Application of Group Theory in Physics (eBook)

Front Cover 1
The Application of Group Theory in Physics 4
Copyright Page 5
Table of Contents 6
PREFACE 10
CHAPTER I. Elements of the Theory of Groups 12
1.Groups 12
2.Subgroups 15
3. Isomorphism and homomorphism of groups 17
Chapter II. Some Specific Groups 19
4. The Permutation Group 19
5. The Rotation Group 22
6. The Pull Orthogonal Group 26
7.The Euclidean Group 27
8.The Point Groups 29
9.The Point Groups of the First Kjnd 31
10.The Point Groups of the Second Kind 34
11. The Translation Group 39
12. Syngonies 41
13. The Symmetry of Crystals 48
Chapter III. 
52 
14. Representation of a Group 52
15. Equivalent representations 54
16. The Averaging Functional 56
17. Reducible Representations 58
18. Irreducible Representations and Orthogonality properties 61
19. The Completeness Theorem 67
20. The Theory of Characters 69
Chapter IV. Operations with Group Representations 73
21. The Product of Representations 73
22. Conjugate Representation 77
23. Real Representations 79
24. The Direct Product 81
25. Symmetrized Multiple Products of Representations 82
26. Decomposition of a Reducible Representation into Irreducible Representations 86
Chapter V. Representations of Certain Groups 93
27. Representations of the Permutation Group Sn 93
28. The Irrreducible Representations of Point Groups 96
29. Representations of Translation Groups 100
30. Representations of Space Groups 102
Chapter VI. Small Oscillations of Symmetrical Systems 114
31. Normal Coordinates and Eigen-Frequencies 114
32. Symmetrical Coordinates 117
33. The Lagrangian in Symmetrical Coordinates 119
34. The Oscillatory Representation 124
35. An Example: the Molecule CHCl3 128
Chapter VII. Second Order Phase Transitions 132
36.Formulation of the Problem 132
37. Active Representations 139
38. An Example 145
Chapter VIII. Crystals 169
39. Sound in Crystals 169
40. Electron Levels in a Crystal 174
41. Tensors in Crystals 177
Chapter IX. Infinite Groups 181
42. Specific Properties of Infinite Groups 181
43. Elements of the Theory of Lie Groups 189
44. Infinitesimal Representation of a Lie Group 200
Chapter X. Representations of the Rotation Groups in Two and Three Dimensions and of 'the Full Orthogonal Group. 203
45, The Irreducible Representations of the Two-dimensional Rotation Group Z 203
46. Classification of the irreducible Representations of Three-Dimensional Rotation Group 204
47. The Matrix Elements of the Irreducible Representations 211
48. Properties of the Irreducible Representations of the Rotation Group 216
49. The Product of Representations of the Rotation Group 221
50. Spinor Algebra 223
51. Tensor Algebra 228
52. Representations of the Pull Orthogonal Group 234
53. Double-Valued Representations of Point Groups 236
Chapter XI. Clebsch-Gordan and Racah Coefficients 238
54. Evaluation of the Clebsch-Gordan Coefficients 238
55. Properties of the Clebsch-Gordan Coefficients 245
56. Racah Coefficients 249
Chapter XII. The Schrödinger Equation 258
57. Conservation Laws 258
58. Classification of States 267
Chapter XIII. Equations Invariant under the Euclidean Group of Motions in Space 271
59. Spherical Harmonics with Spin 271
60, Equations Invariant under the Group of Euclidean Motions in Space 275
61. An Example 279
Chapter XIV. Absorption and Raman Scattering of Light 284
62. Quantum Mechanical Introduction 284
63. Selection Rules for the Absorption of Light by Atoms and Molecules 287
64. Raman Scattering of Light by Atoms and Molecules 294
Chapter XV. Representations of the Lorentz Group 297
65. The Lorentz Group 297
66. Infinitesimal Operators of the Lorentz Group 299
67. Classification of the Irreducible Representations of the Lorentz Group 301
68. Product of Irreducible Representations of the Lorentz Group 303
69. Complex-conjugate Representations 306
70. Spinor Algebra 308
71. Tensor Algebra 310
72. Representations of the Full Lorentz Group 314
Chapter XVI. Relativistically Invariant Equations 318
73, The Wave Function 318
74. Relatiyistically Invariant Equation 320
75. The Lagrangian 326
76. Conservation Laws 329
77. Spin 335
78. The Relativistically Invariant Operation of Time Inversion and the Pauli Theorem 338
79. The Dirac Equation 341
Chapter XVII. Nuclear Reactions 346
80. The Scattering Matrix 346
81. Angular Distribution of the Products of a Nuclear Reaction 350
82. Angular Distribution of the Products of a Nuclear Reaction. (Continued) 352
APPENDICES I: Characters of Irreducible Representations of the Permutation Groups S4,S5,S6,S7 355
APPENDICES II: Characters of Irreducible Representations of Point Groups 357
APPENDICES III: Two-valued Representations of Point Groups 358
APPENDICES IV: Space Groups 359
APPENDICES V: Racah Coefficients 384
Bibliography 386
Subject Index 391