Table of contents :
CONTENTS......Page 8
PREFACE......Page 6
INTRODUCTION: BASIC CONCEPTS......Page 14
1.1.1. Cartesian Coordinate System......Page 16
1.1.2. Polar Coordinate System......Page 17
1,1.3. Spherical Coordinate System......Page 18
1.1.5. Relations Between Different Basis Vectors......Page 20
1.2.1. Vector Components......Page 24
1.2.2. Scalar Product of Vectors......Page 27
1.2.3. Vector Product of Vectors......Page 28
1.2.5. Tensors δik and εikl......Page 29
1.3.1. Operator V......Page 30
1.3.2. Laplace Operator......Page 31
1.3.3. Differential Operations on Scalars and Vectors......Page 32
1.4.1. Description of Rotations in Terms of the Euler Angles......Page 34
1.4.2. Description of Rotations in Terms of Rotation Axis and Rotation Angle......Page 36
1.4.3. Description of Rotations in Terms of Unitary 2x2 Matrices. Cayley-Klein Parameters.......Page 37
1.4.4. Relations Between Different Descriptions of Rotations......Page 39
1.4.5. Rotation Operator......Page 40
1.4.6. Transformation of Cartesian Vectors and Tensors Under Rotations of Coordinate Systems. Rotation Matrix a......Page 41
1.4.7. Addition of Rotations......Page 44
2.1.2. Commutation Relations......Page 49
2.1.4. Total Angular Momentum of a System. Orbital and Spin Angular Momenta......Page 51
2.2.1. Definition......Page 52
2.2.3. Explicit Form......Page 53
2.3.3. Explicit Form......Page 55
2.3.4. Traces of Products of Spin Matrices......Page 56
2.4.2. Explicit Form......Page 57
2.4.3. Properties of LM(S) under Transformations of the Coordinate System......Page 58
2.4.6. Traces of Products of Polarization Operators......Page 59
2.5.2. Commutators and Anticommutators......Page 60
2.5.3. Products of Spin Matrices......Page 61
2.5.5. Rotation Operators......Page 62
2.5.6. Traces of Products of Spin Matrices (S = 1/2)......Page 63
2.6.1. Spin S = 1......Page 64
2.6.2. Explicit Form......Page 65
2.6.3. Products of Spin and Polarization Matrices......Page 68
2.6.4. Functions of Spin Matrices......Page 70
2.6.5. Operators of Coordinate Rotations......Page 71
2.6.6. Traces of Products of Spin Matrices......Page 72
3.1.1. Definition......Page 74
3.1.4. Transformation of Irreducible Tensors Under Inversion of the Coordinate System......Page 75
3.1.7. Direct and Irreducible Tensor Products. Commutators of Tensor Products......Page 76
3.1.8. Scalar Products of Irreducible Tensors......Page 77
3.2.1. Vectors and Irreducible Tensors......Page 78
3.2.2. Cartesian Tensors of Second and Third Ranks......Page 80
3.2.3. Differential Operations as Irreducible Tensor Products......Page 81
3.3.1. Relations Valid for Commuting as well as Non-Commuting Tensors......Page 82
3.3.2. Relations for Commuting Tensors......Page 83
3.3.3. Relations for Non-Commuting Tensors......Page 84
4.1. DEFINITION OF DJMM'(α,β,γ)......Page 85
4.2. DIFFERENTIAL EQUATIONS FOR DJMM'(α,β,γ)......Page 87
4.3.1. Expressions for dJMM'(β) Involving Trigonometric Functions......Page 89
4.3.2. Differential Representations of dJMM'(β)......Page 90
4.3.5. Relations Between dJMM'(β) and Hypergeometric Functions......Page 91
4.4. SYMMETRIES OF dJMM'(β) AND DJMM'(α,β,γ)......Page 92
4.5.1. Definition......Page 93
4.5.2. Explicit form......Page 94
4.5.4. Orthogonality and Completeness......Page 95
4.5.5. Principal Properties......Page 96
4.6.1. The Clebsch-Gordan Series......Page 97
4.6.3. Generalization of the Clebsch-Gordan Expansion......Page 98
4.6.4. Determinant of Matrix DJMM'......Page 99
4.7.2. The Addition Theorem for dJMM'(β)......Page 100
4.7.3. Addition of Two Identical Rotations......Page 101
4.7.6. The Ponzano-Regge sum......Page 102
4.8.1. Relations between DJ and DJ±1......Page 103
4.8.2. Relations Between DJ and DJ±1/2......Page 105
4.8.3. Relations Between DJMM' and DJM±1M'±1......Page 106
4.10. ORTHOGONALITY AND COMPLETENESS OF THE D-FUNCTIONS......Page 107
4.11.1. Integration of Products of DJMM'......Page 109
4.12. INVARIANT SUMMATION OF INTEGRALS INVOLVING DJMM'(α,β,γ)......Page 110
4.13. GENERATING FUNCTIONS FOR dJMM'(β)......Page 111
4.14.1. Definition......Page 112
4.14.2. Explicit Forms......Page 113
4.14.3. Principal Properties......Page 114
4.14.6. Algebraic Relations......Page 115
4.14.8. Integrals Involving X2(ω)......Page 116
4.14.9. Sums Involving x2(ω)......Page 117
4.14.10. XJ(ω) for Particular Values of ω......Page 118
4.15.2. Explicit Forms......Page 119
4.15.3. Principal Properties......Page 121
4.15.6. The Addition Theorem for XJλ(ω)......Page 122
4.15.7. Sums and Infinite Series Involving XJλ(ω)......Page 123
4.15.8. Special Cases of XJλ(ω) for Particular λ......Page 124
4.16. DJMM'(α,β,γ) FOR PARTICULAR VALUES OF THE ARGUMENTS......Page 125
4.17. SPECIAL CASES OF DJMM' FOR PARTICULAR M OR M'......Page 126
4.18.1. Large Angular Momentum......Page 128
4.18.3. Infinitesimal Rotations......Page 129
4.22. SPECIAL CASES OF UJMM'(ω,θ,Φ)......Page 130
5.1.2. Differential Equations......Page 143
5.1.4. Normalization......Page 144
5.1.7. Solutions of Some Differential Equations in Terms of Ylm(J,φ)......Page 145
5.2. EXPLICIT FORMS OF THE SPHERICAL HARMONICS AND THEIR RELATIONS TO OTHER FUNCTIONS......Page 146
5.2.2. Representations of Ylm(J,φ) as a Power Series of Trigonometric Functions of J/2......Page 147
5.2.3. Representations of Ylm(J,φ) as a Power Series of Trigonometric Functions of J......Page 148
5.2.4. Ylm(J,φ) and the Hyper geometric Functions with Arguments Expressed in Terms of IVigonometric Functions of J/2......Page 149
5.2.5.Ylm(J,φ) and the Hypergeometric Functions with Arguments Expressed in Terms of Trigonometric Functions of J......Page 150
5.2.8. Ylm(J,φ) as an Irreducible Tensor Product......Page 151
5.3.2. Ylm(J,φ) in the Form of Definite Integrals......Page 152
5.4. SYMMETRY PROPERTIES......Page 153
5.5.2. Inversion......Page 154
5.5.4. Special Cashes of Cooriiinate-Sygtem Transformations......Page 155
5.6.1. General Relations......Page 156
5.6.2. Expansion of Products of the SphfericarHarmonics......Page 157
5.7. RECURSION RELATIONS......Page 158
5.8.2. First and Second Older Derivatives of Ylm(J,φ)......Page 159
5.8.3. Vector Differentiation Operations......Page 160
5.9.1. Integrals over Total Solid Angle......Page 161
5.9.3. Integrals with Respect to J......Page 162
5.10.2. Sums over l (with fixed m ≥ 0)......Page 163
5.11. GENERATING FUNCTIONS FOR Ylm(J,φ)......Page 164
5.12.1. Ylm(J,φ) for Large l......Page 165
5.12.3. Asymptotic Expression for Fixed m,l→ ∞,J→0 and Finite lJ......Page 166
5.13.1.Ylm(J,φ) for l≤5......Page 168
5.13.3.Ylm(J,φ) for |m| = I, I - 1, l - 2, l - 3, l - 4, l - 5 and Any Integer l......Page 170
5.15.1. Zeros of Ylm(J,φ)......Page 171
5.15.2. Zeros of ∂/∂JYlm(J,φ)......Page 172
5.16.1. Bipolar Spherical Harmonics......Page 173
5.16.2. Tripolar Spherical Harmonics......Page 174
5.17.1. Preliminary Remarks......Page 176
5.17.3. Expansions of Some Functions Which Depend on (r1 .r2)......Page 177
5.17.4. Expansions of Some Functions Which Depend on r = |r1 - r2|......Page 178
5.17.5. Expansions of rn = |r1 - r2|n......Page 179
5.17.6. Expansions of Spherical Waves......Page 180
5.17.7 Expansions of rN YLM (J,φ)......Page 181
6.1.1. Definition......Page 183
6.1.2. Basis Spin Functions......Page 184
6.1.3. Helicity Basis Functions......Page 185
6.1.4. General Spin Functions......Page 187
6.1.5. Polarization Density Matrix......Page 188
6.1.6. Two Particles with Arbitrary Spins......Page 189
6.2.2. Expansions of Products of Basis Functions......Page 191
6.2.4. Transformation of the Basis Functions Under......Page 192
6.2.5. Helicity Basis Functions......Page 193
6.2.6. General Spin Functions for S =1/2......Page 195
6.2.7. Polarization Density Matrix......Page 197
6.3.1. Basis Spin Functions......Page 198
6.3.2. Expansions of Products of Spin Functions......Page 200
6.3.3. Action of Spin Operators on Basis Functions......Page 201
6.3.4. Action of Quadrupole Operators on Basis Functions......Page 202
6.3.5. Transformation of Basis Functions Under Rotations of the Coordinate Systems......Page 203
6.3.6. Helicity Basis Functions for S = 1......Page 204
6.3.7. General Spin Functions for S = 1......Page 207
7.1.1. Definition......Page 209
7.1.2. Components of Tensor Spherical Harmonics......Page 210
7.1.5. Differential Equations......Page 211
7.1.6. Action of Operators Ñ , n and Angular Momentum Operators......Page 212
7.1.7. Sums of Tensor Spherical Harmonics......Page 213
7.1.9. Expansion in a Series of Tensor Spherical Harmonics......Page 214
7.2.2. Components of Spinor Spherical Harmonics......Page 215
7.2.3. Complex Conjugation. Time Reversal......Page 216
7.2.5. Action of V and Angular Momentum Operators......Page 217
7.2.6. Recursion Relations......Page 218
7.2.8. Clebsch-Gordan Series......Page 219
7.2.10. Quadratic Forms of Spinor Spherical Harmonics......Page 220
7.3.1. Definition......Page 221
7.3.2. Components of Vector Spherical Harmonics......Page 224
7.3.3. Complex Conjugation......Page 228
7.3.6. Differential Operations......Page 229
7.3.7. Action of Angular Momentum Operators......Page 231
7.3.8. Algebraic Relations......Page 232
7.3.9. Sums of Vector Spherical Harmonics......Page 233
7.3.10. Clebsch-Gordan Series......Page 235
7.3.11. Addition Theorems for Vector Spherical Harmonics......Page 236
7.3.12. Integrals Involving Vector Spherical Harmonics......Page 239
7.3.14. Expansion in Series of Vector Spherical Harmonics......Page 240
7.3.15. Vector Spherical Harmonics for J = 0 or J = π......Page 242
7.3.16. Vector Spherical Harmonics at J = 0,1......Page 243
7.3.17. Quadratic Forms of the Vector Spherical Harmonics......Page 244
7.4. OTHER NOTATIONS FOR TENSOR SPHERICAL HARMONICS......Page 247
8.1.1. The Clebsch-Gordan Coefficients......Page 248
8.1.2. The Wigner 3jm Symbols......Page 249
8.2. EXPLICIT FORMS OF THE CLEBSCH-GORDAN COEFFICIENTS AND THEIR RELATIONS TO OTHER FUNCTIONS......Page 250
8.2.1. Representations of the Clebsch-Gordan Coefficients in the Form of Algebraic Sums......Page 251
8.2.3. Clebsch-Gordan Coefficients and Finite Differences......Page 252
8.2.5. Representations of the Clebsch-Gordan Coefficients in Terms of of the Hypergeometric Functions......Page 253
8.2.6. Representations of the 3jm Symbols in the Form of Algebraic Sums......Page 254
8.2.7. Quasi-binomial Representations of the 3jm Symbols......Page 255
8.3.4. Integral Representations for Products of the Clebsch-Gordan Coefficients......Page 256
8.4.2. Symmetry Properties of t h e 3jm Symbols......Page 257
8.4.3. Symmetry Properties of the Clebsch-Gordan Coefficients......Page 258
8.4.4. "Mirror" Symmetry......Page 259
8.4.5. Properties of the Vector-Addition Coefficients under Transformations of the Coordinate System and Time Reversal......Page 260
8.5.1. Special Values of Momenta a, b, c......Page 261
8.5.2. Special Values of Momentum Projections......Page 264
8.6.1. General Recursion Relations......Page 265
8.6.2. Arguments α,β,γ Change by 1......Page 266
8.6.3. Arguments Change by 1/2......Page 267
8.6.5. Arguments α,β,γ Change by 1......Page 268
8.6.6. Arguments a, b,α,β Change by 1......Page 269
8.6.7. Arguments c, b, γ,β Change by 1......Page 270
8.6.8. Recursion Relations for the Regge Symbols......Page 271
8.7.2. Sums Involving Products of Two Clebsch-Gordan Coefficients......Page 272
8.7.4. Sums Involving Products of Four Clebsch-Gordan Coefficients......Page 273
8.7.5. Sums Involving Products of the Clebsch-Gordan Coefficients and One 6/ Symbol......Page 274
8.7.7. Some Additional Sums of Products of Two Clebsch-Gordan Coefficients......Page 275
8.8.4. Hyper geometric Function......Page 276
8.9.1. Asymptotic Expressions for a, c > b......Page 277
8.9.3. Semiclassical Formulas for a,b, c » 1......Page 278
8.9.4. Squares of the Clebsch-Gordan Coefficients in the Classical Limit......Page 280
8.11. CONNECTION OP THE CLEBSCH-GORDAN COEFFICIENTS AND THE 3jm SYMBOLS WITH ANALOGOUS FUNCTIONS OF OTHER AUTHORS......Page 281
8.13. NUMERICAL TABLES OF THE CLEBSCH-GORDAN COEFFICIENTS......Page 283
9.1.1. 6jSymbols......Page 303
9.1.2. Racah Coefficients......Page 304
9.1.3. R Symbols......Page 305
9.2.1. Expressions for the 6j Symbols in Terms of Finite Sums......Page 306
9.2.2. Bargmann Formula [53]......Page 307
9.2.3. Relations Between the 6; Symbols and the Generalized Hypergeometric Functions......Page 308
6.2.5. Quasi-Binomial Representation of the 6j Symbols......Page 309
9.3. INTEGRAL REPRESENTATIONS OF THE 6j SYMBOLS......Page 310
9.4.3. Racah Coefficients......Page 311
9.5.1. One of Arguments is Equal to Zero......Page 312
9.5.2. One of Arguments is Equal to the Sum of Two Others......Page 313
9.5.3. One of Arguments is Smaller by Unity than the Sum of Two Others......Page 314
9.5.4. Arguments a, b, d, e are Equal in Pairs......Page 315
9.6.1. Relations in Which Arguments are Changed by 1/2......Page 316
9.6.2. Relations in Which Arguments are Changed by 1......Page 317
9.8. SUMS INVOLVING THE 6j SYMBOLS......Page 318
9.9.2. Asymptotic Expressions for the 6j Symbols......Page 319
9.12. NUMERICAL VALUES OF THE 6j SYMBOLS......Page 323
10.1.1. 9j Symbols as Recoupling Coefficients......Page 346
10.1.2. 9j Symbol and r Symbol......Page 348
10.2.1. Expressions for the 9j Symbols in the Forms of Algebraic Sums......Page 349
10.2.2. Wu Formulas [ 111]......Page 350
10.2.3. 9j Symbols as Sums of Products of the Clebsch-Gordan Coefficients or the 3;m Symbols [110]......Page 352
10.3.2. Representation Involving the Wigner D Functions......Page 353
10.3.3. Integrals Involving Characters of Irreducible Representations of Rotation Group......Page 354
10.4.1. Permutation Symmetry......Page 355
10.4.2. Symmetries of the r Symbol......Page 356
10.4.3. "Mirror" Symmetry......Page 357
10.5.1. General Form of the Recursion Relation......Page 358
10.5.2. Relations Involving Four 9j Symbols......Page 359
10.5.3. Relations Involving Five 9j Symbols [45]......Page 360
10.5.5. Recursion Relations for Some Special Cases......Page 362
10.7. ASYMPTOTIC EXPRESSION FOR A 9j SYMBOL......Page 364
10.8.2. One Degenerate Triad......Page 365
10.3.3. Two Degenerate Triads......Page 366
10.8.5. Four Degenerate Triads......Page 368
10.9.1. One of Arguments Equals Zero......Page 370
10.9.3. One of Triads Equals (1/2, 1/2, 1)......Page 371
10.11. TABLES OF ALGEBRAIC FORMULAS OF THE 9j SYMBOLS......Page 372
10.12. TABLES OF NUMERICAL VALUES OF THE 9j SYMBOLS......Page 373
10.13.2. 12j Symbols of the First Kind (12j(I)-Symbols)......Page 374
10.13.3. 12; Symbols of the Second Kind (12;? (II) Symbols)......Page 380
11.1. GRAPHICAL REPRESENTATION OF FUNCTIONS......Page 425
11.1.1. Basic Elements of Diagrams......Page 426
11.1.2. Diagrams of the Basic Functions of the Theory......Page 427
11.2.1. Multiplication......Page 432
11.2.2. Invariant Summation over Projections......Page 433
11.2.3. Summation over Angular Momentum......Page 434
11.2.4. Invariant Integration over Directions......Page 435
11.2.5. Integration over Rotation Parameters......Page 436
11.3.1. Deformation of Diagrams......Page 437
11.3.2. Change of Node Sign......Page 439
11.3.3. Change of Direction of External Lines......Page 440
11.3.5. Linking Subdiagrams......Page 441
11.3.6. Cutting Diagram into Subdiagrams......Page 442
11.3.7. Graphical Method of Summation......Page 444
11.3.9. Elimination of j = 0 Line......Page 445
11.4.1. General Properties of Diagrams......Page 459
11.4.2. Generalized Wigner-Eckart Theorem in Diagrammatic Form......Page 460
11.4.3. Scheme for the Application of the Graphical Technique......Page 463
12.1. SUMMATION OP PRODUCTS OP 3jm SYMBOLS......Page 465
12.1.2. Sums Involving Products of Two 3jm Symbols......Page 466
12.1.4. Sums Involving Products of Four 3jm Symbols......Page 467
12.1.5. Sums Involving Products of Five 3jm Symbols......Page 469
12.1.6. Sums Involving Products of Six 3jm Symbols......Page 471
12.2.1. Suing Involving One 3mj Symbol......Page 475
12.2.2. Sums Involving Products of Two 3nj Symbols......Page 476
12.2.3. Sums Involving Products of Three 3nj Symbols......Page 479
12.2.4. Sums Involving Products of Four 3nj Symbols......Page 482
13.1.1. Wigner-Eckart Theorem......Page 488
13.1.3. Matrix Elements of Products of Irreducible Tensor Operators......Page 489
13.1.4. Matrix Elements of Operators Which Depend on Variables of Two Subsystems......Page 491
13.1.5. Matrix Elements of Operators Which Depend on Variables of One of the Subsystems......Page 494
13.2.1. Some Introductory Remarks......Page 496
13.2.3. Matrix Elements of the Unit Vector ñ(J,φ) = ñ1(J, φ)......Page 497
13.2.4. Matrix Elements of the Operatpr V(r, J,φ) = V1(r,J,φ)......Page 499
13.2.5. Matrix Elements of the Total Angular Momentum Operator Ĵ = Ĵx......Page 502
13.2.6. Matrix Elements of the Orbital Angular Momentum Operator Ĺ = Ĺ1......Page 505
13.2.7. Matrix Elements of the Spin Angular Momentum Operator Ŝ = Ŝ1......Page 508
13.2.8. Matrix Elements of the Spherical Harmonic Operator ŶLv = ŶLv(J,φ))......Page 509
13.2.9. Matrix Elements of Some Scalar and Vector Products......Page 515
GLOSSARY OF SYMBOLS AND NOTATION......Page 518
REFERENCES......Page 522

##### Citation preview

Quantum Theory of Angular Momentum

This page is intentionally left blank

Quantum Theory of Angular Momentum Irreducible Tensors, Spherical Harmonics, Vector Coupling Coefficients, 3/7/ Symbols

D. A . Varshalovich A. F. loffe Physical-Technical Institute

A . N . Moskalev B. P. Konstantinov Institute of Nuclear Physics

V . K. Khersonskii Special Astrophysical Observatory

World Scientific Singapore • New Jersey • Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd., P O Box 128, Farrer Road, Singapore 9128 USA office: 687 Hartwell Street, Teaneck, NJ 07666 UK office: 73 Lynton Mead, Totteridge, London N20 8DH

Library of Congress Cataloging- in-Publication Data Varshalovich, D. A. (Dirnitrii Aleksandrovich) Quantum theory of angular momentum. Translation of: Kvantovaia teoriia uglovogo momenta. 1. Angular momentum (Nuclear physics) 2. Quantum theory^ I. Moskalev, A. N. II. Khersonskii, V. K. (ValeriiKermanovich) III. Title. QC793.3.A5V3713 1988 530.1*2 86-9279 ISBN 9971-50-107-4 9971-50-996-2 pbk

First published 1988 First reprint 1989

QUANTUM THEORY QF ANGULAR MOMENTUM Copyright ©1988 by World Scientific Publishing Co Pte Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photo­ copying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

Printed in Singapore by Chong Moh Offset Printing Pte Ltd.

\

PREFACE

This book deals with one of the basic topics of quantum mechanics: the theory of angular momentum and irreducible tensors. Being rather versatile, the mathematical apparatus of this theory is widely used in atomic and molecular physics, in nuclear physics and elementary particle theory. It enables one to calculate atomic, molecular and nuclear structures, energies of ground and excited states, fine and hyperfine splittings, etc. The apparatus is also very handy for evaluating the probabilities of radiative transitions, cross sections of various processes such as elastic and nonelastic scattering, different decays and reactions (both chemical and nuclear) and for studying angular distributions and polarizations of particles. Today this apparatus is finding ever increasing use in solving practical problems relating to quantum chemistry, kinetics, plasma physics, quantum optics, radiophysics and astrophysics. The basic ideas of the theory of angular momentum were first put forward by M. Born, P. Dirac, W. Heisenberg and W. Pauli. However, the modern version of its mathematical apparatus was developed mainly in the works of E. Wigner, J. Racah, L. Biedenharn and others who applied group theoretical methods to problems in quantum mechanics. At present a number of good books on the theory of angular momentum have been already published. The general principles and results of the theory may be found in the books by M. Rose [31], A. Edmonds [16], U. Fano and G. Racah [18], A. P. Yutsis, I. B. Levinson and V. V. Vanagas [44], A. P. Yutsis and A. A. Bandzaitis [45], D. Brink and G. Satcher [9]. Nevertheless, many formulas and relationships essential for practical calculations have escaped these books and are either scattered in various editions, or included as appendices in papers discussing somewhat disparate topics, making them generally inaccessible. Even greater difficulties arise when one tries to use the results, as each author employs his own phase conventions, initial definitions and symbols. The authors of this book aimed at collecting and compiling ample material on the quantum theory of angular momentum within the framework of a single system of phases and definitions. This is why, in addition to the basic theoretical results, the book also includes a great number of formulas and relationships essential for practical applications. This edition is the translated version of our book published in the USSR in 1975. In the course of its preparation we have tried to comply with a number of suggestions from our readers. For instance, each chapter opens with a comprehensive listing of its contents to ease the search for information needed. We also included some new results relating to different aspects of angular momentum theory which have recently appeared in journals. Unfortunately the limited volume of the present book prevented us from covering all the aforementioned results. We offer sincere apologies to the authors whose results we failed to include. The monograph is a kind of handbook. Consequently the material is presented in concise form. Most of the formulas and relationships are given without proof. Their full derivation may be found in the literature v

VI

Preface

listed at the end of the text. Some results which have become generally known are given without references, and for this we also apologize. The sequence adopted is as follows: chapter, section and subsection. Many chapters are self-contained and can be read independently of the others. Sections have double numbering: the first figure denotes the number of the chapter, the second, the number of the section. Equations are numbered within the confines of the section they are included in. When referring to an equation from the same section only the number of the equation is given, e.g., (3), (27); when reference is made to an equation from another section the numbers of the chapter, section and equation are given, e.g., Eq. 4.2.(17). A similar system is adopted when referring to individual subsections, e.g., Sec. 1.2.5. For convenience the book also contains a glossary of all symbols used in the text with references to the pages where their corresponding definitions are given. The list of references is divided into parts: the first part lists books and reviews; the second, papers on different subjects; tthe third, tables; the fourth, references added during translation. The authors hope that many specialists will find in the book some fresh and interesting information. The material is prepared and arranged so as to make it useful to those less familiar with theory and for students of physics. These readers can effectively use the monograph as a supplementary text to their main courses. For those who wish to thoroughly familiarize themselves with the fundamentals of angular momentum theory we recommend the excellent new book by L. Biedenharn and J. Louck [132] Angular Momentum in Quantum Physics. Theory and Applications. The authors wish to express their deep appreciation to D. G.Yakovlev who took the trouble of reading the English translation of the book and gave some valuable suggestions on its preparation.

Leningrad

D. A. Varshalovich A. N. Moskalev V. K. Khersonskii

CONTENTS

Preface

v

Introduction: Basic Concepts

1

C h a p t e r 1. E l e m e n t s of Vector and Tensor Theory

3

1.1. 1.2. 1.3. 1.4.

Coordinate Systems. Basis Vectors Vectors. Tensors Differential Operations Rotations of Coordinate System

C h a p t e r 2. Angular M o m e n t u m O p e r a t o r s 2.1. Total Angular Momentum Operator 2.2. Orbital Angular Momentum Operator 2.3. Spin Angular Momentum Operator 2.4. Polarization Operators 2.5. Spin Matrices for S = \ 2.6. Spin Matrices and Polarization Operators for S = 1 C h a p t e r 3. Irreducible Tensors 3.1. Definition and Properties of Irreducible Tensors 3.2. Relation Between the Irreducible Tensor Algebra and Vector and Tensor Theory 3.3. Recoupling in Irreducible Tensor Products Chapter 4. Wigner D-Functions 4.1. Definition of DJMM,(a,^,7) 4.2. Differential Equations for DJMM, (a, £, 7) 4.3. Explicit Forms of the Wigner D-Functions 4.4. Symmetries of dJMM,{p) and DJMM,(a,0,7) 4.5. Rotation Matrix U^M, in Terms of Angles w, 6 , \$ 4.6. Sums Involving D-Functions 4.7. Addition of Rotations 4.8. Recursion Relations for DJ^M, vii

3 11 17 21 36 36 39 42 44 47 51 61 61 . . 65 69 72 .72 74 76 79 80 84 87 90

Contents

Vlll

4.9. 4.10. 4.11. 4.12. 4.13. 4.14. 4.15. 4.16. 4.17.

Differential Relations for D^M, (a, £,7) 94 Orthogonality and Completeness of the D-Functions 94 Integrals Involving the 2?-Functions 96 Invariant Summation of Integrals Involving D^^, (a, 0,7) 97 Generating Functions for diiAi,(0) 98 Characters xJ(^R) of Irreducible Representations of Rotation Group 99 Generalised Characters, X\(R)> of Irreducible Representations of the Rotation Group 106 DJMW (a, 0} 7) for Particular Values of the Arguments 112 Special Cases of D3MW (a, 0,7) for Particular M or M' 113

4.18. 4.19. 4.20. 4.21. 4.22.

Asymptotics of ^ A / A ^ () under Transformations of Coordinate Systems 141 5.6. Expansions in Series of the Spherical Harmonics 143 5.7. Recursion Relations . 145 5.8. Differential Relations 146 5.9. Some Integrals Involving Spherical Harmonics 148 5.10. Sums Involving Spherical Harmonics . , 150 5.11. Generating Functions for y/m(t?,v?) 151 5.12. Asymptotic Expressions for yjm(t?,vp) 152 5.13. Yim{#, — e^ - 7 = ^ , e 0 = e r cos t? — e\$ sin #, e_! = e r 4 = sin t?c""^ + e* 4 = cos t?e~*v - e„ 4=«" f > •

Elements of Vector and Tensor Theory

9

Spherical Contravariant and Polar Basis Vectors 1 1 i e + 1 = - e r —7=8int?c""i,p - e#—7= cost?e"~iV> + e«p—=6""'^, e° = e r cos 1? — e# sin 1?, 1 1 i e""1 = er—psint?eiV> + e# -7= cos t?etv> + e^—pctv?.

(39)

Spherical Covariant and Spherical Contravariant Basis Vectors e+i = - e ~ \

e+1 = -e_i,

e 0 = e°, e_! = - e

e° = e 0 ,

+1

,

e

_1

(40)

= -e+i.

Spherical Covariant and Helicity Covariant Basis Vectors e+i—e+1

1 + cosfl . e

e0

sinr? ■„ 1-coetf

x

(42) 1-CQ31? , „

\Z5

2

'

Spherical Covariant and Helicity Contravariant Basis Vectors =

eo =

e 1

_e+i1 +

(43)

cos ?

2

' e-^+e>o!ye-^_e'^1-c08,?e-'V. v/2

2

Spherical Contravariant and Helicity Contravariant Basis Vectors sin c + i ^ + i i + c08^-.* * c - - i l - c o s t ? c _ < t , c >o 2 v/2 2 ..,sint? , . ,_isint? n n e° = e ' + 1 — ^ + e'° costf- e' 1—7=-,

2

v^2

2

... (44)

10

Quart tumTh eory of A ngular Momen turn Equations (41)-(44) may be written in a more compact form using the Wigner D-functions (see Chap. 4).

V

V

e" = ("I)" £ &-»A*> »> + e , + eeo0—T=T ~ee -1i

1—

'

W" -

2

, sint? _4.^ . . sint? ,„ e'°0 = - e + i —p-e"*^ + e 0 cos t?n + e_x - y = V , e

/_i

= -e+1

1 + cost?

. sint? 1 - c o s t ? iutv e %* - e 0 —— - e_i e ^

(50)

Elements of Vector and Tensor Theory

11

Helicity Covariant and Spherical Contravariant Basis Vectors _ e+i-

+1l-cosi?

. .tsint? • « 1 sin t? . ■ n B0 = - e + 1 — p r e , , p + e 0 cost? + e 1 - y r - e - " p , '_, =- —e- + i 1 +

C08

sin,? *.«•«> e K —O eo —7=—. e. x l - c o s t f 2 v/2 2

, v (51) e_ ^ .

Helicity Contravariant and Spherical Contravariant Basis Vectors

2 V2 2 .n .ismi? ,-i/» n « 1 sin # .^ e'° = - e + 1 --7=-etv? + e° costf + e"1—7=r-e~tv% \/2 \/2 •„ nsintf ,1-f-cosfl x +1l-co8tf 2 y/2 2

/ * (52) -,

Equations (49)-(52) may be written in a more compact form using the Wigner D-functions (see Chap. 4).

e"* = 5 3 ( - I ) M ^ - M ( ^ * . ° K = E(- 1 )" +PI, --M^^. o ) eV .

(53)

(*.,!/ = ±1,0). Helicity Covariant and Helicity Contravariant Basis Vectors

ei=e'°,

e'° = ei f 1

e'.^-e'* ,

1.2.

(54)

e'-^-eLx.

VECTORS. TENSORS

Vectors and tensors are usually defined by transformation properties of their components under rotations of coordinate systems. The transformation rule for cartesian components of vectors and tensors is given below in Sec. 1.4 (Eqs. (46)-(51)). The transformation properties of spherical components of vectors and irreducible tensors are discussed in Chap. 3. 1.2.1.

Vector C o m p o n e n t s

Any vector can be expanded in terms of basis vectors, i.e., written as

a

a

The expansion coefficients Aa are called the covariant components of the vector, and Aa are the contravariant vector components i 4 a = A - e a , Aa=A-ea. (2)

12

Quantum Theory of Angular Momentum

In a cartesian coordinate system one has A = Axex + Ayey + A*e* = Axex + A y e y + A*e*.

(3)

The covariant cartesian components of a vector coincide with the contravariant ones. In a polar coordinate system A = A r e r + A#e# + ^^e^ = Arer + A°e# + A^e^.

(4)

The convariant pol^r components coincide with the contravariant ones. For a spherical coordinate system A = A + 1 e + 1 + A°e0 + A~le-!

= 4 + i e + 1 + A0e° + A-ie~l.

(5)

The relations between covariant and contravariant spherical components are given by AM = (-1)"A-",

A" = (-1)M_ M ,

(A* = ±1,0).

(6)

If A is a real vector^ i.e., if A* = A, then Al = A»,

A- = A»

(/x = ± l , 0 ) .

(7)

If A is a complex vector, then A; = ( A T .

A"* = (A*)M>

(M = ±I,O).

(8)

An expansion of a real vector A in terms of spherical basis vectors is written as

A = £ 4.e" = £ **«M = £ W M

H

f*

^X

V

= E **.< = E *;* - E ^eM* = E ^*e" A*

A*

M

M

= £ ( - 1 ) " A_MeM = E ( - i r A - V .

(9)

An expansion of an arbitrary vector A in terms of helicity basis vectors is given by A = A , + 1 e' + 1 + ii'Vo + A'^eLx = A ' + 1 e ' + 1 + ^ e ' ° + i l ^ e ' " 1 .

(10)

The helicity components of a vector satisfy the same relations (6)-(9) as the spherical components. The relations between vector components in different bases are the same as the relations between basis vectors. These relations are given by Eqs. l.l(29)-l.l(54) in which one should replace e a —► Aa and e°* —► Aa. In particular, A+1 = -A-1

= -~(AX

+iAy),

A0 = A° = AZ, A_i = -A+1

= -j={Ax - iAy),

Ax = ±{A-x

- A+1) = ^(A'1

-

Ay = ^=(A^1

+ A+1) = ^(A-1

+ A+1),

AZ = A0 = A0.

A+1), (11)

Elements of Vector and Tensor

Theory

13

The matrices of transformations between cartesian, contravariant spherical and polar components of vectors are given in Tables 1.1 and 1.2. Spherical components of a real vector A which contains no derivatives and is independent of spin variables are IA IS

A

^

±i=:F|A|

m

^

7r

,±*V

A*1 = =F |A

AQ = | A | c o s # ,

sint? =F»V

7T

(12)

A° = |A|cost9,

where #,

2/>2)

e%ik = e%ki = Skii = 0,

(3 components), (i,k = x,y,z)

Gxyz = Syzx = Gzxy = ~^xzy = Syxz

=

(18 components), e

~~ zyx

= 1

(6 Components).

The components e ^ are invariant with respect to rotations and inversion of coordinate systems.

(35)

Elements of Vector and Tensor Theory The tensor e^i has the following properties: The product of two tensors eiM and er8t may be the form of a determinant

s

e

ikl r»t

Sir

Sis

Skr

Sks

Su Skt

Sir

S{8

Su

= SirSk8Sit + Si9SktSir + SitSkrSl8 - SirSktSi8 - Si8SkrSlt -

By summing over a pair of indices, one obtains 2_^e%kiei8t

=

t

Sk\$ Skt Su Su

Sk»Su -

SktSi8.

Summation over two pairs of indices yields 2_^SikiSikt = 2£/ti,k

Finally, the summation over three pairs of indices gives

For an arbitrary 3x3 matrix ||Ai*|| (i9k — x,y,z)

22,AxiAykAxi€iki

the following relation holds

= det||A tfc || =

i,kj

1.3.

A

A

**xx AyX AZx

xx

A

xy Ayy AZy

^xz AyZ Azz

DIFFERENTIAL OPERATIONS 1.3.1.

Operator V

The operator V (nabla) is the basic vector differential operator. Cartesian components of V are given by d y

dx*

d

d

dy*

dz'

These components may be expressed in terms of polar coordinates as _ . _ d cost? cos