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