.1. Figure 4.7 shows the effective potential (V eff ) for a relativistic harmonic oscillator given by Eqs. (4.13) and (4.14). Quantum particles could pass from the classical region around x = 0, to the classical regions at x →± ∞, by tunneling through the inaccessible regions associated with V eff . Strictly speaking, there are no stationary states for a

4.3 Particles, Antiparticles, and Exotic Quantum States with Negative …

61

Veff(x) = V - (E-V)2/2mc2 V (x) = kx2/2 E = mc2

x Fig. 4.7 Effective potential of a relativistic harmonic oscillator. If k = 1 and E = mc2 , the nonrelativistic harmonic oscillator (discontinuous line) has a stationary state, but the relativistic one (continuous line) has a resonance because the quantum particle can tunnel through a potential barrier with a finite width

relativistic quantum oscillator. For small particle energies, there could be resonances with the energies given by Eq. (4.18); however, the particle is not bound in the ultrarelativistic limit. Something similar occurs for a relativistic quantum particle in an infinite one-dimensional well. The energies given by Eq. (3.81) do not correspond to stationary states, but to resonances because relativistic quantum particles cannot be spatially localized inside an infinite well. Nevertheless, we should be aware that in this Section, we have only considered the PPGP equation that corresponds to the states with E T = E + mc2 . However, the introduction of the special theory of relativity in quantum mechanics produces fundamental consequences. One of them is that there is not a wave, but there are two waves associated with a relativistic quantum particle. In contrast with nonrelativistic quantum particles, relativistic quantum particles could exist in two kinds of quantum states. The total energy of the particle is E T = E + mc2 in the states studied in this Section, but it is E T = E ' − mc2 in the exotic states. Another fundamental consequence is the existence of antiparticles. These two topics are discussed from different perspectives in Annexes A, C, D, and E. For these reasons, a complete description of the Klein’s paradox requires to also consider the complementary PPGP equation (Eq. 3.64) that corresponds to the exotic states with E T = E ' − mc2 . For simplicity, this will be completed in the next Section for the limit V >> E + 4mc2 .

4.3 Particles, Antiparticles, and Exotic Quantum States with Negative Kinetic Energy Elemental particles come in pairs. For example, the electron and the positron have the same mass m and magnitude of their charges |q| = e. However, the sign of their charge is opposite to each other where electrons have a charge −e and positrons have a charge +e. The proton is not an elemental particle, but the antiproton exists. Both have the same mass mp but contain opposite charges. The proton has a charge of +e and the

62

4 Other One-Dimensional Problems

antiproton a charge of −e. For simplicity, let us imagine an only-matter world where electrons and protons only exist, and an only-antimatter world where positron and antiprotons only exist. Hypothetically, in the only-matter world can exist Hydrogen atoms formed by a proton and an electron. In the hypothetical only-antimatter world, can exist anti-Hydrogen atoms formed by an antiproton and a positron. We also could hypothesize an inclusive world formed by electrons, positrons, protons, antiprotons, Hydrogen atoms and anti-Hydrogen atoms. The current laws of Physics predict a perfectly inclusive world, but nobody truly knows why or how we live in an almost only matter world. For simplicity, let us assume that the finite step potential studied in the previous Section is produced by the sum of the Coulomb potentials associated to a large arrangement of electrons; this is the external world producing the potential step in which a beam of electrons is moving. In this hypothetical only-matter world, the potential V > 0 because at x = 0, the external electrons producing the potential instantaneously pull backward the electrons form the incoming beam. This is the problem that we solved in the previous Section. An idealized physical situation realizing this problem for the limit case V >> E + 4mc2 is sketched in Fig. 4.6. The wavefunction associated to the beam of electrons can be found by solving the time independent PPGP equation (Eq. 3.56) for a particle (electron) moving in the external potential V produced by the distribution of electrons in the Faraday’s cages: ) ( ℏ2 d 2 E−V m. − ϕ + V ϕ = Eϕ, with μ = 1 + 2μ d x 2 2mc2

(4.54)

In general, the incident plane wave associated to the input beam of free particles is partially reflected and partially transmitted. However, as discussed in the previous Section, there is not a reflected wave if V >> E + 4mc2 . The frequency of the waves (w = E/ℏ) are equal in both Faraday’s cages, but their wavenumbers k i /= k t . As sketched in Fig. 4.8, the current laws of Physics predict and deem it to be impossible to distinguish an only-matter world from an only-antimatter world. Therefore, if we substitute all the electrons in the beam and in the Faraday’s cages by positrons, the problem that we must solve is the one solved in the last Section. In this onlyantimatter world, the potential V > 0 because at x = 0, the external positrons producing the potential instantaneously pull backward the positrons forming the incoming beam. The time independent PPGP equation, that corresponds to a beam of antiparticles (positrons) moving in the external potential V produced by the distribution of positrons in the hypothetical Faraday’s cages, is also Eq. (4.54). It is worth noting that in the physical situations sketched in Figs. 4.6 and 4.8, we not only changed the beam of electrons to a beam of positrons, but we changed the external matter world surrounding the electron beam to an external antimatter world surrounding the beam of positrons. In contrast with the pure matter or antimatter worlds, we could have a beam of positrons moving in a step potential produced by the sum of the Coulomb potentials associated to a large arrangement of electrons in the world that we live in. This is as sketched in Fig. 4.9. In this inclusive world, the potential V < 0 because at x =

4.3 Particles, Antiparticles, and Exotic Quantum States with Negative …

63

V(x) Isolator

V >> E + 4mc2

E>0

+-

Incident beam of positrons

x

Transmitted beam of positrons

Faraday cage with an excess of positrons

Faraday cage at V = 0

x=0 Fig. 4.8 Klein’s paradox in antimatter world

0, the external electrons producing the potential instantaneously pushed forward the positrons forming the incoming beam. Consequently, the time independent PPGP equation to be resolved in this case is not Eq. (4.54), but Eq. (3.70): −

] [ E a − (−V ) ℏ2 d 2 m. ϕ − V ϕ = E ϕ , with μ = 1 + a a a a a 2μa d x 2 2mc2

(4.55)

The only difference between the physical situations sketched in Figs. 4.6 and 4.9 is the change of the electron beam to the positron beam. The external distribution

V(x) Faraday cage with an excess of electrons Incident beam of positrons

Ea > 0

Transmitted beam of positrons

-+ x

E'=-Ea

-V -V > −V is totally transmitted

64

4 Other One-Dimensional Problems

of electrons responsible for the potential remains the same. This is the sole reason why Eq. (4.55) is different than Eq. (4.54). However, if the external distribution of electrons were substituted by the external distribution of positrons sketched in Fig. 4.8, then Eq. (4.54) would be the PPGP equation that corresponds to the beam of positrons. Note that there is not a reflected beam in Fig. 4.9 because it corresponds to the limit E >> V (Eq. 4.37) in the case E > V discussed in the previous Section. As discussed in Sect. 3.5, there is a relationship (ϕ a = χ*) between the solutions of Eq. (4.55) and the solutions of Eq. (3.66): ) ( ℏ2 d 2 E' − V ' ' m, and E ' = −E a . − ' 2 χi + V χi = E χi , with μ = −1 + 2μ d x 2mc2 (4.56) If the physical situation is the one illustrated in Fig. 4.6; then Eq. (4.56) is the complementary PPGP equation of Eq. (4.54) for the particles (electrons) in the input beam. The exotic states χ i are states of the electrons in the incident beam that complement the states ϕ i given by Eq. (4.28). Region X < 0 By solving the same wave equation (Eq. 4.26 for x < 0), we obtain the quantum states (ϕi = ϕia ) with K = E > 0 that corresponds to both, a beam formed by free electrons in an only-matter world (ϕi , Fig. 4.6) and a beam formed by free positrons in an only-antimatter world (ϕia , Fig. 4.8): ( ) ℏ2 d 2 E − ϕi = Eϕi , with μ1 = 1 + m > m, and E > 0. 2μ1 d x 2 2mc2

(4.57)

The solutions of Eq. (4.57) corresponding to the input beam are given by Eq. (4.29). They are common plane waves traveling from left to right in the same direction of p = èk 1 : √ Ψi (x, t) = Ψia (x, t) = Ae

i(k1 x−wt)

, with k1 =

2μ1 E > 0. ℏ

(4.58)

Like classical particles, the electrons or positrons in the incident beam have μ1 > m, and K = E > 0 when they are in quantum states that are solutions of Eq. (4.57). As discussed in Sects. 3.1 and 3.3 and as sketched in Fig. 3.3, due to Eq. (3.28) the free particle’s energy (E) in the exotic quantum states is E ' = −E a = −E < 0. By solving the same wave equation (Eq. (4.56) for x < 0), we obtain the exotic quantum states (χi = χia ) with μ’1 < -m, E ' = −E < 0 and K ' = −K < 0 corresponding to both, a beam formed by free electrons in an only-matter world (χi , Fig. 4.6) and a beam formed by free positrons in an only-antimatter world (χia , Fig. 4.8): −

ℏ2 d 2 χi = E ' χi , with μ'1 = −μ1 < −m, and E ' = −E. 2μ'1 d x 2

(4.59)

4.3 Particles, Antiparticles, and Exotic Quantum States with Negative …

65

After substituting E' by −E on μ' 1 (Eq. 4.56). we obtain: ( )2 ∂2 χi + k1' χ i = 0, con k1' = 2 ∂x

√

2(−μ1 )(−E) = k1 = ℏ

√ 2μ1 E > 0. (4.60) ℏ

Therefore, as discussed in Sect. 3.5, the solutions of Eq. (4.60) that correspond to the exotic states of the input beam, are given by the complex conjugate of Eq. (4.58). They are exotic plane waves traveling from left to right, where they travel in the opposite direction of p = −èk 1 : Ωi (x, t) = Ωia (x, t) = A∗ e−i(k1 x−wt) .

(4.61)

In general, as discussed in the previous Section, there is a plane wave reflected by the potential discontinuity at x = 0 in addition to the incident plane wave. The reflected plane waves associated to the particle and antiparticle states in the exclusive worlds, with E T = E + mc2 and E Ta = E a + mc2 , are given by Eq. (4.32): Ψr (x, t) = Ψra (x, t) = Ce−i (k1 x+wt) .

(4.62)

Nevertheless, Ω r = (Ψ ra )* and Ω r = Ω ra . With that being said, the reflected plane waves associated to the particle and antiparticle exotic states in the exclusive ' worlds, with E T = E' − mc2 and E Ta = E a − mc2 , are: Ωr (x, t) = Ωra (x, t) = C ∗ ei(k1 x+wt) .

(4.63)

These are also exotic plane waves traveling from right to left; that is in the opposite direction of p = èk 1 . Equations (4.57)–(4.59) are valid for both, an input beam of free electrons in an only-matter world and an input beam of free positrons in an onlymatter world. As discussed in the previous Section, if V >> E + 4mc2 , then there is not reflection due to the sharp discontinuity of the potential at x = 0. Therefore C = C* = 0 in Eqs. (4.62) and (4.63). This means if V >> E + 4mc2 , then in the region x < 0, only the incident exotic plane waves need to be considered (Eq. 4.61). Interestingly, if the physical situation is the one illustrated in Fig. 4.6; then like Ψ i , Ω i * corresponds to a plane wave that travels from left to right in the direction p = èk 1 : Ωi∗ (x, t) = Ψia (x, t) = Aei(k1 x−wt) .

(4.64)

This is because Ω i * = Ψ ai corresponds to a quantum state where μ1 > m, E > 0, and K = E > 0, which is associated to the antiparticle that has the same mass (m) as the particles forming the input beam. Ψ ai is the wavefunction that corresponds to a beam of positrons with p = èk 1 when the positrons are in a quantum state where μ1 > m, E > 0, and K = E > 0. Note that: I ∗I IΩ I = |Ψai |. i

(4.65)

66

4 Other One-Dimensional Problems

Therefore, the density of positrons in the input beam of ρa = |Ψ ai |2 , is equal to the density of vacated exotic electronic states Ω i (|Ω i |2 = |Ω i *|2 ). In the physical situation sketched in Fig. 4.6, there is an observable input beam of electrons. There are no observable positrons in this case. This is because the electron quantum state (Ψ i ) given by Eq. (4.58), and all the exotic electronic states (Ω i ) given by Eq. (4.61), are occupied by electrons. In contrast, in the physical situations sketched in Figs. 4.8 and 4.9, there is an observable input bean of positrons. There are no observable electrons in this case. This is because all electron quantum states (Ψ i ) given by Eq. (4.58) and the exotic electron state (Ω i ) given by Eq. (4.61) are empty. An empty exotic electron state (Ω i ) given by Eq. (4.61) determines there is a hole in Dirac’s Sea. This hole is observed as a positron in the quantum state (Ψ ai ) given by Eq. (4.58). Region X ≥ 0 In what follows, we will assume the incident beam is a beam of electrons and the physical situation is the one sketched in Fig. 4.6 (E > 0, V >> E + 4mc2 ). As discussed in the previous Section, the wavefunction Ψ t corresponding to the transmitted beam of electrons is given by Eq. (4.51): Ψt (x, t) = Bei (k2 x−wt) , with k2 > 0 and B = A.

(4.66)

These are common plane waves that travel from right to left in the direction p = èk 2 . We are interested in finding the exotic electronic states, Ω t with E T = E' − mc2 , that complement the electronic states Ψ t with E T = E + mc2 . For finding Ω t , we could start by noting that the continuity of Ω at x = 0 requires that E' = −E and w must be the same everywhere. Like Ω i (Eq. 4.61), the wavefunction associated to the transmitted beam of quantum particles with E T = E' − mc2 should be an exotic plane wave traveling from left to right: Ωt (x, t) = χt (x)e+iwt .

(4.67)

With χt satisfying the following time independent equation: −

) ( ℏ2 d 2 E' − V ' ' m, and E ' = −E. χ + V χ = E χ , with μ = −1 + t t t 2 2μ'2 d x 2 2mc2 (4.68)

Due to Eq. (3.28), E' = −E a = −E, as discussed in Sects. 3.1 and 3.3 and as sketched in Fig. 3.4. Therefore, by conveniently substituting E' by −E a in Eq. (4.68), we obtain: −

ℏ2 d 2 χt − V χt = E a χt , with μ2a = −μ'2 2μ 2a d x 2 ] [ E a − (−V ) m, and E a = E. = 1+ 2mc2

(4.69)

4.3 Particles, Antiparticles, and Exotic Quantum States with Negative …

67

The complex conjugate of Eq. (4.69) is: ] [ E+V ℏ2 d 2 ∗ ∗ ∗ m > m. χ − V χt = Eχt , with μ2a = 1 + − 2μ 2a d x 2 t 2mc2

(4.70)

As discussed in Sect. 3.5, ϕ ta = χt * is the spatial part of the antiparticle (positron) quantum state with E Ta = E a + mc2 and E a = E. Therefore, Eq. (4.70) is equivalent to: −

ℏ2 d 2 ϕta − V ϕta = Eϕta . 2μ 2a d x 2

(4.71)

Equation (4.71) is the time independent PPGP equation for an antiparticle moving in the potential (−V ) with total energy E Ta = E a + mc2 and E a = E. This is the physical situation sketched in Fig. 4.9 at x ≥ 0 where E a = E >> −(V + 4mc2 ). Thus: √ d2 2μ2a (E + V ) 2 > 0. (4.72) ϕ + k ϕ = 0, with k = ta 2a 2a ta dx2 ℏ Therefore, Ψ ta is the following common plane wave that travels from left to right in the same direction of p = èk 2a : Ψta (x, t) = Ba ei(k2a x−wt) .

(4.73)

As discussed in Sect. 3.5, Ω t can be obtained from Ψ ta as: Ωt (x, t) = (Ψta )∗ = Ba∗ e−i(k2a x−wt) .

(4.74)

Note that the same result can be obtained by directly solving Eq. (4.68). From Eqs. (4.61) and (4.74), and the continuity of Ω t at x = 0, follows that Ba * = A*. So, it follows as: Ωt (x, t) = A∗ e−i(k2a x−wt) .

(4.75)

Like Ω i given by Eq. (4.61), the quantum states Ω t are exotic plane waves traveling from left to right, so in the opposite direction of p = −èk 2a . The exotic quantum states Ω i are all occupied in the physical situation sketched in Fig. 4.6 because at x < 0 only free electrons are observable. However, a large potential energy (V >> E + 4mc2 ) is available at x ≥ 0 for exciting an electron from the exotic state Ω t with E T = E' − mc2 to a particle’s state (Ψ e ) with E T = E + mc2 . This process simultaneously creates a hole in the Dirac’s Sea and an observable electron. The hole corresponds to an observable positron. The creation of an electron-positron pair requires the consumption of an amount of energy equal to ΔE = 2E + 2mc2 . This

68

4 Other One-Dimensional Problems

energy is supplied by the external potential. Note that an external source of energy is then necessary for maintaining the potential constant in these circumstances. As sketched in Fig. 4.10, the creation of particle-antiparticle pairs results in two beams that, due to the conservation of the linear momentum, should travel in opposite directions in the region x ≥ 0. The created beam of antiparticles (positrons) corresponds to Ψ ta given by Eq. (4.73) with Ba = A. Therefore, at x ≥ 0, the number of positrons per unit length in the positron beam is equal to the number of electrons per unit length in the electron beam given by Eq. (4.66). Both beams travel in the same direction from left to right. Eventually, these beams should annihilate each other producing highly energetic radiation. The electrons in the created beam of particles contain the same energy E, but opposite linear momentum than the positrons. Therefore, the created electrons move in the opposite direction than the positrons. These electrons contain the same energy E as the electrons in the transmitted beam (Eq. 4.66). Therefore, the electrons in the created beam of particles are in the following quantum state with E T = E + mc2 : Ψe (x, t) = Ae−i (k2 x+wt) .

(4.76)

As sketched in Fig. 4.10, this beam of electron travels from right to left which is in the same direction of p = −èk 2 . This beam passes without reflection to the region x < 0. This is because this is the inverse situation of the Klein’s paradox discussed in the previous Section (Fig. 4.6). As a result, the same number of particles leaves and enters the region x < 0 per unit time.

V(x)

Faraday cage with an excess of electrons Transmitted beam of observable electrons

Incident beam of observable electrons

V >> E + 4mc2

E>0

Emitted radiation

x Transmitted beam of observable electrons

Faraday cage at V = 0

-E

- +

-V > E + 4mc2 . However, mc2 ≈ 0.5 meV, so it requires more than two million volts of potential difference between the two Faraday’s cages in Fig. 4.10. This explains why the experiment that corresponds to the physical situation sketched in Fig. 4.10 has not yet been realized. However, it looks like the realization of this experiment is within reach using currently available modern technology. Finally, let us go back one more time to the problem of a relativistic particle in an infinite one-dimensional well. As discussed in Sect. 3.6, if the width of the well is exceedingly smaller than the Compton wavelength associated to the particle, even the ground state energy of the particle could be E n =1 > mc2 (Eq. 3.81) . Therefore, the number of particles (one) inside of the well will remain constant because the same number of particles would leave and enter the well per unit time. In this sense, the results obtained in Sect. 3.6 remain valid. If an infinite well were squeezed to an extremely small spatial size, a particle would remain inside the well. In addition, the well would emit high energy radiation. This would require a permanent source of energy for maintaining the potential constant in these circumstances.

Bibliography 1. 2. 3. 4. 5. 6. 7. 8.

W. Greiner, Relativistic quantum mechanics: wave equations (Spring-Verlag, New York, 1990) D.J. Griffiths, Introduction to quantum mechanics (Prentice Hall, USA, 1995) D. Bohm, Quantum theory, 11th edn. (Prentice -Hall, USA, 1964) L.A. Poveda, L. Grave de Peralta, J. Pittman, B. Poirier, A non-relativistic approach to relativistic quantum mechanics: the case of the harmonic oscillator. Found. Phys. 52, 29 (2022) L. Grave de Peralta, Quasi-relativistic description of a quantum particle moving through onedimensional piecewise constant potentials. Results Phys. 18, 103318 (2020) Klein’s paradox (Wikipedia). https://en.wikipedia.org/wiki/Klein_paradox. Accessed 10 Apr 2023 Step potential, non-relativistic case, (Wikipedia). https://en.wikipedia.org/wiki/Solution_of_ Schr%C3%B6dinger_equation_for_a_step_potential. Accessed 10 Apr 2023 Rectangular potential barrier, non-relativistic case, (Wikipedia). https://en.wikipedia.org/wiki/ Rectangular_potential_barrier. Accessed 10 Apr 2023

Chapter 5

Quantum Mechanics in Three Dimensions

Abstract Real quantum particles exist in a three-dimensional world. In this Chapter, we extend to three dimensions from the one-dimensional models discussed in previous Chapters. This includes a particle confined in a cubic infinite well, and a spinless relativistic particle in a Coulomb potential.

Real quantum particles exist in a three-dimensional world. In this Chapter, we will extend to three dimensions from the one-dimensional models discussed in previous Chapters.

5.1 The Infinite Cubic Well The simplest three-dimensional potential to study is the infinite cubic well: V (x, y, z) =

if 0 < x < L , 0 < y < L , 0 < z < L 0, . +∞, otherwise

(5.1)

With the significant matters discussed in the previous Chapter, for finding the wavefunctions and energies of a particle inside an infinite well, we can proceed like we did in Sect. 3.6. Inside the well, the PPGP equation to solve is: iℏ

ℏ2 ∂ Ψ(x, y, z, t) = − ∇ 2 Ψ(x, y, z, t), if 0 < x < L , 0 < y < L , 0 < z < L . ∂t 2μ

(5.2) The null boundary conditions are (Ψ = 0) at the edges of the cube. In Eq. (5.2), μ is given by Eq. (3.52): and the Laplacian operator is: ( ) E−V ∂2 ∂2 ∂2 μ= 1+ m, and ∇ 2 = 2 + 2 + 2 . 2 2mc ∂x ∂y ∂y © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L. Grave de Peralta et al., Relativistic and Non-Relativistic Quantum Mechanics, Undergraduate Lecture Notes in Physics, https://doi.org/10.1007/978-3-031-37073-1_5

(5.3)

71

72

5 Quantum Mechanics in Three Dimensions

The spatial part of the stationary solutions of Eq. (5.2) satisfies the following equations: −

ℏ2 2 ∇ ϕ(x, y, z) = Eϕ(x, y, z), if 0 < x < L , 0 < y < L , 0 < z < L . 2μ (5.4)

We are looking for a solution of Eq. (5.4) of the form: ϕ(x, y, z) = X (x)Y (y)Z (z).

(5.5)

We can obtain that X(x), Y (y), and Z(z) are solutions of the one-dimensional infinite well problem each: −

ℏ2 2 ∇ Σ(ξ ) = EΣ(ξ ), if 0 < ξ < L . 2μ

(5.6)

With null boundary conditions Σ(0) = Σ(L) = 0. In Eq. (5.6): E = Ex + E y + Ez .

(5.7)

In Sects. 2.5 and 3.6, two different approximations to the exact problem (Eq. (5.6) with null boundary conditions) were solved. By substituting Eqs. (2.54), (3.75) and (3.81) in Eqs. (5.5) and (5.7), we obtain: ( ) 23 ( ) ( ) ( ) 2 sin kn x x sin kn y x sin kn z z , ϕn x ,n y ,n z (x, y, z) = L nξ π with kn ξ = , n ξ = 1, 2, . . . L

(5.8)

And: ⎡ ⎢ E n x ,n y ,n z = ⎢ ⎣(

n 2y n 2x )+( ) / / n 2 ( )2 n 2 ( )2 1 + 1 + 4x λLC 1 + 1 + 4y λLC ⎤

+( 1+

/

n 2z 1+

⎥ π 2 ℏ2 )⎥ ⎦ m L2 . n 2z ( λC )2 4

L

In the non-relativistic limit, L >> λC , thus:

(5.9)

5.1 The Infinite Cubic Well

73

) π 2 ℏ2 ( E n x ,n y ,n z = n 2x + n 2y + n 2z . 2m L 2

(5.10)

The energy of the ground state corresponds to the unique combination of nx = ny = nz = 1 that results in nx 2 + ny 2 + nz 2 = 3. The ground state is then a nondegenerated state. However, there are three different combinations of these quantum numbers given the next energy value that corresponds to nx 2 + ny 2 + nz 2 = 6. Therefore, the first excited state is a degenerated state with a degeneration equal to three.

5.1.1 The Coulomb Potential In the Hydrogen atom, the electron moves in the Coulomb potential produced by the proton. While it is not perfect, an electron moving in the following Coulomb potential should be a very good approximation to what really happens to the electron in a Hydrogen-like atom with atomic number Z: VC (r ) = −

√ Z e2 , r = x 2 + y2 + z2. 4π ∊o r

(5.11)

Consequently, the energies E and the wavefunctions ϕ corresponding to the stationary states of the electron, can be found solving the following time independent PPGP equation: −

ℏ2 ∇ 2 ϕ(r, θ, φ) + VC (r )ϕ(r, θ, φ) = Eϕ(r, θ, φ), with μ(r ) 2μ(r ) ] [ E − VC (r ) m. = 1+ 2mc2

(5.12)

Note that the effective mass of the particle that is moving in a Coulomb potential is a local function on r. However, if μ(r) ≈ m in the non-relativistic limit, then Eq. (5.12) reduces to the time independent Schrödinger equation: −

ℏ2 2 ∇ ϕ(r, θ, φ) + VC (r )ϕ(r, θ, φ) = Eϕ(r, θ, φ). 2m

(5.13)

While Eq. (5.13) is only a non-relativistic approximation of Eq. (5.12), it is historically important. In 1925, Schrödinger demonstrated that the most notable features of the spectrum of the Hydrogen atom can be explained by using the energy values obtained from Eq. (5.13). For this reason, we will discuss how to solve both Eqs. (5.12) and (5.13). The Coulomb potential is an example of a central potential, so it is convenient to rewrite Eqs. (5.12) and (5.13) using the Laplace operator in spherical coordinates:

74

5 Quantum Mechanics in Three Dimensions

( ) ( ) [ ] Lˆ 2 ∂ 1 ∂2 1 ∂ 1 ∂ 2 ∂ 2 2 ˆ r − 2 2 , with L = −ℏ sinθ + . ∇ = 2 r ∂r ∂r ℏr sinθ ∂θ ∂θ sin 2 θ ∂φ 2 (5.14) 2

In Eq. (5.14), an important new operator was introduced. The operator L 2 is the operator angular momentum square. Further on in this Section, this operator will be described more in exact detail. For now, it is important to note that the operator L 2 forms a part of the Laplace operator written in spherical variables; therefore, it will be important in any problem including a central potential. Looking for a solution of Eq. (5.13) of the form: ϕ(r, θ, ϑ) = R(r )Y (θ, ϑ), with R(r ) =

u(r ) . r

(5.15)

Substituting ϕ by RY in Eq. (5.13), dividing by RY, and multiplying the resulting equation by the left by −2mr 2 /è2 , we can rewrite Eq. (5.13) in the following way: [

) ( ] ( ) 1 Lˆ 2 1 d 2mr 2 2dR r + 2 (E − VC ) − Y = 0. R dr dr ℏ Y ℏ2

(5.16)

Therefore, each term on the left side of Eq. (5.16) should be equal to a constant when, per convenience, we will choose as ±l(l + 1). We then obtain: Lˆ 2 Y = l(l + 1)ℏ2 Y.

(5.17)

( ) 2mr 2 d 2dR r + 2 (E − VC )R = l(l + 1)R. dr dr ℏ

(5.18)

And:

Substituting R by u/r in Eq. (5.18), so that dR/dr = [r(du/dr) − u]/r 2 , and (d/ dr)[r 2 (dR/dr)] = rd 2 u/dr 2 , we obtain: −

ℏ2 d 2 u ℏ2 l(l + 1) + V u = Eu, with V = V + . e f f e f f C 2m dr 2 2m r 2

(5.19)

Equation (5.19) depicts a one-dimensional time independent Schrödinger equation for a particle moving in the potential V eff .

5.1.2 The Angular Equation Equation (5.17) is the eigen-equation of the L 2 operator. The eigenvalues are L 2 = l(l + 1)è2 and the eigenfunctions referenced as spherical harmonics (Y lm ):

5.1 The Infinite Cubic Well

75

Y (θ, φ) = Ylm l (θ, φ), with l = 0, 1, 2, . . . , and m l = −l, −l + 1 . . . + l (5.20) In Eq. (5.20): Ylm l (θ, φ) = Alm l eim l φ Plm l (cosθ ).

(5.21)

The normalization constant is given by: / Alm l = ∊

(2l + 1)(l − |m l |)! (−1)m l if m l ≥ 0 . , with ∊ = 1 if m l < 0 4π (l + |m l |)!

(5.22)

The associated Legendre polynomials are defined as: )|m l |/2 d |m l | )l ( 1 dl ( 2 x −1 . Plm l (ξ ) = 1 − ξ 2 P with P = (ξ ), (ξ ) l l |m | l l l dξ 2 l! dξ

(5.23)

It is worth noting that the spherical harmonics are orthogonal to each other: ∫2π ∫π 0

[

]∗ Ylm l (θ, φ) Yl ' m l' (θ, φ)sinθ dθ dφ = δll ' δm l m l' .

(5.24)

0

In Fig. 5.1 are plotted the probability densities (ρ = |Y|2 ) that correspond to the first few spherical harmonics: ρ00 =

1 3 3 , ρ10 = cos 2 θ, ρ11 = ρ1(−1) = sin 2 θ. 4π 4π 8π

Note that ρ00 is spherically symmetrical. l = 0, ml = 0

l = 1, ml = 0

l = 1, ml = +1 l = 1, ml = -1

Fig. 5.1 Probability densities that correspond to few spherical harmonics (Eq. 5.25)

(5.25)

76

5 Quantum Mechanics in Three Dimensions

5.1.3 The Radial Equation It is useful for solving Eq. (5.19) for bound states with E < 0, to introduce the following variables: √ κ=

−2m E Z me2 > 0, τ = κr, and τo = . ℏ 2π ∊o ℏ2 κ

(5.26)

By using Eq. (5.26), it allows for rewriting Eq. (5.19) in the following way: ] [ l(l + 1) d 2u τo u. + = 1− dτ 2 τ τ2

(5.27)

Conveniently proposing: u(τ ) = τ l+1 e−τ v(τ ).

(5.28)

We obtain that v(τ) satisfies the following differential equation: τ

dv d 2v + [τo − 2(l + 1)]v = 0. + 2(l + 1 − τ ) dτ 2 dτ

(5.29)

Looking for a solution of Eq. (5.29) as a finite power series: v(τ ) =

jmax ∑

ajτ j.

(5.30)

j=0

After plugging Eq. (5.30) into Eq. (5.29), we obtain: ] 2( j + l + 1) − τ0 aj. = ( j + 1)( j + 2l + 2) [

a j+1

(5.31)

If j = jmax , then: a jmax +1 = 0 ⇒ 2( jmax + l + 1) − τ0 = 0.

(5.32)

τ0 = 2n, with n = jmax + l + 1 = 1, 2, 3 . . .

(5.33)

Therefore:

The energies of the stationary states of the electron in the Coulomb potential are found from Eqs. (5.26) and (5.33) to be:

5.1 The Infinite Cubic Well

En = −

77

1 Z 2 α2 2 e2 ≈ . mc , with α = 2n 2 4π ∊o ℏc 137

(5.34)

This is the famous Bohr formula. Bohr obtained it in 1913 using semiclassical arguments. In 1925, Schrödinger obtained the same result starting from Eq. (5.13). This could be considered the first big triumph of quantum mechanics. It is worth noting that in electron-volts (eV) units, E 1 ≈ −13.6 eV, and mc2 ≈ 0.5 meV for Z = 1. With this being said, E 1 ∆E). If a photon is absorbed, an electron could jump from the valence band to the conduction band. As illustrated in Fig. 7.3b, this would result in the simultaneous existence of an occupied state in the conduction band and an empty state on the valence band. The occupied state in the conduction band is a mobile electron. This is because it is surrounded by numerous empty states in the conduction band. The empty state in the valence band is a mobile hole because the hole is surrounded by numerous occupied states in the valence band. If an external electric field were then applied to the material, both the electron and the hole would move in opposite directions. Two electrical currents would be observed where one would be formed by electrons and the other formed by holes. There is a clear similitude between Figs. 3.2 and 7.3. The set of full energy bands in a solid crystal correspond to the Dirac Sea in relativistic quantum mechanics. The formation of the electron–hole pair in a solid crystal resembles the formation of an electron–positron pair. We should note that a hole is the absence of an electron; thus,

114

7 Identical Particles Electron states

(a)

(b)

E

E

Conduction band Electron

ΔE

Energy gap

ΔE Hole

Valence band

Fig. 7.3 a Energy bands and energy band gap in an isolator, b excitation of an electron–hole pair

like the positron, a hole carries a positive charge. Like positrons, holes are observable. Holes produce an important contribution to the total current in semiconductors. This similitude justifies the existence of a healthy theoretical interaction between the theory and methods of solid-state physics and relativistic quantum mechanics. For instance, the validity of the Klein’s paradox (Sect. 4.2) has been confirmed in experiments with quasi-particles in solid crystals.

Bibliography 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

W. Greiner, Relativistic Quantum Mechanics: Wave Equations (Spring, New York, 1990) D.J. Griffiths, Introduction to Quantum Mechanics (Prentice Hall, USA, 1995) I. Morison, Introduction to Astronomy and Cosmology (Wiley, U. K, 2008) R. López-Boada, L. Grave de Peralta, Some consequences of a simple approach for constructing a theory of a relativistic Fermi gas. J. Modern Phys. 12, 1966 (2021) Pauli’s exclusion principle (Wikipedia), https://en.wikipedia.org/wiki/Pauli_exclusion_pri nciple. Accessed 10 Apr 2023 Periodic Table (Wikipedia), https://en.wikipedia.org/wiki/Periodic_table. Accessed 10 Apr 2023 Fermi gas (Wikipedia), https://en.wikipedia.org/wiki/Fermi_gas. Accessed 10 Apr 2023 Chandrasekhar mass (Wikipedia), https://en.wikipedia.org/wiki/Chandrasekhar_limit. Accessed 10 Apr 2023 C. Kittel, Introduction to Solid State Physics, 8th ed. (Wiley, 2005) Energy band structure of the electrons in solid crystals (Wikipedia), https://en.wikipedia.org/ wiki/Electronic_band_structure. Accessed 10 Apr 2023 A hole in the Dirac’s Sea of electrons (Wikipedia), https://en.wikipedia.org/wiki/Electron_ hole. Accessed 10 Apr 2023

Chapter 8

Some Consequences of Relativity for Quantum Mechanics

Abstract Most of the results that are presented in this book are widely known results proposed and discovered by a multitude of physicists. However, we introduced some controversial but exciting topics in the last Chapter. Only time and experiments will judge the validity of the ideas discussed in this chapter. These topics were included for celebrating the first century of quantum mechanics, for illustrating that relativistic quantum mechanics remains an open field of research, and for emphasizing that the pleasure of discovery and the critical thinking skills should be systematically cultivated.

In previous Chapters, several consequences for quantum mechanics, due to the introduction of special relativity on it, have been discussed. The need for new relativistic equations that have solutions with K, < 0, the existence of antiparticles, and the consequences of trying to confine a quantum particle in an infinite well are more relevant. This Chapter was designed for developing the critical thinking abilities of the readers. Therefore, some more controversial consequences of relativity in quantum mechanics are discussed. Annexes C and I contain additional information about this topic. Almost a century after quantum mechanics was developed, relativistic quantum mechanics remains an open field of research. Only time and experiments will judge the correction of the ideas discussed in this Chapter.

8.1 Atoms Cannot Be Too Heavy In Sects. 5.2 (Eq. (5.43)) and 6.5 (Eq. (6.45)), we discussed that atoms with Z ≥ 137 cannot exist in their ground state. This result is an important consequence of relativity for quantum mechanics. The size of the Hydrogen atom can be qualitatively obtained by realizing that, in the Hydrogen atom, the electron is approximately trapped in a localized spherical region of radius r. Therefore, the Bohr radius (r B ) can easily be obtained as the value of r that minimizes the sum of the particle-in-a-box kinetic energy (Eq. (2.54)), plus the potential energy of the slow-moving electron in the Hydrogen atom: © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L. Grave de Peralta et al., Relativistic and Non-Relativistic Quantum Mechanics, Undergraduate Lecture Notes in Physics, https://doi.org/10.1007/978-3-031-37073-1_8

115

116

8 Some Consequences of Relativity for Quantum Mechanics

ESch (r) ≈

e2 h2 . − 2me r 2 4π ε0 r

(8.1)

The first term of E Sch (r), corresponds to the non-relativistic kinetic energy of the ground state of a trapped and slow-moving particle with the electron mass (m = me ). The second term corresponds to the potential energy associated with the Coulombic attraction between a particle, with a charge equal to the electron charge (−e), and a positive charge +e placed at r = 0. E Sch (r) has a minimum when: (8.2) Therefore, the size of the Hydrogen atom is approximately 1/α ≈ 137 times the electron’s reduced Compton wavelength, which confirms the initial slow-moving assumption. On the other hand, the ground-state electron moves at relativistic speeds in Hydrogen-like atoms which the nuclear charge Z >> 1. Therefore, to obtain a better qualitative estimate of the size of Hydrogen-like atoms, we should use Eq. (3.81) for modifying Eq. (8.1) in the following way: (8.3) E PPGP (r) has a minimum when: (8.4) In Eq. (8.4), a = r B /Z is the Bohr radius for a Hydrogen-like atom with Z > 1, the electron moves at relativistic speeds; this results in the square root factor in Eq. (8.4) becoming significant. As shown in Fig. 8.1, the relativistic correction to the size of the ground state of Hydrogen-like atoms becomes significant when a ≈ ňC . Moreover, the size of the Hydrogen-like atom becomes undefined when Z > 1/α ≈ 137. This could be interpreted as a prediction about the impossibility of the natural existence of elements with Z > 137. While no element with Z > 118 has ever been discovered, the current opinion of the superheavy elements experts is that an atomic nucleus with Z ≤ 172 may be possible. However, these assessments are based on stability arguments pertaining to atomic nuclei, rather than the electronic structure arguments discussed here.

8.2 A Possible Frontier Between the Classical and the Quantum World Fig. 8.1 (Discontinuous) Non-relativistic and (continuous) relativistic estimates of the radius (in reduced Compton wavelength units) of the quantum field of the electron with E T = E + mc2 in Hydrogen-like atoms

117

rZ/ƛC

Z

8.2 A Possible Frontier Between the Classical and the Quantum World Quantum mechanics was originally developed for explaining phenomena at the atomic scale. Nevertheless, the only property of the particle explicitly included in the Schrödinger equation is the mass. This has been interpreted by some physicists as suggesting that there is a quantum field (wavefunction) associated to any object with mass. Using this interpretation, there is even a wavefunction associated to the whole Universe. However, semi-quantitative arguments are presented suggesting that the existence of a clear frontier, between the classical and the quantum world, is a consequence of including relativity in quantum mechanics. Any extended classical bodies of mass m should gravitationally interact with itself. Black holes exist due to the gravitational interaction between the different parts of their original and spatially distributed mass. The enormous pressure existing inside planets acquires the same origin. If the same happens to any extended quantum body of mass m, the energy of a slow-moving free particle with mass spread out over a finite spatial region, would be given by the following modification of Eq. (8.1): ESch (r) ≈

Gm2 h2 . − 2 2mr r

(8.5)

In Eq. (8.5), the gravitational interaction of the particle with itself substitutes for the Coulomb interaction included in Eq. (8.1). Due to its null size, the gravitational term in Eq. (8.5) would have to be removed if the quantum particle could not interact with itself. In this case, the kinetic energy term of Eq. (8.5) would not have a local minimum which results in an infinitely spatial extended plane wave as the wavefunction for a free particle. In contrast, E Sch (r) has a minimum when: ( m )3 h2 P r = aG = = l , with lP = P Gm3 m

/

hG , c3

/ and mP =

hc . G

(8.6)

118

8 Some Consequences of Relativity for Quantum Mechanics

In Eq. (8.6), l P and mP are the Planck’s length and mass, respectively. At relativistic speeds, Eq. (8.5) should be substituted by: (8.7) E PPGP (r) has a minimum when: (8.8) As shown in Fig. 8.2, a notable consequence of combining quantum mechanics with the special theory of relativity, is the existence of a critical mass mc = mP above which the size of the particle becomes undefined. This critical mass could be interpreted as the frontier between the quantum and the classical matter world. It should be noted that this critical mass value (mP ≈ 22 μg) is quite small for having to consider the full complexity of quantum mechanics in the daily life. In contrast, it is quite large when compared to molecular masses, and the quantum experiments that have been accomplished to date. Interestingly, biological cells, including human neurons, could still be quantum objects. In any event, the experimental confirmation or rejection of this hypothesis would have fundamental consequences for quantum mechanics and cosmology. In particular, the confirmation of the existence of mc could mean that there is not a universal wavefunction, that the Schrödinger quantum cat does not exist, and that the world that surrounds us is as classical as it seems to be. Nevertheless, it is important to realize that huge classical bodies can be formed by numerous quantum particles. The Fermi-gas stars that are studied in Sect. 7.2, are good examples of this. We can make some general comments about the stability of the matter. As discussed in Sects. 2.5, 5.2, and 6.5, quantum mechanics explains the stability of atoms. However, as discussed in Sect. 8.1, the stability of atoms disappears when the combined effects of special relativity and electrostatics overcome the stability provided by quantum mechanics effects, thus producing the collapse of superheavy Fig. 8.2 (Discontinuous) Non-relativistic and (continuous) relativistic estimates of the radius (in Planck length units) of the quantum field with E T = E + mc2 for a particle that interacts gravitationally with itself

rm/lP

m/mP

8.3 About the Observed Asymmetry Between Matter and Antimatter

119

atoms. As discussed in this Section, the stability of single quantum particles, a kind of zero-order stability, disappears when the combined effects of special relativity and gravity overcome the stability provided by quantum mechanics effects. This is when the collapse to a point of the quantum field of elemental particles happens. While elemental quantum particles with a mass larger than the Planck mass may not exist, massive cosmological bodies formed by an extremely large number of quantum particles do exist. As discussed in Sect. 7.2, the stability of the white dwarfs and neutron stars is an instance of the so-called stability of the second kind. This stability also disappears when the combined effects of special relativity and gravity overcome the quantum effects that are associated to Pauli’s exclusion principle, which makes stability possible.

8.3 About the Observed Asymmetry Between Matter and Antimatter In Sect. 8.2, we explore what could happen if the particle quantum states with E T = E + mc2 were bound by the action of a self-attractive gravitational potential. We obtained that this imposes a maximum possible mass to a quantum particle. In this Section, it is explored what could happen if the antiparticle quantum states with, E Ta = E a , − mc2 , were bounded by the action of a self-repulsive electric potential. We will see that this imposes a maximum possible charge to an antimatter quantum particle. Atoms and Antiatoms The existence of a minimum value of r, in Eq. (8.3), is in correspondence with the existence of bound states with E T = E + me c2 for a particle (electron) of charge −e; that is moving in the attractive Coulomb potential produced by a particle with charge +Ze placed at r = 0. For instance, this particle could be the nucleus of a Hydrogen-like atom. The energies of these bound states are given by Eqs. (5.43) and (6.45) for particles with spin s = 0 and s = 1/2, respectively. As sketched in Fig. 8.3, the probability density (ρ = |ϕ|2 ) of the ground particle states, with E T = E + me c2 , are spatially localized around r = 0. As it will be shown below, the occupied exotic particle states with E T = E, − me c2 are unbound and thus are spatially delocalized quantum states. As sketched in Fig. 8.4b, the quantum states with E Ta = E a + me c2 , corresponding to the antiparticle (positron) with charge + e, and while moving in the repulsive Coulomb potential produced by a particle with charge + Ze placed at r = 0, are unbound antiparticle states. Thus, as sketched in Fig. 8.4c, the corresponding probability density for the antiparticle (ρa = |ϕa |2 ) is spatially delocalized. This corresponds to the non-existence of a value of r that is locally minimum to the following equation for the antiparticle:

120

8 Some Consequences of Relativity for Quantum Mechanics

(a)

ET = E + mec2

Bound particle states

(b)

ρ(r)

+mec2 0 -mec2

r

+Ze

Occupied unbound particle states

Spatially localized probability density corresponding to the particle’s ground state

ET = Eꞌ - mec2

Fig. 8.3 a Occupation of quantum states and b particle’s probably density of the electron in a Hydrogen-like atom

/

h2 Ze2 , EPPGPa (r) ≈ + (γ + 1)me r 2 4π ε0 r

with γ =

( 1+

λC r

)2 .

(8.9)

The first term of E PPGPa (r) corresponds to the relativistic kinetic energy of the antiparticle with m = me . The second term corresponds to the potential energy associated with the Coulombic repulsion between the antiparticle with charge + e and a positive charge + Ze placed at r = 0. The energy of the antiparticle (positron) in the nucleus of a matter Hydrogen-like atom is then E Ta = E a + me c2 with E a > 0. Due to the Hole Theory described in Sects. 3.3 and 3.5, the antiparticle’s energy (E a ) and its wavefunction (ϕa ) are linked by the relations of E, = −E a and |ϕa |2 = |χ|2 , to energy (E, ) and wavefunction (χ ) of the exotic particle states with E T = E, Particle states

Empty bound particle states

ET = E + mec2

ETa = Ea + mec2

+mec2 0 -mec2

Unbound particle states

(b)

-mec2 ET = Eꞌ - mec2

Unbound antiparticle states

Spatially delocalized probability density of the occupied unbound antiparticle state

ρa(r)

+mec2 0

(a)

Antiparticle states

Occupied bound antiparticle states

ETa = Eꞌa - mec2

+Ze

r

(c)

Fig. 8.4 Occupation of quantum states and antiparticle’s probably density of a positron interacting with the nucleus of a Hydrogen-like atom

8.3 About the Observed Asymmetry Between Matter and Antimatter

121

− me c2 . Therefore, if |ϕa |2 is spatially delocalized, then the probability density (|χ|2 ) that corresponds to the particle exotic states with E T = E, − me c2 is also spatially delocalized. This determines the unbound character of the exotic particle’s states E T = E, − me c2 sketched in Figs. 8.3a and 8.4a. The complimentary equation of Eq. (8.3) for the exotic particle states with E T = E, − me c2 , can be obtained from Eqs. (3.40) and (3.64): Ze E , − 4πε Ze2 h2 , 0r , with γ = −1 + − . ≈ , 4π ε0 r mc2 (γ − 1)me r 2 2

, EPPGP (r)

(8.10)

Like Eq. (8.9), there is not a value of r that is a local minimum of Eq. (8.10) because E, = −E a < 0. Therefore, both γ, and the kinetic term in Eq. (8.10) are negative. Consequently, there is no local minimum of E, PPGP (r). This corresponds to the spatial delocalization of the exotic particle states with E T = E, − me c2 sketched in Figs. 8.3a and 8.4a. More importantly, as sketched in Fig. 8.5a, b, the repulsive interaction of a positron and the nucleus of a matter Hydrogen-like atom results in bound exotic antiparticle states with E Ta = E, a − me c2 . Figure 8.5 illustrates the Hole Theory for antiparticle states. There is a single empty antiparticle exotic state with E Ta = E, a − me c2 but is not occupied antiparticle state with E Ta = E a + me c2 in Fig. 8.5b. Therefore, as sketched in Figs. 8.5c, this is perceived as a single anti-antiparticle, meaning a single observable particle has energy with E = –E a ,. Notably, Fig. 8.5c, d are identical to Fig. 8.3a, b. Consequently, the energies E a , are minus the energies given by Eqs. (5.43) and (6.45) for particles with spin s = 0 and s = 1/2, respectively. Comparing Fig. 8.5a, d, we can conclude that the electron states (ϕ) with E T = E + me c2 and the exotic positron states (χa ) with E Ta = E, a − me c2 , are spatially localized by the interaction of the electron and positron with a matter nucleus with Z protons, respectively. This corresponds to the notion from the Hole Theory that Antiparticle states

ρꞌa(r)

(b)

(a)

Particle states

ETa = Ea + mec2

(c)

ET = E + mec2

ρ(r)

(d)

+mec2

Spatially localized probability density of the empty bound antiparticle state

0

+mec2

-mec2

0

+Ze

r

-mec2

ETa = Eꞌa - mec2

+Ze

r

Spatially localized probability density of the occupied bound particle state

ET = Eꞌ - mec2

Fig. 8.5 A particle is the antiparticle of the antiparticle

122

8 Some Consequences of Relativity for Quantum Mechanics

a particle is the antiparticle of the antiparticle. Quantitatively, this means that by changing E by −E, a , in the time independent PPGP equation for a spin-(s = 0) particle with E T = E + me c2 (Eq. (5.12)): ( ) Ze2 E + 4πε h2 2 Ze2 r 0 ϕ = Eϕ, with μ = 1 + − ∇ ϕ− m. 2μ 4π ε0 r 2mc2

(8.11)

We obtain: ( ) Ze2 −Ea, + 4πε Ze2 h2 2 , 0r ϕ = −Ea ,ϕ, with μa = − 1 + − ( , )∇ ϕ − m. 4π ε0 r 2mc2 2 −μa (8.12) Or: ( ) Ze2 , E − h2 2 Ze2 a 4πε r 0 − , ∇ χa + m, and χa = ϕ ∗ . χa = Ea, χa , with μ,a = −1 + 2μa 4π ε0 r 2mc2 (8.13) Equation (8.13) is the complimentary time independent PPGP equation for the antiparticle with E Ta = E, a − me c2 . This is not an evident result. The antiparticle (positron) exotic quantum states (χa ) with E Ta = E, a − me c2 are spatially localized by the repulsive interaction between the antiparticle with charge + e and a matter nucleus with Z protons. This suggests that the antiparticle exotic states with E Ta = E, a − mc2 , that correspond to an antiparticle with mass m and charge q, could also be spatially localized by the repulsive electrostatic interaction of an antiparticle with itself. However, we should be aware that, theoretically, the universe would not change if all the particles were interchanged by particles and vice versa. Which one, the electron or the positron, we call particle, and which one antiparticle should be only an arbitrary convention. This implies that we can rewrite the previous paragraph in the following way: the particle (electron) exotic quantum states (χ) with E T = E, − me c2 are spatially localized by the repulsive interaction between the particle with charge −e and an antimatter nucleus with Z antiprotons. This means, as depicted in Fig. 8.6, the complementary Schrödinger-like (Eq. (3.64)) and Pauli-like (Ec. (6.43)) PPGP equations have bound solutions if the potential is a repulsive Coulomb potential. Particles and Antiparticles The above discussion refers to the interaction of particles and antiparticles with an external Coulomb potential. We will focus now on describing the hypothetical repulsive electrostatic interaction of an antiparticle with itself. In this case, Eq. (8.7) should be substituted by:

8.3 About the Observed Asymmetry Between Matter and Antimatter Positron states

Electron states

ET = E + mec2

ETa = Ea + mec2

r

Spatially localized probability density of the bound exotic positron states

ρꞌ(r)

-mec2

-mec2

ETa = Eꞌa - mec2

(d)

+mec2

+mec2 0

+Ze

(c)

(b)

(a)

0

ρꞌa(r)

123

r

-Ze

Spatially localized probability density of the bound exotic electron states

ET =Eꞌ - mec2

Fig. 8.6 A repulsive Coulomb potential bound the exotic (a-b) positron and (c-d) electron states

(8.14) E, PPGPa (r) has a maximum when: (8.15) Therefore, ňC /ξ−2 → ňC (the reduced Compton wavelength), when |q| → qP ≈ 11e (the Planck charge). Moreover, r → 0, when |q| → qP . Therefore, as shown in Fig. 8.7, a notable consequence from combining quantum mechanics with the special theory of relativity, is the existence of a critical charge, |qc | = qP . Above this charge, the size of the antiparticle becomes undefined because the exotic quantum field χa collapses to a point. This critical charge could be interpreted as the frontier between the quantum and the classical antimatter world. Fig. 8.7 Relativistic estimate of the radius (in Planck length units) of the exotic quantum field χa with E Ta = E, a − mc2 for an antiparticle that interacts electrostatically with itself

rq/ƛC

q/qP

124

8 Some Consequences of Relativity for Quantum Mechanics

It should be noted that no interaction of a particle with itself is included in the Klein-Gordon and Dirac equations. Therefore, the hypothesis that an antiparticle interacts electrostatically with itself is an addition to current ideas in relativistic quantum mechanics. Nevertheless, a notable consequence for elemental quantum particles follows from the discussions presented in this Section and the previous Section. Elemental quantum particles are born in pairs such as the electron and the positron. Therefore, no elemental quantum particles should have m > mP or |q| > qP . In fact, there is no known violation of this rule. Interestingly, the Planck units of mass, length, and charge naturally appear as limit values in Eqs. (8.6) and (8.15). The potential energy term in Eq. (8.14) corresponds to the electrical interaction of a particle with itself. This potential energy would be extremely large if the quantum particle could be extremely small. This suggests that self-interacting quantum particles cannot have a radius smaller than a certain value. There should be an internal source of energy responsible for the potential energy of an isolated self-interacting particle. Special theory of relativity may suggest that the mass of the particle could be the particle’s internal source of energy. Intriguingly, this hypothesis implies that the minimum possible value of r, for a particle with the Planck charge (qP ), is the Planck length (lP ). This is because: mc2 =

qP2 . 4π ε0 lP

(8.16)

At this point, we can focus our attention on the evident asymmetry between matter and antimatter that exist in the world surrounding us. As stated in Sect. 3.3, particles and antiparticles are formed in pairs, so therefore, the number of particles in the Universe should be equal to the number of antiparticles. The fact that we seem to live in a Universe where there are many more particles than antiparticles is an unsolved mystery. In addition, no one knows why we live in a world formed almost exclusively by matter atoms and molecules. No antimatter life seems to exist in the Universe. Most physicists believe that the reason for this huge discrepancy between theory and reality is related to some type of unknown asymmetry between matter and antimatter that occurred at the beginnings of the Universe. But nobody knows for sure what this hypothetical asymmetry was and how it occurred. Interestingly, by hypothesizing that a quantum antiparticle could interact electrostatically with itself, we could find a simple explanation for the observed asymmetry between matter and antimatter. We should add that the symmetry break-down requires that a quantum particle does not interact with itself, or it does in a different manner than the antiparticle does. We could propose that the antiparticle self-repulsive electrostatic interaction results in the validity of Eq. (8.14). In contrast, the particle electrostatic self-interaction results in the following equation: (8.17)

Bibliography

125

This equation does not have a local minimum. For this reason, the hypothesized electrostatic self-interaction would break the particle-antiparticle symmetry. In this way, we could first easily explain why we are not surrounded by antimatter atoms, molecules, and antimatter living creatures. Living creatures and molecules are formed by quantum atoms. If we have learned something in this course, it is that the stability of atoms is a consequence of quantum mechanics. A stable atom requires a quantum nucleus and a quantum “cloud” of particles attracted by the nucleus. However, an antimatter nucleus and a cloud of positrons with, |q| > qP ≈ 11e, could not be quantum, but classical. Indeed, no antimatter atom with Z > 11 has ever been observed. This explains why we are not surrounded by antimatter atoms, molecules, and antimatter living creatures. We are surrounded by matter atoms, molecules, and matter living creatures because the condition “classical if |q| > qP ≈ 11e” does not apply to matter. Second, we could hypothesize an explanation for the relative abundance of matter in the Universe. In the spirit of the Big Bang theory, about the origin of the Universe, we could hypothesize that all the existing elemental quantum particles were created in pairs from random fluctuations of the quantum vacuum at the very beginning of the Universe. Mass density fluctuations with m > mP could have collapsed in the so-called primordial black holes. The existence of such primordial black holes has been previously hypothesized, but currently there is no observational evidence of their existence. Similarly, fluctuations of charged antimatter with |q| > qP could have collapsed in primordial electric sinks. The existence of such primordial electric sinks could explain the relative abundance of charged matter in the rest of the Universe. It is worth noting that most physicists working in high energy physics, relativistic quantum mechanics, and quantum field theory, believe that elemental quantum particles are mathematical points. Therefore, they believe that quantum particles cannot interact with themselves. Consequently, they rather believe that the reason for the current discrepancy, between theory and reality, is related with some unknown asymmetry between matter and antimatter that occurred at the beginnings of the Universe. They support research looking for the culprits of such a birth phenomenon. Nevertheless, in science, it is not the number of scientists that believe in the same theory what makes correct a theory, but the experiments that prove whether a theory is valid or not. The scientific method suggests that if a hypothesis is capable of explaining the world as is it seems to be, then this hypothesis should be seriously considered. As we stated at the beginning, this Chapter was designed for developing the critical thinking abilities of the readers, and for introducing them to the pleasure of discovery that every Science lover should feel. We expect to have achieved our goal!

Bibliography 1. L. Grave de Peralta, L.A. Poveda, B. Poirier, Making relativistic quantum mechanics simple. Eur. J. Phys. 42, 055404 (2021) 2. R. López-Boada, L. Grave de Peralta, Some consequences of a simple approach for constructing a theory of a relativistic Fermi gas. J. Modern Phys. 12, 1966 (2021) 3. Planck units (Wikipedia), https://en.wikipedia.org/wiki/Planck_units. Accessed 10 Apr 2023

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4. Primordial black holes (Wikipedia), https://en.wikipedia.org/wiki/Primordial_black_hole, https://en.wikipedia.org/wiki/Penrose_interpretation. Accessed 10 Apr 2023 5. Antimatter (Wikipedia), https://en.wikipedia.org/wiki/Antimatter. Accessed 10 Apr 2023 6. Why are we not surrounded by antimatter? Why are we surrounded by macroscopic classical objects? (YouTube). https://www.youtube.com/@luisgravedeperalta

Annexes

In this set of Annexes, either some advanced topics are covered, or some topics are revisited from an alternative point of view, which allows for a better understanding of the topic. In Annex A, a precise mathematical derivation of the PPGP equations is presented. In addition, the relationship between the PPGP equations for a particle and its antiparticle, if both are moving in the same external world, is discussed. In Annex B, the Dirac equation is presented and discussed. Heuristic discussions related to materials presented in Chap. 8 are presented in Annexes C and I. An in-depth discussion of the antiparticle PPGP equations is presented in Annex D. The solution of the complementary PPGP equation for a particle confined in a one-dimensional infinite well is presented in Annex E. Annex F contains a general discussion about the mathematical apparatus of quantum mechanics. The superposition principle and its relationship with the PPGP equations is the topic discussed in Annex G. Finally, Annex H is directed to quantum mechanics instructors. The pedagogical values of the approach followed by the authors in this book are discussed.

Annex A: Schrödinger-Like and Pauli-Like Relativistic Wave Equations In this Annex, a precise mathematical derivation of the PPGP equations is presented. In addition, the relationship between the PPGP equations for a particle and its antiparticle, if both are moving in the same external world, is discussed. Theorem I By solving the following Schrödinger-like equations (Chap. 3): iℏ

[ ] E − V (r ) ℏ2 ∂ Ψ=− ∇ 2 Ψ + V (r )Ψ, with μ(r ) = 1 + m. ∂t 2μ(r ) 2mc2

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L. Grave de Peralta et al., Relativistic and Non-Relativistic Quantum Mechanics, Undergraduate Lecture Notes in Physics, https://doi.org/10.1007/978-3-031-37073-1

(A.1)

127

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And: iℏ

[ ] ℏ2 ∂ E ' − V (r ) Ω = − ' ∇ 2 Ω + V (r )Ω, with μ' (r ) = −1 + m. (A.2) ∂t 2μ (r ) 2mc2

We obtain the exact solutions, Ψ and E, and Ω and E', of the Klein-Gordon (KG) equation with total energy E T = E + mc2 and E T = E ' − mc2 , respectively. Demonstration: The KG equation for a spin-(s = 0) particle, with mass m, in the external central potential V (r) is: [ ]2 ∂ iℏ − V (r ) ψ K G = −ℏ2 c2 ∇ 2 ψ K G + m 2 c4 ψ K G . ∂t

(A.3)

The time independent equations that correspond to Eqs. (A.1)–(A.3) are: −

ℏ2 i ∇ 2 ϕ + V (r )ϕ = Eϕ, with Ψ = ϕe− ℏ Et . 2μ(r )

ℏ2 i ' ∇ 2 χ + V (r )χ = E ' χ , with Ω = χ e− ℏ E t . ' 2μ (r ) { } i −ℏ2 c2 ∇ 2 φ = [E T − V (r )]2 − m 2 c4 φ, with ψ K G = φe− ℏ E T t . −

(A.4)

(A.5) (A.6)

By substituting E T by E + mc2 in Eq. (A.6), and after some algebraic manipulations, we obtain: { } ℏ2 2 [E − V (r )]2 − φ = Eφ. ∇ + V (r ) − 2m 2mc2

(A.7)

But also, Eq. (A.7) can be obtained by pre-multiplying both sides of Eq. (A.4) by μ(r)/m and substituting μ(r) by its value in Eq. (A.1). This is the first half of the demonstration. Similarly, by substituting E T by E ' − mc2 in Eq. (A.6), and after some algebraic manipulations, we obtain: {

]2 } [ ' E − V (r ) ℏ2 2 ∇ + V (r ) + φ = E ' φ. 2m 2mc2

(A.8)

But also, Eq. (A.8) can be obtained by pre-multiplying both sides of Eq. (A.5) by μ' (r)/m and substituting μ' (r) by its value in Eq. (A.2). This completes the demonstration.

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129

Theorem II By solving the following Pauli-like equations (Chap. 6): [ ] ( ( e ) e ) 1 ∂ σ . p − A Ψ + V Ψ, V = e Ao . Ψ = σ. p − A c 2μ c ∂t ∆

iℏ

∆

∆

∆

(A.9)

And: ] ( [ ( ∂ e ) e ) 1 Ω = σ. p − A σ A Ω + V Ω. . p − ∂t c 2μ' c ∆

iℏ

∆

∆

∆

(A.10)

We can obtain the spinors and energies, Ψ and E, as well as Ω and E', that correspond to the exact solutions of the Dirac equation and total particle energies E T = E + mc2 and E T = E ' − mc2 , respectively. In Eqs. (A.9) and (A.10), A and Ao represent the vector and scalar potentials of electromagnetism. Note that the effective masses μ(r) and μ' (r) in Eqs. (A.9) and (A.10) are given by their values in Eqs. (A.1) and (A.2), respectively. Demonstration It is well-known that the Dirac equation (Annex B) is the correct Lorentz covariant equation that describes the interaction of a spin-(s = 1/2) particle, with mass m and charge e, with an external electromagnetic field. The stationary solutions of the Dirac equation are bi-spinors, which means they possess four components. They are of the form: ( ) ( ) ( ) ( ) χ1 Ψ ϕ1 ϕ −i EℏT t , and χ = . (A.11) ΨD = , with ϕ = = e ϕ2 χ2 Ω χ It is also widely known that the time independent Dirac equation is equivalent to the following system of two coupled spinor equations (Annex B): ( ( ) ) ( e ) cσ . p − A χ = E T − mc2 − e Ao ϕ, with σ = σx , σ y , σz . c ( ( ) e ) cσ . p − A ϕ = E T + mc2 − e Ao χ . c ∆

∆

∆

∆

(A.12)

∆

(A.13)

In Eq. (A.12), σi are the Pauli matrices. Eq. (A.13) can be rewritten as: ( c e ) A ϕ = χ. σ . p − E T + mc2 − e Ao c ∆

∆

Substituting χ given by Eq. (A.14) in Eq. (A.12), we obtain:

(A.14)

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] ( [ ( ( ) c e ) e ) cσ . p − A σ . p − A ϕ = E T − mc2 − e Ao ϕ. 2 c E T + mc − e Ao c (A.15) ∆

∆

∆

∆

Finally, making V = eAo and E T = E + mc2 in Eq. (A.15), we obtain the time independent Pauli-like PPGP equation: [ ] ( ( e ) e ) 1 1 c2 σ . p − A ϕ + V ϕ = Eϕ, with σ. p − A = . c 2μ c 2μ E + 2mc2 − V (A.16) ∆

∆

∆

∆

Consequently, the time dependent Pauli-like PPGP equation is Eq. (A.9). This is the first half of the demonstration. Similarly, Eq. (A.12) can be rewritten as: (

( e ) c ) σ . p − A χ = ϕ. c E T − mc2 − e Ao ∆

∆

(A.17)

Substituting ϕ given by Eq. (A.17) in Eq. (A.13), we obtain: ] [ ( ( ( ) c e ) e ) ) σ . p − A χ = E T + mc2 − e Ao χ . cσ . p − A ( 2 c c E T − mc − e Ao (A.18) ∆

∆

∆

∆

Finally, making V = eAo and E T = E ' − mc2 in Eq. (A.18), we obtain the complementary time independent Pauli-like PPGP equation: ] ( [ ( e ) 1 c2 e ) 1 ' σ. p − A σ A χ + V χ = E . . p − χ , with = c 2μ' c 2μ' E − 2mc2 − V (A.19) ∆

∆

∆

∆

Consequently, the complementary time dependent Pauli-like PPGP equation is Eq. (A.10). This completes the demonstration. It is worth noting, as stated in Sect. 6.4, that the Dirac wavefunctions given by Eq. (A.11) are bi-spinors. This means that for the total energy of the particle (E T ), Ψ D includes a spinor ϕ and a spinor χ. If E T = E + mc2 , Theorem II states that ϕ can be obtained by solving the Pauli-like PPGP equation (Eq. A.9). After ϕ is known, we can use Eq. (A.14) for obtaining χ. In contrast, if E T = E ' − mc2 , Theorem 2 states that χ can be obtained by solving the complementary Pauli-like PPGP equation (Eq. (A.10). After χ is known, we can use Eq. (A.17) for obtaining ϕ. This topic is covered in more detail in Annex G.

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131

Theorem III The particle’s wave equations that correspond to E T = E ' − mc2 , can be obtained from the antiparticle’s wave equations that correspond to E Ta = E a + mc2 . Also, the antiparticle’s wave equations that correspond to E Ta = E' a − mc2 , can be obtained from the particle’s wave equations that correspond to E T = E + mc2 . Demonstration for Particles with Spin-(s = 0) An antiparticle is just another particle. A particle and its antiparticle contain the same mass m but opposite electric charge. Therefore, if the particle and the antiparticle are moving in the same external central potential, then the PPGP equations that correspond to a spin-(s = 0) charged particle are Eqs. (A.1) and (A.2). But because of the charge difference, the following equations correspond to the antiparticle: ] [ ∂ ℏ2 E a + V (r ) m. Ψa = − ∇ 2 Ψa − V (r )Ψa , with μa (r ) = 1 + ∂t 2μa (r ) 2mc2 (A.20) ] [ ℏ2 E ' + V (r ) ∂ m. ∇ 2 Ωa − V (r )Ωa , with μa' = −1 + a iℏ Ωa = − ' ∂t 2μa (r ) 2mc2 (A.21)

iℏ

A comparison between Eqs. (A.1) and (A.2) and Eqs. (A.20) and (A.21) reveals that V changes to -V. This is because the particle and the antiparticle have charges of opposite signs, but they are moving in the same external world. The complex conjugate of Eq. (A.20) is the following equation: −iℏ

∂ ∗ ℏ2 ∇ 2 Ψa∗ − V (r )Ψa∗ . Ψa = − ∂t 2μa (r )

(A.22)

∂ ∗ ℏ2 Ψa = ∇ 2 Ψa∗ + V (r )Ψa∗ . ∂t 2μa (r )

(A.23)

Or: iℏ

But E' = −E a , so therefore: ] [ ] [ E ' − V (r ) E a + V (r ) m = −1 + m = μ' (r ). −μa (r ) = −1 − 2mc2 2mc2

(A.24)

Substituting μa (r) by −μ' in Eq. (A.23), we obtain: iℏ

∂ ∗ ℏ2 Ψa = − ' ∇ 2 Ψa∗ + V (r )Ψa∗ ∂t 2μ (r )

(A.25)

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Annexes

Therefore, (Ψ a * ) satisfies Eq. (A.2). This means: Ω = Ψa∗

(A.26)

This is the first half of the demonstration. Similarly, the complex conjugate of Eq. (A.1) is the following equation: −iℏ

∂ ∗ ℏ2 Ψ =− ∇ 2 Ψ ∗ + V (r )Ψ ∗ ∂t 2μ(r )

(A.27)

ℏ2 ∂ ∗ ∇ 2 Ψ ∗ − V (r )Ψ ∗ Ψ = ∂t 2μ(r )

(A.28)

Or: iℏ

But E ' a = −E, so therefore: ] [ ] [ E a' + V (r ) E − V (r ) m = −1 + m = μa' (r ) −μ(r ) = −1 − 2mc2 2mc2

(A.29)

Substituting μ(r) by −μ' a in Eq. (A.28), we obtain: iℏ

∂ ∗ ℏ2 Ψ =− ' ∇ 2 Ψ ∗ − V (r )Ψ ∗ ∂t 2μa (r )

(A.30)

Therefore, Ψ * satisfies Eq. (A.21). This means: Ωa = Ψ ∗

(A.31)

This completes the demonstration. It is worth noting that Eqs. (A.26) and (A.31) imply that: |Ω|2 = |Ψa |2 and |Ωa |2 = |Ψ|2

(A.32)

Demonstration for Particles with Spin-(s = 1/2) If a particle with mass m and its antiparticle are moving in the same external electromagnetic field, then the PPGP equations that correspond to a spin-(s = 1/2) charged particle are Eqs. (A.9) and (A.10). But because of the charge difference, the following equations correspond to the antiparticle: [ ] ( ] [ ( E a + V (r ) ∂ e ) 1 e ) σ . p + A Ψa − V Ψa , with μa (r ) = 1 + Ψa = σ . p + A m. ∂t c 2μa c 2mc2 ∆

iℏ

∆

∆

∆

(A.33)

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133

] ( [ ] [ ( e ) 1 ∂ E a' + V (r ) e ) ' σ Ωa = σ . p + A . p m. A Ω − V Ω , with μ = −1 + + a a a ∂t c 2μa' c 2mc2 ∆

iℏ

∆

∆

∆

(A.34) A comparison of Eqs. (A.9) and (A.10) and Eqs. (A.33) and (A.34) reveals that V changes to -V and −eA/c changes to +eA/c. This is because the particle and the antiparticle have charges of opposite signs, but they are moving in the same external world. The complex transpose conjugate of Eq. (A.33) is depicted in the following equation: −iℏ

] ( { ( [ } e ) † † ∂ † e ) 1 σˆ . pˆ + A Ψa − V Ψa† . Ψa = σˆ . pˆ + A ∂t c 2μa c

(A.35)

But: { ( ] ( ] ( [ } [ ( e ) 1 e ) † e ) 1 e ) σˆ . pˆ + A σˆ . pˆ + A σˆ † . − pˆ + A = σˆ † . − pˆ + A c 2μa c c 2μa c (A.36) The Pauli matrices given by Eq. (6.19) are Hermitic. This means that they are equal to their transpose conjugates: ] [ σˆ † = σˆ x† , σˆ y† , σˆ z† =

[(

) ( ) ( )]∗ 01 0 i 1 0 = σˆ . , , 10 −i 0 0 −1

(A.37)

Therefore, utilizing Eqs. (A.36) and (A.37), we can rewrite Eq. (A.35) in the following way: ] ( [ ( e ) ∂ † e ) 1 σˆ . pˆ − A Ψa† − V Ψa† . Ψa = σˆ . pˆ − A ∂t c 2μa c

(A.38)

] ( [ ( e ) ∂ † e ) 1 σˆ . pˆ + A Ψa† + V Ψa† . Ψa = −σˆ . pˆ + A ∂t c 2μa c

(A.39)

−iℏ Or: iℏ

But E' = −E a . Therefore, utilizing Eq. (A.24) and substituting μa (r) by −μ' in Eq. (A.39), we obtain: iℏ

[ ] ( ( 1 e ) † ∂ † e ) ˆ ˆ Ψa = σˆ . pˆ + A A Ψa + V Ψa† σ . p + ∂t c 2μ' (r ) c

(A.40)

Therefore, (Ψ a † ) satisfies Eq. (A.10). This means: Ω = Ψa† .

(A.41)

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Annexes

This is the first half of the demonstration. Similarly, the complex transposed conjugate of Eq. (A.9) is the following equation: −iℏ

{ ( [ ] ( } e ) † † ∂ † e ) 1 σˆ . pˆ − A Ψ + V Ψ †. Ψ = σˆ . pˆ − A ∂t c 2μ c

(A.42)

But: { ( [ ] ( } [ ] ( ( e ) 1 e ) † e ) 1 e ) † σˆ . pˆ − A σˆ . pˆ − A σˆ † . − pˆ − A . = σˆ . − pˆ − A c 2μ c c 2μ c (A.43) Therefore, utilizing Eqs. (A.43) and (A.37), we can rewrite Eq. (A.42) in the following way: [ ] ( ( e ) ∂ † e ) 1 ˆ ˆ σˆ . pˆ + A Ψ † + V Ψ † . −iℏ Ψ = σ . p + A ∂t c 2μ c

(A.44)

[ ] ( ( e ) ∂ † e ) 1 σˆ . pˆ + A Ψ † − V Ψ † iℏ Ψ = −σˆ . pˆ + A ∂t c 2μ c

(A.45)

Or:

But E ' a = −E. Therefore, utilizing Eq. (A.29) and substituting μa (r) by −μ' in Eq. (A.45), we obtain: [ ] ( ( 1 e ) e ) ∂ σˆ . pˆ + A Ψ† − VΨ† iℏ Ψ† = σˆ . pˆ + A ' ∂t c 2μa (r ) c

(A.46)

Therefore, (Ψ † ) satisfies Eq. (A.34). This means: Ωa = Ψ † .

(A.47)

This completes the demonstration. It is worth noting that Eqs. (A.41) and (A.47) imply that: I I2 I I2 |Ω|2 = IΨa† I and |Ωa |2 = IΨ † I .

(A.48)

The meaning of Eq. (A.48) could be illustrated with some instances. As shown in Fig. 8.3, the electron in a Hydrogen-like atom has bound quantum states where the electron’s total energy is E T = E + mc2 . These electronic bound states are spatially localized around the nucleus of the Hydrogen-like atom. As illustrated in Fig. 8.5, due to Eq. (A.48), the repulsive interaction of a positron with the nucleus of a Hydrogenlike atom produces bound quantum states where the positron’s total energy is E Ta

Annexes

135

= E ' a − mc2 . These positronic bound states are also spatially localized around the nucleus of the Hydrogen-like atom. In contrast, as shown in Fig. 8.4, the repulsive interaction of a positron with the nucleus of a Hydrogen-like atom produces unbound quantum states where the positron’s total energy is E Ta = E a + mc2 . These positronic unbound states are spatially delocalized. As illustrated in Figs. 8.3–8.5, due to Eq. (A.48), the attractive interaction of the electron with the nucleus of a Hydrogen-like atom produces unbound quantum states where the electron’s total energy is E T = E ' − mc2 . These electronic unbound states are also spatially delocalized.

Annex B: Dirac Equation The Dirac equation for a free particle with spin-(s = 1/2), charge e and mass m, that is interacting with an external electromagnetic field is: ( e ) ∂ Ψ D = cα . p + A Ψ D + e Ao Ψ D + mc2 β Ψ D . c ∂t ∆

iℏ

∆

∆

(B.1)

In Eq. (B.1), A and Ao are the vector and scalar potentials of electromagnetism (V = eAo ). Ψ D is the 4-components wavefunction given by Eq. (A.11). The operator β and each component of the vectorial operator α are 4 × 4 matrices: (

) ( ) I 0 10 β= , with I = . 01 0 −I ) ( ) ( 0 σi α = α x , α y , α z , with α i = , i = x, y, z. σi 0 ∆

∆

∆

∆

(B.2)

∆

∆

∆

∆

∆

∆

∆

(B.3)

In Eq. (B.2), the x, y, and z components of the operator α are defined in terms of the respective Pauli matrix given by Eq. (6.19). Looking for stationary solutions of the Dirac equation of the form: ( ΨD =

Ψ Ω

) =

( ) ( ) ( ) χ1 ϕ1 ϕ −i EℏT t , and χ = . , with ϕ = e ϕ2 χ2 χ

(B.4)

Substituting Eq. (B.4) in (B.1), we obtain: ( ) ( ) ( ) ( ) ( e ) ϕ ϕ ϕ ϕ = cα . p + A + mc2 β . + e Ao χ χ χ χ c ∆

ET

∆

∆

Utilizing Eq. (B.2), we can rewrite Eq. (B.5) in the following way:

(B.5)

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Annexes

)( ) ( ( ) ( ) ( e ) ϕ ϕ ϕ 2 I 0 cα . p + A . − mc = [E T − e Ao ] χ χ χ c 0 −I ∆

∆

∆

∆

(B.6)

But: (

∆

I 0 0 −I

mc2

∆

)( ) ( ) ϕ ϕ 2 = mc . χ −χ

(B.7)

This allows to rewrite Eq. (B.6) as: ( ) ( ( ) ) ( e ) ϕ ϕ ϕ − mc2 . = [E T − e Ao ] cα . p + A χ −χ χ c ∆

∆

(B.8)

Utilizing Eq. (B.3), we obtain: ]( ) ( )] [ ( ) [∑ z ( e ) ϕ 0 cσ i pi + ec Ai ϕ ( ) cα . p + A ai = , with a i = cσ i pi + ec Ai 0 χ χ c ∆

∆

∆

∆

∆

∆

∆

∆

i=x

(B.9) Or: ( ) ( ) ( ( e ) ϕ e ) χ cα . p + A . = σ. p + A ϕ χ c c ∆

∆

∆

∆

(B.10)

Utilizing Eq. (B.10), we can rewrite Eq. (B.8) in the following way: ( ) ( ( ) ) ( e ) χ ϕ ϕ cα . p + A − mc2 . = [E T − e Ao ] χ −χ ϕ c ∆

∆

(B.11)

This is equivalent to the system of two spinor equations (Eqs. A.12 and A.13): ( [ ] e ) σ . p + A χ = E T − e Ao − mc2 ϕ. c ( [ ] e ) σ . p + A ϕ = E T − e Ao + mc2 χ. c ∆

∆

∆

(B.12)

∆

(B.13)

Annex C: Classical Versus Quantum As discussed in Sect. 2.7, the observed stability of the atoms is thoroughly explained

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137

Fig. C.1 a There are no waves associated to a classical particle, b but two waves associated to a relativistic quantum particle

in non-relativistic quantum mechanics. This is accomplished by assuming that there is a wave associated with a quantum particle. This wave is described by a wavefunction that ends up as a solution of a wave equation. As discussed in Chap. 2, if the particle has spin-(s = 0), the wave equation is the Schrödinger equation. However, as discussed in Chap. 6, if the particle has spin-(s = 1/2), the wave equation is the Pauli equation. The introduction of the special theory of relativity in quantum mechanics produces fundamental consequences. One of them is that there is not a wave, but there are two waves associated to a relativistic quantum particle. In contrast with nonrelativistic quantum particles, relativistic quantum particles can exist in two kinds of quantum states. The total energy of the relativistic quantum particle is E T = E + mc2 when the particle is in the first kind of quantum state. These states are associated with wave functions that are solutions of a wave equation, which is directly related to a nonrelativistic wave equation. If the relativistic quantum particle has spin-(s = 0), the wave equation relates to the Schrödinger-like PPGP equation (Eq. A.1). In the nonrelativistic limit, Eq. (A.1) coincides with Schrödinger equation. However, if the relativistic quantum particle has spin-(s = 1/2), the wave equation is related to the Pauli-like PPGP equation (Eq. A.9). In the nonrelativistic limit, Eq. (A.9) coincides with Pauli equation. The PPGP equations can be formally obtained from the Schrödinger and Pauli equations by substituting the mass of the particle (m) by its relativistic effective mass μ, which is given by Eq. (A.1). This facilitates the simultaneous study of nonrelativistic and relativistic quantum mechanics. The total energy of the relativistic quantum particle is E T = E ' − mc2 when the particle is in in the second kind of quantum states. We call these states “exotic” because they do not exist in nonrelativistic quantum mechanics. Exotic states are associated with complementary wave equations. If the relativistic quantum particle has spin-(s = 0), the complementary wave equation is the complementary Schrödingerlike PPGP equation (Eq. A.2). However, if the relativistic quantum particle has spin(s = 1/2), the complementary wave equation is the complementary Pauli-like PPGP equation (Eq. A.10). The relativistic effective mass of the quantum particle in the complementary wave equations is not μ, but μ' given by Eq. (A.2).

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As illustrated in Fig. C.1, the most important and notable difference between a classical and a relativistic quantum particle is that there is not any wave associated with a classical particle, but there are two waves associated with a relativistic quantum particle. It is worth noting that there is never a single wave associated with a relativistic quantum particle. For instance, Eqs. (B.12) and (B.13) are valid for the electron, which is a spin-(s = 1/2) quantum particle. Therefore, χ does not exist if ϕ does not exist, and vice versa. Antiparticles are particles (other particles). Therefore, everything that we have discussed above about relativistic quantum particles is also valid for relativistic quantum antiparticles. There are two waves associated with a relativistic quantum antiparticle. However, an antiparticle without waves associated to it is then considered to be a classical antiparticle. We could then argue that something transcendental may occur if a relativistic quantum particle or antiparticle loses for any reason a wave, and then both waves associated to it. Interestingly, as it will be discussed below, there are good reasons to believe that relativistic quantum particles could get “naked” in extreme physical conditions. Atoms Cannot Be Too Heavy As discussed in Sect. 8.1, atoms with Z > 137 do not no exist. There is an attractive Coulombic interaction between the electron and the nucleus in a Hydrogen-like atom. This interaction produces bound states where the total energy of the electron is E T = E + mc2 . As sketched in Fig. 8.3, the electrons’ wavefunctions in these bound states (Ψ) are spatially localized around the nucleus. As shown in Fig. 8.1, the spatial localization of Ψ increases when Z increases. Eventually, Ψ collapses to a point when Z is too large. When this happens, the relativistic quantum particle loses the two waves associated to it (Ψ and Ω). The “naked” particle without waves cannot form a stable atom. As discussed in Sect. 2.7, this is a fundamental result of relativistic quantum mechanics. Consequently, there are no atoms with Z > 137. As shown in Fig. 8.1, the collapse of Ψ only occurs if the electron in the Hydrogenlike atom is theoretically described using relativistic quantum mechanics. This indicates that the inexistence of atoms with Z > 137 could be considered as an observational fact supporting the validity of both special theory of relativity and quantum mechanics. Quantum Objects Cannot Be Too Massive Large bodies with m > mP ≈ 22 μg are classical objects. This is a hypothesis presented in Sect. 8.2. This hypothesis argues that there should be a gravitational attractive interaction between different regions in a massive body. If this large body were a quantum object, then two waves (Ψ and Ω) should be associated to it. Consequently, the hypothetical internal interaction should produce spatially localized quantum states (Ψ), where the total energy of the quantum object is E T = E + mc2 . As shown in Fig. 8.2, the spatial localization of Ψ increases when m increases. Eventually, when m is too large, Ψ collapses to a point. When this happens, the relativistic quantum object loses the two waves associated to it (Ψ and Ω). The “naked” body without waves cannot be a quantum object, but a classical one.

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As shown in Fig. 8.2, the collapse of Ψ only occurs if the quantum object is theoretically described using relativistic quantum mechanics. It seems like we are often surrounded by macroscopic bodies that have an apparent classical nature. Our everyday experiences could then be considered as an enormous volume of observational facts supporting (1) the validity of the proposed hypothesis (a quantum object should gravitationally interact with itself), (2) special theory of relativity, and 3) quantum mechanics. We are not Surrounded by Antimatter Antimatter does not surround us, which is an observational fact. It is also a huge mystery because current physical theories predict that we should be surrounded by antimatter, but we are not. Nevertheless, an interesting hypothesis about this is discussed in Sect. 8.3. This hypothesis states that charged antimatter bodies containing |q| > qP ≈ 11e should be classical objects. In contrast, charged matter objects with any electrical charge should be quantum objects. The hypothesis places the rupture of the theoretical symmetry between matter and antimatter in the electrostatic interaction of a charged relativistic quantum body with itself. An antimatter charged relativistic quantum object should interact electrostatically with itself in a different manner than how a matter charged relativistic quantum body interacts electrostatically with itself. The hypothesis argues that there should be an electrostatic repulsive interaction between different regions in an extended antimatter object with charge q. If this antimatter body were a quantum object, then two waves (Ψ a and Ωa ) should be associated to it. Consequently, the hypothetical internal interaction should produce spatially localized “exotic” quantum states (Ωa ), where the total energy of the antimatter object is E Ta = E ' a − mc2 . We should note that the argument revolves around the spatial localization of exotic antimatter states Ωa . These states do not exist in nonrelativistic quantum mechanics. Therefore, this argument implies the theoretical framework of relativistic quantum mechanics. The hypothesis argues that the spatial localization of the wavefunction Ωa is produced by a repulsive interaction. This may be perceived with some skepticism, but this is because we have argued that the spatial localization of the wavefunction Ψ is produced by an attractive interaction. Certainly, as sketched in Fig. 8.4, the repulsive Coulomb interaction between a positron and the nucleus of a Hydrogenlike atom produces the spatial delocalization of the positronic wavefunction Ψ a . However, as sketched in Fig. 8.5, the same repulsive interaction produces the spatial localization of the exotic positronic wavefunction Ωa . In relativistic quantum mechanics, attractive external interactions may produce the spatial localization of Ψ and Ψ a , but the spatial delocalization of the exotic states Ω and Ωa . In contrast, repulsive external interactions may produce the spatial localization of the exotic states Ω and Ωa , but the spatial delocalization of Ψ and Ψ a . The equations of relativistic quantum mechanics do not include a possible interaction of a particle or antiparticle with itself. However, the hypothesis that we are considering explores the possibility of such interactions.

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As shown in Fig. 8.7, the spatial localization of the exotic state Ωa increases when |q| increases. Eventually, when |q| is too large, Ωa collapses to a point. When this happens. the antimatter quantum object loses the two waves associated to it (Ωa and Ψ a ). The “naked” antimatter body without waves cannot be a quantum object, but a classical one. We should emphasize and elaborate on how the theoretical symmetry between matter and antimatter is broken in this hypothesis. The internal repulsive Coulomb interaction spatially delocalizes both the matter wavefunction Ψ and the antimatter wavefunction Ψ a . However, this repulsive interaction spatially localizes the antimatter exotic wavefunction Ωa , but spatially delocalizes the matter wavefunction Ω. This hypothesis does not explain the reason Mother Nature behaves in this way. However, it correctly predicts that Mother Nature should act as it seems to be. The rest of the argument is straightforward. Antimatter atoms with Z ≥ 12 cannot exist because a cloud formed by 12 positrons would be a “naked” antimatter object without waves, which is a classical object. Charged classical antimatter cannot form antimatter atoms. Antimatter atoms are needed for the existence of antimatter living beings. We are surrounded by matter, but not by the existence of antimatter. This is because of how the condition “classical if |q| > qP ≈ 11e” does not apply to matter. As stated above, the collapse to a point of Ωa only occurs if the antimatter object is theoretically described using relativistic quantum mechanics. We are not surrounded by antimatter. This everyday experience could then be considered as a humongous aggregate of observational facts supporting the validity of (1) the proposed hypothesis (a charged quantum object should electrically interact with itself), (2) special theory of relativity, and (3) quantum mechanics. Primordial Black Holes and Antimatter Electrical Sinks Primordial black holes with a relatively small mass may exist. Primordial black holes may have been created around 13 billion years ago, at the beginning of our universe. Mass fluctuations with m > mP could have produced their formation. As discussed above, these hypothetical mass fluctuations may have formed primordial relativistic quantum objects. If their masses were larger than Planck’s mass, then the collapse to a point of Ψ may have created primordial black holes. At present, there is no observational evidence of the existence of primordial black holes. Nevertheless, the possible existence of these small mass black holes is a research topic of great interest in the modern day. Similarly, primordial antimatter electric sinks may have been created around 13 billion years ago, at the beginning of our universe. Antimatter charge fluctuations with |q| > qP could have produced their formation. As discussed above, these hypothetical antimatter charge fluctuations may have formed primordial relativistic quantum objects. If their charges were larger than Planck’s charge, then the collapse to a point of Ωa may have created primordial electric sinks. This may explain the existence of an excess of charged matter in the rest of the universe.

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Annex D: Antiparticles As discussed in Sects. 3.3 and 4.3, an antiparticle is another particle. Historically, the existence of electrons was widely accepted before positrons were discovered in a cloud chamber in 1932. It was later proposed that there exists a relationship between positrons and the theoretically predicted Dirac’s quantum states, where the electron total energy is E T = E ' − mc2 . This relationship is illustrated in Fig. 3.3. As mentioned in Sect. 3.2, it was initially difficult to accept the existence of quantum states where the kinetic energy of a particle is negative. This motivated Schrödinger to put forth his undivided attention on a nonrelativistic wave equation. Today, we know that electrons and positrons are always created in pairs. Which one we call the particle, the electron, or the positron, is just an historical accident. Electrons are commonly referred to as particles because they were discovered before the positron. However, it would be strictly correct to refer to positrons as particles and electrons as the antiparticles associated to them. It is worth noting that if the hypothesis discussed in the previous Annex were to be correct, then we should solely be surrounded by matter. This would transform the historical accident into a necessity. Electrons are not antiparticles, but particles because we are surrounded and formed by them. The positron is the antiparticle of the electron, so positrons must be scarce. This is why they were discovered later. We could differentiate a charged particle from a charged antiparticle because we are not surrounded and formed by antiparticles, but by only particles instead. Nevertheless, we will only consider the interaction of particles and antiparticles with the external world from now on. In this framework, relativistic quantum mechanics predict the existence of a complete matter-antimatter symmetry. Today, we know that there is a relationship between the “exotic” particle’s states (Ω), where E T = E' − mc2 , and the antiparticle’s states (Ψ a ) where E Ta = E a + mc2 . As expected from the theoretical matter-antimatter symmetry, there is also a relationship between the exotic antiparticle’s states (Ωa ), where E Ta = E ' a − mc2 , and the particle’s states (Ψ) where E T = E + mc2 . These relationships are stated in Theorem III in Annex A. This means that given a common external world, if we know E a and Ψ a , we could obtain E' and Ω in the following way (Eqs. A.26 and A.41): E ' = −E a , and

Ω = Ψa∗ if s = 0 . Ω = Ψa† if s = 1/2

(D.1)

Also, if we know E and Ψ, we could obtain E' a and Ωa in the following way (Eqs. A.31 and A.47): E a' = −E, and

Ωa = Ψ ∗ if s = 0 . Ωa = Ψ † if s = 1/2

(D.2)

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Therefore, given a common external world, we only need to solve the PPGP equations that correspond to the particle and antiparticle of “regular” quantum states. If s = 0, these equations are (Eqs. A.1 and A.20): ] [ ℏ2 E − V (r ) ∂ 2 m. ∇ Ψ + V (r )Ψ, with μ(r ) = 1 + iℏ Ψ = − ∂t 2μ(r ) 2mc2

(D.3)

And: iℏ

] [ ℏ2 ∂ E a + V (r ) m. Ψa = − ∇ 2 Ψa − V (r )Ψa , with μa (r ) = 1 + ∂t 2μa (r ) 2mc2 (D.4)

If s = 1/2, these equations are (Eqs. A.9 and A.33): [ ] ( ( e ) e ) 1 ∂ σ . p − A Ψ + V Ψ, V = e Ao . iℏ Ψ = σ . p − A c 2μ c ∂t ∆

∆

∆

∆

(D.5)

And: ] ( [ ( e ) 1 ∂ e ) σ . p + A Ψa − V Ψa . Ψa = σ . p + A c 2μa ∂t c ∆

iℏ

∆

∆

∆

(D.6)

All these equations are Schrödinger-like or Pauli-like equations. This makes it possible to simultaneously learn relativistic and nonrelativistic quantum mechanics.

Annex E: A Relativistic Quantum Particle Confined in a Small Spatial Region As discussed in Chap. 2, the simplest model for explaining the stability of atoms is a quantum particle confined in a one-dimensional infinite well. This model captures the essential properties of a quantum particle, including that it has a wave associated to it. This wave is described by a wavefunction that can be obtained by solving a wave equation. The spatial localization of the wave associated to the quantum particle results in a discrete set of possible energy values. The minimum amount of energy possible is E = K > 0. Consequently, like an electron in the Hydrogen atom, the spatially confined quantum particle cannot lose all its kinetic energy. In Sect. 3.6, the model was improved from the consideration that the wave equation that should be solved was not the Schrödinger equation, but the Schrödingerlike PPGP equation instead. For pedagogical simplicity, some naïve suppositions were introduced in Sect. 3.6. First, like in nonrelativistic quantum mechanics, it was assumed that the wavefunction of the relativistic quantum particle should be null outside of the infinite well. This naïve assumption was addressed in Chap. 4. It was

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Fig. E.1 A charged spin-(s = 0) particle a and b its antiparticle, that interact with the same external world, experience potentials with opposite signs

discussed that, in contrast with the mass of the particle, the effective mass of a relativistic quantum particle (μ) is not constant. The Klein paradox was discussed, and it occurs because μ < 0 if V > 2mc2 . Second, like in nonrelativistic quantum mechanics, it was assumed that there is a single wavefunction (Ψ ) associated to a relativistic quantum particle. Consequently, the total energy of the particle was assumed to be E T = E + mc2 . We know now that there also is a second wavefunction (Ω) associated to a relativistic quantum particle. The total energy of the particle in these exotic states is not E T = E + mc2 , but instead E T = E ' − mc2 . In these exotic states the kinetic energy of the particle is negative. For completing the relativistic quantum mechanics description of a charged particle (with spin s = 0) in a one-dimensional infinite well, we should solve the complementary Schrödinger-like PPGP equation (Eq. A.2): iℏ

] [ ℏ2 ∂ E ' − V (x) m. (E.1) Ω = − ' ∇ 2 Ω + V (x)Ω, with μ' (x) = −1 + ∂t 2μ (x) 2mc2

Alternatively, if we assume a charged antiparticle that experiences the same external world than the charged particle, we should solve the Schrödinger-like PPGP equation for the antiparticle (Eq. A.20): iℏ

] [ ∂ ℏ2 E a + V (x) m. Ψa = − ∇ 2 Ψa − V (x)Ψa , with μa (x) = 1 + ∂t 2μa (x) 2mc2 (E.2)

The potentials V (x) and −V (x) are sketched in Fig. E.1a, b, respectively. The energies (E a ) of the antiparticle in the state equal to Ψ a is E a > − V (x) everywhere. In addition, the antiparticle’s relativistic effective mass is μa > 0 everywhere. Therefore, there are no bound solutions of Eq. (E.2). Consequently, a continuous energy spectrum is associated to the antiparticle. Utilizing Eq. (D.1), we can then predict that there are no bound solutions of Eq. (E.1). There is a continuous amount of possible energy values, E ' < 0, if the particle is in an exotic state Ω. Therefore, in the exotic

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states Ω, the kinetic energy of the relativistic quantum particle is negative inside of the infinite well. The energy distribution of the electronic states in the Hydrogen atom is sketched in Figs. 8.3a and 8.5c. Unsurprisingly, if we retain the naïve assumption that Ψ is null outside of the infinite well, then the energy distribution of particle’s states in the infinite one-dimensional well is also like the one sketched in these figures. Like the Coulomb potential in the Hydrogen atom, V (x), corresponds to an attractive interaction of the particle with the external world. This attractive external interaction spatially localizes ψ but delocalizes Ω. Particle’s Ψ states are bound, but exotic particle’s Ω states are unbound. There is then a discrete set of energy values, E T = E + mc2 , but a continuous amount of energy values, E T = E ' − mc2 . As sketched in Fig. 8.4a, to an empty exotic particle state Ω corresponds the energy E ' < 0. Thus in Eq. (D.1), E' < V (x) and μ' < 0 everywhere. This corresponds to an unbound hole in the Dirac’s Sea. The relativistic quantum antiparticle tends to sink to the Ψ a state with minimum positive energy (Fig. 8.4b), while a hole (an empty exotic particle’s state) tends to float in the Dirac’s Sea to the exotic particle’s state Ω with maximum negative energy (Fig. 8.4a). As sketched in Fig. 8.5, a particle is the antiparticle of the antiparticle. The antiparticle’s wavefunctions Ψ a can be obtained by solving Eq. (E.2). As discussed above, the solutions of Eq. (E.2) correspond to unbound antiparticle’s states and a continuous set of energy values, E Ta = E a + mc2 . The antiparticle’s wavefunctions Ω a can be obtained by solving the following equation (Eq. A.21): ] [ ℏ2 E a' + V (x) ∂ 2 ' m. (E.3) ∇ Ωa − V (x)Ωa , with μa = −1 + iℏ Ωa = − ' ∂t 2μa (r ) 2mc2 The potential −V (x) is sketched in Fig. E.1b. We do not need to solve Eq. (E.3) because we already know the solution of the particle wave equation (Eq. A.1): iℏ

] [ ℏ2 E − V (r ) ∂ Ψ=− ∇ 2 Ψ + V (r )Ψ, with μ(r ) = 1 + m. ∂t 2μ(r ) 2mc2

(E.4)

In Sect. 3.6, assuming that Ψ is null outside of the infinite well, Eq. (E.4) was solved. It was found that Eq. (E.4) has bound solutions (Ψ ) and there is a discrete set of energies, E T = E + mc2 . The energy that corresponds to a bound state Ψ is E > 0; therefore, E > V (x) inside of the well but E < V (x) outside of it. Due to Eq. (D.2), Eq. (E.3) has bound solutions (Ω a = Ψ *) and there is a discrete set of energies, E T = E ' a − mc2 . In contrast with the particle states, the energy that corresponds to a bound exotic antiparticle state Ω a is E ' a = −E < 0. Therefore, E ' a < V (x) inside of the well, but E ' a > V(x) outside of it. As sketched in Fig. 8.5b, an empty exotic antiparticle state Ω a corresponds to a bound hole in the Dirac’s Sea. The relativistic quantum particle tends to sink to the Ψ state with minimum positive energy (Fig. 8.5c), while a hole (an empty exotic antiparticle’s state) tends to float in the Dirac’s Sea to the exotic antiparticle state Ω a with maximum negative energy (Fig. 8.5b).

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Fig. E.2 Distribution of energy states that correspond to a a charged spin-(s = 0) particle and b its antiparticle moving in the same external world. The external world confines the particle in an infinite well

The energy distribution of the positronic states, which are a result of the repulsive interaction of a positron with the nucleus of a Hydrogen atom, is sketched in Fig. 8.5b. Foreseeable, if we retain the naïve assumption that Ψ is null outside of the infinite well, and if the antiparticle interacts with the same infinite one-dimensional well that traps the particle, then the energy distribution of antiparticle’s states is also like the one sketched in Fig. 8.5b. Like the repulsive Coulomb interaction between a positron and the nucleus of a Hydrogen atom, −V (x) corresponds to a repulsive interaction of the antiparticle with the external world. This repulsive external interaction spatially delocalizes Ψ a but localizes the exotic antiparticle states of Ω a . Antiparticle’s Ψ a states are unbound, but exotic antiparticle’s Ω a states are bound. There is then a continuous set of energy values, E Ta = E a + mc2 , but discrete set of energy values, E Ta = E ' a − mc2 . The discussion above is summarized in Fig. E.2. As sketched in Fig. E.2a, the attractive interaction of a charged particle with the potential V (x), which is produced by the external world that surrounds the particle, spatially localizes the particle states Ψ . This produces a discrete set of energy values, E T = E + mc2 . The same external attractive interaction spatially delocalizes the exotic particle states Ω. This produces a continuous set of energy values, E ' T = E ' − mc2 . An observable particle trapped in the infinite well corresponds to all the exotic particle states Ω being occupied by particles. In addition, the observed particle occupies the bound particle state Ψ with minimum energy in the ground state. As sketched in Fig. E.2b, the repulsive interaction of the charged antiparticle with the same external world that surrounds the particle, spatially delocalizes the antiparticle states Ψ a . This produces a continuous set of energy values, E Ta = E a + mc2 . The same external repulsive interaction spatially localizes the exotic antiparticle states Ω a . This produces a discrete set of energy values, E ' Ta = E ' a − mc2 . An observable antiparticle, that is in the same infinite well that traps the particle, corresponds to

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all the exotic antiparticle states Ω a being occupied by antiparticles. In addition, the observed antiparticle occupies an unbound antiparticle state (Ψ a ).

Annex F: Mathematical Formalism of Quantum Mechanics In this Annex, without being extremely careful about the mathematical intricacies involved in this subject, we want to give the reader a general idea about the mathematical apparatus of quantum mechanics. For simplicity, we will refer to nonrelativistic quantum mechanics where there is a wave associated with a quantum particle. This wave is mathematically represented by a wavefunction. Wavefunctions are solutions of linear wave equations and are multi-dimensional vectors of a Hilbert space. Several mathematical topics should be extensively covered to give a practical meaning to the previous sentences. Linear Algebra in Quantum Mechanics [In quantum mechanics, a vector space consists of a set of vectors (wavefunctions |α〈, | β〈, |γ 〈,…), together with a set of scalars (complex numbers a, b, c, …) which are subject to two operations of vector addition and scalar multiplication]. The vector addition satisfies the following properties. The sum of any two vectors is another vector: |α〈 + |β〈 = |γ〈.

(F.1)

The vector addition is commutative: |α〈 + |β〈 = |β〈 + |α〈.

(F.2)

|α〈 + (|β〈 + |γ 〈) = (|α〈 + |β〈) + |γ〈.

(F.3)

And associative:

There exist a zero (or null) vector, |0〈, with the property that for every vector |α〈: |α〈 + |0〈 = |α〈.

(F.4)

And for every vector |α〈, there is an associated inverse vector |-α〈, such that: |α〈 + | − α〈 = |0〈.

(F.5)

The scalar multiplication satisfies the following properties. The product of any scalar with any vector is another vector: a|α〈 = |β〈.

(F.6)

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Scalar multiplication is distributive with respect to vector addition: a(|α〈 + |β〈) = a|α〈 + a|β〈.

(F.7)

Also, scalar multiplication is distributive with respect to scalar addition: (a + b)|α〈 = a|α〈 + b|α〈.

(F.8)

It is also associative with respect to the ordinary multiplication of scalars: a(b|α〈) = (ab)|α〈.

(F.9)

From these properties it follows that: 0|α〈 = |0〈 and | − α〈 = (−1)|α〈.

(F.10)

[Always remember that, in quantum mechanics, wavefunctions are vectors of a linear algebra and the scalars are complex numbers]. A linear combination of the vectors |α〈, |β〈, |γ〈, … is an expression of the form: a|α〈 + b|β〈 + c|γ 〈 + . . .

(F.11)

A vector | λ > is said to be linearly independent of the set of vectors |α〈 , |β〈, |γ〈, … if it cannot be written as a linear combination of these vectors. By definition, a set of vectors is linearly independent if each one is linearly independent of all the rest. [For example, any polynomial equation can be represented as a linear combination of other polynomials. If we choose our set of linearly independent vectors to include all real polynomials, {φ} = a + bx + cx 2 + · · · , then the equation f (x) = x + 3x 2 can be formed using coefficients a = 0, b = 1, c = 3 and all other coefficients equal to 0. This is precisely what it means to be a linear combination. Note that the components of this linear combination cannot be represented as a linear combination themselves. This means x cannot be written as an addition (combination) of x 2 or x 3 and so on]. A collection of vectors is said to span the “space formed by all vectors” if every vector can be written as a linear combination of the members of this set. A set of linearly independent vectors that span a space is considered to be a basis. The number of vectors in any basis is called the dimension of the space. [For instance, the wavefunctions given by Eq. (2.54) in Chap. 2 (which are solutions of the time independent Schrödinger equation for a one-dimensional infinite well), are the vectors: / |ϕ n 〈 = ϕn (x) =

( nπ ) 2 sin x , with n = 1, 2, . . . L L

(F.12)

These vectors form a basis of the space formed by all the wavefunctions φ(x) in the form:

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|φ〈 = φ(x) =

+∞ ∑

cn ϕn (x).

(F.13)

n=1

The dimension of this space of functions is infinite because there are infinite linearly independent functions ϕ n (x)]. For the moment, let us assume that the dimension of a space (N) is finite. With respect to a prescribed basis: |θ 1 〈, |θ 2 〈, . . . |θ N 〈.

(F.14)

Any given vector: |α〈 =

N ∑

an |θ 1 〈.

(F.15)

n=1

It is uniquely represented by the ordered N-tuple of its components: |α〈 ↔ (a1 , a 2 , . . . , a N ).

(F.16)

This is of practical importance because it is often easier to work with the component than with the abstract vectors themselves. For instance, to add vectors, we add their corresponding components: |α〈 + |β〈 ↔ (a1 + b1 , a2 + b2 , . . . , a N + b N ).

(F.17)

To multiply by a scalar, you multiply both components: c|α〈 ↔ (ca1 , ca2 , . . . , ca N ).

(F.18)

The null vector is represented by a string of zeros: |0〈 ↔ (0, 0, . . . , 0)

(F.19)

| − α〈 ↔ (−a1 , −a2 , . . . , −a N )

(F.20)

And:

Inner Products The inner product of two vectors (|α〈 and |β〈), is a complex number (which we write 〈α|β〈), with the following properties: 〈β|α〈 = 〈α|β〈∗ , and 〈α|α〈

= 0 if |α〈 = |0〈 . > 0 if |α〈 /= |0〈

(F.21)

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And: 〈α|(b|β〈 + c|γ 〈)〈 = b〈α|β〈 + c〈α|γ 〈.

(F.22)

A vector space with an inner product is called an inner product space. Because Eq. (F.21), the inner product of any vector with itself is a nonnegative number, meaning its corresponding square root is real. We call this the norm of the vector: ||α〈| =

√

〈α|α〈.

(F.23)

A vector is said to be normalized if its norm is equal to 1. Two vectors whose inner product equals zero are called orthogonal. A collection of mutually orthogonal normalized vectors is called an orthonormal set: 〈αi |αi 〈 = δi j .

(F.24)

It is always possible and convenient to work using an orthonormal basis. If this is the case, then the inner product of two vectors can be written very neatly in terms of their components: 〈α|β〈 = a1∗ b1 + a2∗ b2 + · · · + a ∗N b N .

(F.25)

The norm becomes: ||α〈| =

/

|a1 |2 + |a2 |2 + · · · + |a N |2 .

(F.26)

And the components of vector given by Eq. (F.15) are: an = 〈θn |α〈.

(F.27)

[Therefore, the wavefunctions given by Eq. (F.12) form an inner product space of dimension that is infinite. These functions form an orthonormal set because: ( ' ) ( ∫ L ∫ nπ ) nπ 2 L ∗ x sin x d x = δn ' n . sin 〈ϕn ' |ϕn 〈 = [ϕn ' (x)] ϕn (x)d x = L 0 L L 0 (F.28) Any other vector, |φ〈, can be represented as: |φ〈 = φ(x) =

+∞ ∑ n=1

cn ϕn (x), with cn = 〈θn |φ〈 =

+∞ ∑ n=1

cn 〈ϕn ' |ϕn 〈.

(F.29)

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Linear Transformations [In quantum mechanics, operators are linear transformations that transform any vector (wavefunction) of an inner product space into another vector (function) of the same space: ∆

T |α〈 = |γ 〈.

(F.30)

The transformation is linear if and only if for any two vectors and scalars: ∆

∆

∆

T (a|α〈 + b|β〈) = aT |α〈 + bT |β〈.

(F.31)

In linear algebra, the theory of linear transformations is related to the theory of matrices. [In this book, we have primarily used operators in their differential form (Table 2.1 in Chap. 2 contains some instances). This corresponds to the formulation of wave mechanics in quantum mechanics. Quantum mechanics can also be formulated in an equivalent matrixial form. Historically, Heisenberg’s matrixial formulation of quantum mechanics was discovered shortly before the Schrödinger wave-based proposal. Below, we will summarize some important results of the theory of linear transformations in its matrixial form]. Let us suppose an inner product space of finite dimension (N), and an orthonormal set that allows for writing any vector (function) of the space as: |φ〈 = φ(x) =

N ∑

d j |θ j 〈, with d j = 〈θ j |φ〈, and 〈θ j ' |θ j 〈 = δ j ' j .

(F.32)

j=1

There is then a one-to-one correspondence between φ(x) and the list of N complex numbers (d 1 , d 2 , …, d N ). We can then represent a linear transformation of the vector |φ〈 in two equivalent forms: ∆

T |φ〈 = |γ 〈 =

N ∑

c j |θ j 〈.

(F.33)

j=1

Or: ⎡

⎤ ⎡ d1 c1 ⎢ d2 ⎥ ⎢ c2 ⎥ ⎢ T⎢ ⎣ ... ⎦ = ⎣ ... φN cN

∆

⎤ ⎥ ⎥. ⎦

(F.34)

The second form suggests that the linear transformation can be represented by a matrix. From Eq. (F.32), it follows that:

Annexes

151 ∆

〈θi |T|φ〈 =

N ∑

∆

d j 〈θi |T|θ j 〈.

(F.35)

j=1

From Eq. (F.33), it follows that: 〈θi |γ 〈 = ci .

(F.36)

Therefore, from Eqs. (F.33) to (F.36), it follows that: ∆

〈θi |T|φ〈 =

N ∑

∆

d j 〈θi |T|θ j 〈 = 〈θi |γ 〈 = ci .

(F.37)

j=1

Or, in matrix notation: ⎡

⎤ ⎡ d1 c1 ⎢ d2 ⎥ ⎢ c2 ⎥ ⎢ T⎢ ⎣ ... ⎦ = ⎣ ... dN cN

∆

⎛

⎤

T11 ⎜ T21 ⎥ ⎥, with T = ⎜ ⎝ ... ⎦ TN 1 ∆

T12 T22 ... TN 1

⎞ . . . T1N . . . T2N ⎟ ⎟, and Ti j = 〈θi |T|θ j 〈. ... ... ⎠ . . . TN N (F.38) ∆

The study of linear transformations then reduces to the theory of matrices. The sum and multiplication of linear transformations are defined and have the properties of the sum and multiplication of matrices: (

) ) ( S + T |α〈 = S |α〈 + T |α〈 ⇐⇒ S + T = S i j + Ti j .

∆

∆

∆

∆

∆

∆

∆

ij

(F.39)

The product of two linear transformations is the net effect of performing them in succession. The multiplication of matrices is not commutative, so the order in which the linear transformations are applied is important: (

) ( ( ) ( ) ) S T |α〈 = S T |α〈 /= T S |α〈 = T S |α〈 .

∆∆

∆

∆

∆∆

∆

∆

(F.40)

The elements of the matrix resulting from the multiplication of two matrices are: (

)

∆∆

ST

ij

=

N ∑

∆

S ik Tk j .

The commutator of two matrices is: ] [ S,T = S T − T S . ∆∆

(F.41)

k=1

∆∆

∆∆

(F.42)

152

Annexes

Two matrices commutative if and only if their commutator is null. [Heisenberg’s uncertainty principle is a consequence of the non-commutativity of the multiplication of matrices. In wave quantum mechanics, a quantum particle cannot simultaneously have well-determined values of its position and linear momentum because these operators do not commute: ∆∆

x p ϕ(x) = −iℏx

∂ ∂ ϕ(x) /= p x ϕ(x) = −iℏ [xϕ(x)]. ∂x ∂x ∆∆

(F.43)

The transpose of a matrix is the same set of elements, but with rows and columns interchanged: ⎛

T11 ⎜ T21 T =⎜ ⎝ ... TN 1

∆

T12 T22 ... TN 2

⎞ ⎛ . . . T1N ∼ ⎜ . . . T2N ⎟ ⎟ ⇒T= ⎜ ⎠ ⎝ ... ... . . . TN N

T11 T12 ... T1N

T21 T22 ... T2N

⎞ . . . TN 1 . . . TN 2 ⎟ ⎟. ... ... ⎠ . . . TN N

(F.44)

Notice and note the hat ~ used for symbolizing the transpose matrix. Also, make a note that the transpose of a column matrix is a row matrix and vice versa: ⎤ φ1 ⎢ φ2 ⎥ ⎥ ∼ |φ〈 ↔ ⎢ ⎣ . . . ⎦ ⇒ |φ〈 ↔ (φ1 , φ2 , . . . , φ N ) φN ⎡

(F.45)

The transpose of a product of matrices is the product of the transposes in reverse order: ∆

∆∆

∼∼

If P = S T then P˜ =TS .

(F.46)

The complex transpose conjugate of a matrix (T† ) is the complex conjugate of its transpose: ⎛

T11 ⎜ T21 Tˆ = ⎜ ⎝ ... TN 1

T12 T22 ... TN 1

⎞ ⎛ ∗ . . . T1N T11 ( )∗ ⎜ T ∗ . . . T2N ⎟ ⎟ ⇒ T † = T˜ = ⎜ 12 ⎝ ... ... ... ⎠ ∗ . . . TN N T1N

∗ T21 ∗ T22 ... ∗ T2N

⎞ . . . TN∗ 1 . . . TN∗ 2 ⎟ ⎟ ... ... ⎠ . . . TN∗ N

(F.47)

T † is also known as the Hermitian conjugate or adjoint of the matrix T. A square matrix is Hermitian if it is equal to its Hermitian conjugate: Tˆ = T † For instance, the following matrix is Hermitian:

(F.48)

Annexes

153

⎛

⎞ 1 i i T = ⎝ −i 1 −i ⎠. −i i 1

∆

(F.49)

The Hermitian conjugate of a product of matrices is the product of the adjoints in reverse order: If Pˆ = Sˆ Tˆ then P † = T † S †

(F.50)

Equation (F.48) gives a definition of a Hermitian transformation. An alternative definition of a Hermitian linear transformation involves the inner product of any two vectors |α〈 and |β〈 in the following way: T † α|β = α|Tˆ β

(F.51)

This means that a linear transformation is Hermitian if and only if the inner product of the vectors T † |α〈 and |β〈 (〈T † α| β〈) is equal to inner product of the vectors: ( ) |α〈 and T |β〈 〈α|T β〈 . ∆

∆

Most of the operators (linear transformations) used in quantum mechanics are Hermitian. For instance, in Chap. 2, the time independent Schrödinger equation for a particle in the one-dimensional infinite well is: ∆

H ϕn (x) = E n ϕn (x).

(F.52)

〈ϕn ' |H ϕn (x)〈 = E n 〈ϕn ' |ϕn 〈 = E n δn ' n .

(F.53)

Therefore: ∆

Also: ∆

〈H ϕn (x)|ϕn ' 〈 = E n 〈ϕn |ϕn ' 〈 = E n δn ' n .

(F.54)

This implies that the total energy operator is Hermitian: Hˆ = Hˆ †

(F.55)

Eigenvectors and Eigenvalues Eigenvector equations are equations of the form: ∆

T |α〈 = λ|α〈.

(F.56)

154

Annexes

The vectors (|α〈) and scalars (λ) satisfying Eq. (F.56) are name eigenvectors and eigenvalues, respectively. [Time independent Schrödinger equations are eigenvector equations. A large part of most introductory courses of quantum mechanics is dedicated to the development of mathematical skills needed for solving eigenvector equations]. The eigenvectors and eigenvalues of Hermitian transformations (operators) have three important properties, which will be explained below. First, the eigenvalues of a Hermitian transformation are real. Demonstration Let λ be an eigenvalue of Eq. (F.56), with |α〈 /= |0〈. Therefore: ∆

〈α|T α〈 = 〈α|λα〈 = λ〈α|α〈.

(F.57)

Meanwhile, if the transformation is Hermitian, then: 〈α|Tˆ α〈 = T † 〈α|α〈 = λ∗ 〈α|α〈

(F.58)

But due to Eq. (F.21), 〈α|α〈 /= 0, so λ = λ* , and hence λ is real. [In quantum mechanics, as discussed Sect. 2.3, a Hermitian operator is associated to each observable physical magnitude. The reason for this is that measurements of physical magnitudes always return as real numbers. The real eigenvalues of the corresponding eigenvector equation are the only possible values of the physical magnitude associated to a Hermitian operator]. Second, the eigenvectors of a Hermitian transformation belonging to distinct eigenvalues are orthogonal. Demonstration Suppose that: ∆

∆

T |α〈 = λ|α〈, and T |β〈 = μ|β〈, with λ /= μ.

(F.59)

Then: ∆

〈α|T |β〈 = 〈α|μβ〈 = μ〈α|β〈.

(F.60)

If the transformation is Hermitian: 〈α|Tˆ |β〈 = T † 〈α|β〈 = λ〈α|β〈 = λ∗ 〈α|β〈

(F.61)

But λ = λ* (from the first property), and λ /= μ, by assumption, so < α|β > = 0. Third, the eigenvectors of a Hermitian transformation span the vector space. This property will be given here without demonstration.

Annexes

155

[These properties together, with the linearity of the Schrödinger equation, justify the assertion made in Chap. 2.58 about Eq. (2.7) being the general solution of Eq. (2.49). This also justifies Eq. (2.59)]. Hilbert Space A complete inner product space is called a Hilbert space. Complete could mean that the space is formed by all square-integrable functions on the interval −1 < x < +1. Technically, it is called L2 (−1, +1). More generally, the set of all square-integrable functions in the interval a < x < b is L2 (a, b). The Hilbert space L2 (−∞, +∞) is most often used in quantum mechanics. [For instance, Eq. (F.28) shows that the wavefunction ϕn (x) are vectors of the Hilbert space L2 (−∞, +∞)].

Annex G: About the Non-linearity of the PPGP Equations The mathematical formalism of quantum mechanics (linear algebra, linear transformations, inner product spaces, and Hilbert spaces), requires linear wave equations. The Schrödinger equation is a linear equation. Also, the Klein-Gordon and Dirac equations are linear. However, the PPGP equations are not strictly linear. For instance, let us consider the Schrödinger-like PPGP equation for a spin-(s = 0) particle in a constant potential (Eq. 3.86): ) ( ℏ2 d 2 ∂ E−V m. iℏ Ψ = − Ψ + V Ψ, with μ = 1 + ∂t 2μ d x 2 2mc2

(G.1)

This equation is not linear because μ depends on E. If Ψ 1 and Ψ 2 are respectively distinct solutions of Eq. (G.1) with E = E 1 and E = E 2 , respectively, then the effective mass of the relativistic quantum particle is different in these quantum states: ) ) ( ( E2 − V E1 − V m / = μ m. = 1 + If E 1 /= E 2 ⇒ μ1 = 1 + 2 2mc2 2mc2

(G.2)

Therefore, Ψ 1 satisfies the following equation: iℏ

∂ ℏ2 d 2 Ψ1 = − Ψ1 + V Ψ1 . ∂t 2μ1 d x 2

(G.3)

While Ψ 2 satisfies a different equation: iℏ

ℏ2 d 2 ∂ Ψ2 = − Ψ2 + V Ψ2 . ∂t 2μ2 d x 2

However, in general, a linear combination of these two solutions:

(G.4)

156

Annexes

Ψ = aΨ1 + bΨ2 .

(G.5)

It is a solution of neither Eq. (G.3) nor (G.4), and it is not a solution of Eq. (G.1). Nevertheless, as demonstrated in Annex A, we can find the solution of the KleinGordon equation with E T = E + mc2 by solving Eq. (G.1). This means that the wavefunction Ψ given by Eq. (G.5) is a solution of the Klein-Gordon equation. Similarly, as discussed in Annex A, by solving the following complementary Schrödinger-like PPGP equation: ) ( ℏ2 d 2 ∂ E' − V ' m. iℏ Ω = − ' 2 Ω + V Ω, with μ = −1 + ∂t 2μ d x 2mc2

(G.6)

We can find two distinct solutions Ω 1 and Ω 2 of Eq. (G.6) with E ' = E ' 1 and E ' = E ' 2 , which are also solutions of the Klein-Gordon equation with E T = E ' − mc2 . The effective mass of the relativistic quantum particle is different in these quantum states: ) ) ( ( E 2' − V E 1' − V ' ' ' ' m /= μ2 = −1 + m. (G.7) If E 1 /= E 2 ⇒ μ1 = −1 + 2mc2 2mc2 Therefore, Ω 1 satisfies the following equation: iℏ

∂ ℏ2 d 2 Ω1 = − ' Ω1 + V Ω1 . ∂t 2μ1 d x 2

(G.8)

While Ω 2 satisfies a different equation: iℏ

∂ ℏ2 d 2 Ω2 = − ' Ω2 + V Ω2 . ∂t 2μ2 d x 2

(G.9)

However, in general, a linear combination of these two solutions: Ω = aΩ1 + bΩ2 .

(G.10)

It is a solution of neither Eq. (G.8) nor (G.9), and it is not a solution of Eq. (G.6). Nevertheless, the wavefunction Ω, given by Eq. (G.10), is a solution of the KleinGordon equation. Consequently, a general solution of the Klein-Gordon equation can be written as a linear superposition of all the distinct solutions of Eqs. (G.1) and (G.6). For similar reasons, the Pauli-like PPGP equation for a spin-(s = 1/2) charged particle, in a constant electrostatic potential, is also non-linear: ) [ ] ( 1 E−V ∂ σ . pΨ + V Ψ, V = e Ao , with μ = 1 + m. Ψ = σ.p 2μ 2mc2 ∂t (G.11) ∆

iℏ

∆

∆

∆

Annexes

157

In this case, μ is constant, so Eq. (G.11) can be simplified to: ( )2 σ.p ∂ ℏ2 Ψ= Ψ + V Ψ = − ∇ 2 Ψ + V Ψ. ∂t 2μ 2μ ∆

iℏ

∆

(G.12)

Therefore, each component of the spinor Ψ satisfies de Grave de Peralta (GP) equation: iℏ

∂ ℏ2 ψi = − ∇ 2 ψi + V ψi , with Ψ = ∂t 2μ

(

) ψ1 , i = 1, 2. ψ2

(G.13)

In the non-relativistic limit, E − V