Group Theory and Quantum Mechanics 038706740X, 9780387067407

Citation preview

B. L. van der Waerden

Group Theory and Quantum. Mechanics

With 10 Figures

Springer-Verlag New York Heidelberg Berlin 1974

Die. Grundlehren der mathema tischen Wissenschaften in Einzeldarstellungen mit besonderer Beri.icksichtigung der Anwendungsgebiete

Band 214

H erausgegeben von

S. S. Chern J. L. Doob J. Douglas,jr. A. Grothendieck E. Heinz F Hirzebruch E. Hopf W Maak S. MacLane W Magnus M. M. Postnikov F K. Schmidt D. S. Scott K. Stein Geschaftsfuhrende H erausgeber

B. Eckmann J. K. Moser B. L. van der Waerden

B. L. van der Waerden

Group Theory and Quantum. Mechanics

With 10 Figures

Springer-Verlag New York Heidelberg Berlin 1974

Preface

The German edition of this book appeared in t 932 under the title "Die gruppentheoretische Methode in der Quantenmechanik". Its aim was, to explain the fundamental notions of the Theory of Groups and their Representations, and the application of this theory to the Quantum Mechanics of Atoms and Molecules. The book was mainly written for the benefit of physicists who were supposed to be familiar with Quantum Mechanics. However, it turned out that it was also used by mathematicians who wanted to learn Quantum Mechanics from it. Naturally, the physical parts were too difficult for mathematicians, whereas the mathematical parts were sometimes too difficult for physicists. The German language created an additional difficulty for many readers. In order to make the book more readable for physicists and mathematicians alike, I have rewritten the whole volume. The changes are most notable in Chapters 1 and 6. In Chapter 1, I have tried to give a mathematically rigorous exposition of the principles of Quantum Mechanics. This was possible because recent investigations in the theory of self-adjoint linear operators have made the mathematical foundation of Quantum Mechanics much clearer than it was in t 932. Chapter 6, on Molecule Spectra, was too much condensed in the German edition. I hope it is now easier to understand. In Chapter 2- 5 too, numerous changes were made in order to make the book more readable and more useful.

B. L. VAN Zurich, February 1974

D ER WAERDEN

Prof. Dr. B. L. van der Waerden Mathematisches [nstitut der Universitiit Ziirich

Translation of the German Original Edition : Die Grundlehren der mathematischen Wissenschaften Band 37, Die Gruppentheoretische Methode in der Quantenmechanik. Publisher: Verlag von Julius Springer, Berlin 1932

AMS Subject Classification (1970) 81 --02,81 A 09, 81 A 78

ISBN 0- 387- 06740- X Springer-Verlag New York Heidelberg Berlin ISBN 3- S40- 06740- X Springer-Verlag Berlin Heidel berg New York

Library of Congress Cata loging in Publica ti o n Data Wae rd cn. Ba rtel Lec nert va n def. 1903 G ro up th eory a nd qu a ntum mec ha nics. (Die G run d leh ren der mathemati schcn Wi sscnschart en in Ein zcldarstellun gc n mit bcso ndcrc r Bcrlicksichti gun g def Anwe ndun gsgebiete, 8d. 2 14 ) Tra nslat io n of Die gruppcn th corcli sc hc Meth ode in der Quan tenmcchan ik . Bibliography : p. I. Quantum th eory. 2. Gro ups. The o ry of. I. Titl e. II. Series: Di e Gru ndl ehrcn

def mathemati sc hen Wisscnsc hafte n in Einze ldars lcllun gc n. Bd. 2 14)

QC I74. 17.G7 W313

530. 1'2

74- 139 14

This work is subject to copyri ght. All rights are reserved, whether the whole or part of th e material is concerned, specifically those of tran slation, reprinting, re· use of illu stratio ns, broadcasting, reproducti on by photocopying ma chine o r similar means, a nd storage in d a ta bank s. Under § 54 of the German Copyright Law where copies a re made for other than private use, a fee is pa yable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer. Verlag Berlin · Heidelberg t 974. Printed in Germany. Typesetting, printing and bookbinding : Briihlsche Universitatsdruckerei, GieOen.

Table of Contents

Chapter 1. Fundamental Notions of Quantum Mechanics § § § § § § § § 'v

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

Chapter 2. Groups and Their Representations

32

§ 9. § 10. § 11 . § 12. § 13. § 14. § 15.

32 40 46 53 59 61

Linear Transformations . . . . . . . Groups . . . . . . . . . . . . . . Equivalence and Reducibility of Representations Representations of Abelian Groups. Examples. Uniqueness Theorems . . . . . . . . . . . Kronecker's Product Transformation . . . . The Operators Commuting with all Operators Representation . . . . . . . . § 16. Representations of Finite Groups § 17. Group Characters. . . . . . .

"

1 4 9 12 16 19 25 27

Wave Functions Hilbert Spaces . . . . . Linear Operators . . . . Hypermaximal Operators Separation of Variables One Electron in a Central Field Perturbation Theory. . . . . Angular Momentum and Infinitesimal Rotations.

. . . of a Given

66 71 78

Chapter 3. Translations, Rotations and Lorentz Transformations

82

§ 18. Lie Groups and their Infinitesimal Transformations A. Lie Groups . . . . . . . . . . . . . . . . B. One-dimensional Lie Groups and Semi-Groups . C. Causality and Translations in Time D. The Lie Algebra of a Lie Group . . . . . . . E. Representations of Lie Groups . . . . . . . . § 19. The Unitary Groups SU(2) and the Rotation Group (93

82 82 83 86 87 89 90

.

VIII

Table of Contents

§ 20. Representations of the Rotation Group (93 § 21. Examples and Applications . . . . . A. The Product Representation Qj x Qr B. The Clebsch-Gordan Series C. Applications of (21.1) . . D . The Reflection Character . § 22. Selection and Intensity Rules . § 23. The Representations of the Lorentz Group A. The Group SL(2) and the Restricted Lorentz Group . B. Infinitesimal Transformations . . . . . . . . . C. The Relation between World Vectors and Spinors . I Chapter IV. The Spinning Electron . . . . . . .

96 101 101 102 107 109 110 114 114 117 120 123

123 § 24. The Spin. . . . . . . . . . . . . . . . § 25. The Wave Function of the Spinning Electron 125 A. Pauli's Pair of Functions (11'1 , 11'2) . 125 B. Transformation of the Pair (11'1, 11'2) 126 C. Infinitesimal Rotations . . . . . 128 D. The Angular Momenta . . . . . 129 E. The Doublet Splitting of the Alkali Terms l31 C. The Inversion s l32 132 § 26. Dirac's Wave Equation § 27. Two-Component Spinors. . . l37 A. Dirac's Equation Rewritten l37 B. Weyl's Equation . . . . . 140 § 28. The Several Electron Problem. Multiplet Structure. Zeeman Effect . . . . . . . . . . . . . . . . . . . . . . . . 141 Chapter V. The Group of Permutations and the Exclusion Principle 148

§ 29. § 30. § 31. § 32. § 33.

The Resonance of Equal Particles . . . . . . . . The Exclusion Principle and the Periodical System. The Eigenfunctions of the Atom . . . . . . . . . The Calculation of the Energy Values . . . . . . Pure Spin Functions and their Transformation under Rotations and Permutations . . . . . . . . . . . . § 34. Representations of the Symmetric Group fI" . . . . . . . .

148 157 161 171 174 182

Chapter VI. Molecule Spectra. . . . . . .

188

§ 35. The Quantum Numbers of the Molecule § 36. The Rotation Levels . . . . . § 37. The Case of Two Equal Nuclei . . . .

188 195 202

Chapter I.

Fundamental Notions of Quantum Mechanics

§ 1. Wave Functions According to Wave Mechanics, a pure state! of a mechanical system is defined at any time by a wave function P. The mechanical systems considered in this book are systems such as atoms or molecules, each consisting of a finite number of particles (electrons and nuclei). The wave function P is a complex-valued function of the coordinates of the particles, dependent on time, which is supposed to satisfy Schrodinger's equation h 8P ( 1.1) HP+- =O. 8t

i

As long as spin is neglected, the coordinates occurring in P are the orthogonal coordinates qg or x g , Yg , Zg of the f particles ; the index g goes from 1 to f. The Hamiltonian or energy operator H is defined as follows: Let 11 = I1g be the mass of anyone of the particles. The classical Hamiltonian is an expression of the form (1.2)

T

2 + U = ,L.., 1 2 (Px2 + py2 + pz) + U(q) ,

11

9

U(q) being the potential energy function. In this classical expression, one

has to replace the momentum components Px' Py' pz by differential operators

o

h

h

ox '

o

h 8y ' -

o

fu'

thus obtaining the operator 2

H

2

8

2

=~-

h 2ii

=L-

h -211g Ll g + U(q)

9

(

OX2

8

+ 8l +

2

8

OZ 2

)

+ U(q)

2

The notion " pure sta te", as opposed to " mixture", is due to J. von Neumann. See e.g. his " Principles of Quantum Mechanics". I

2

I. Fundamental Notions of Quantum Mechanics

The wave functions 'l' are supposed to be integrable in the sense of Lebesgue 2 and to have a finite square integral over q-space (1.3)