| Preface |
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xi | |
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1 | (24) |
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1.1 Discovery of the Pauli Exclusion Principle and Early Developments |
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1 | (10) |
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1.2 Further Developments and Still Existing Problems |
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11 | (14) |
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21 | (4) |
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2 Construction of Functions with a Definite Permutation Symmetry |
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25 | (25) |
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2.1 Identical Particles in Quantum Mechanics and Indistinguishability Principle |
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25 | (4) |
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2.2 Construction of Permutation-Symmetric Functions Using the Young Operators |
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29 | (7) |
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2.3 The Total Wave Functions as a Product of Spatial and Spin Wave Functions |
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36 | (14) |
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2.3.1 Two-Particle System |
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36 | (5) |
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2.3.2 General Case of N-Particle System |
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41 | (8) |
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49 | (1) |
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3 Can the Pauli Exclusion Principle Be Proved? |
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50 | (14) |
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3.1 Critical Analysis of the Existing Proofs of the Pauli Exclusion Principle |
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50 | (6) |
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3.2 Some Contradictions with the Concept of Particle Identity and their Independence in the Case of the Multidimensional Permutation Representations |
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56 | (8) |
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62 | (2) |
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4 Classification of the Pauli-Allowed States in Atoms and Molecules |
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64 | (42) |
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4.1 Electrons in a Central Field |
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64 | (10) |
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4.1.1 Equivalent Electrons: L-S Coupling |
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64 | (7) |
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4.1.2 Additional Quantum Numbers: The Seniority Number |
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71 | (1) |
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4.1.3 Equivalent Electrons: j--j Coupling |
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72 | (2) |
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4.2 The Connection between Molecular Terms and Nuclear Spin |
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74 | (8) |
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4.2.1 Classification of Molecular Terms and the Total Nuclear Spin |
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14 | (65) |
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4.2.2 The Determination of the Nuclear Statistical Weights of Spatial States |
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79 | (3) |
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4.3 Determination of Electronic Molecular Multiplets |
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82 | (24) |
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4.3.1 Valence Bond Method |
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82 | (5) |
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4.3.2 Degenerate Orbitals and One Valence Electron on Each Atom |
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87 | (4) |
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4.3.3 Several Electrons Specified on One of the Atoms |
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91 | (2) |
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4.3.4 Diatomic Molecule with Identical Atoms |
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93 | (5) |
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98 | (2) |
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100 | (4) |
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104 | (2) |
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5 Parastatistics, Fractional Statistics, and Statistics of Quasiparticles of Different Kind |
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106 | (29) |
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5.1 Short Account of Parastatistics |
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106 | (3) |
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5.2 Statistics of Quasiparticles in a Periodical Lattice |
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109 | (12) |
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5.2.1 Holes as Collective States |
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109 | (2) |
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5.2.2 Statistics and Some Properties of Holon Gas |
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111 | (6) |
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5.2.3 Statistics of Hole Pairs |
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117 | (4) |
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5.3 Statistics of Cooper's Pairs |
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121 | (3) |
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5.4 Fractional Statistics |
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124 | (11) |
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5.4.1 Eigenvalues of Angular Momentum in the |
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Three- and Two-Dimensional Space |
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124 | (4) |
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5.4.2 Anyons and Fractional Statistics |
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128 | (5) |
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133 | (2) |
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Appendix A Necessary Basic Concepts and Theorems of Group Theory |
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135 | (26) |
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A.1 Properties of Group Operations |
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135 | (6) |
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135 | (2) |
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137 | (1) |
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A.1.3 Isomorphism and Homomorphism |
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138 | (1) |
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A.1.4 Subgroups and Cosets |
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139 | (1) |
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A.1.5 Conjugate Elements. Classes |
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140 | (1) |
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A.2 Representation of Groups |
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141 | (20) |
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141 | (1) |
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142 | (3) |
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A.2.3 Reducibility of Representations |
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145 | (2) |
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A.2.4 Properties of Irreducible Representations |
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147 | (1) |
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148 | (1) |
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A.2.6 The Decomposition of a Reducible Representation |
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149 | (2) |
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A.2.7 The Direct Product of Representations |
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151 | (3) |
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A.2.8 Clebsch--Gordan Coefficients |
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154 | (2) |
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A.2.9 The Regular Representation |
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156 | (1) |
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A.2.10 The Construction of Basis Functions for Irreducible Representation |
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157 | (3) |
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160 | (1) |
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Appendix B The Permutation Group |
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161 | (21) |
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161 | (6) |
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B.1.1 Operations with Permutation |
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161 | (3) |
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164 | (1) |
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B.1.3 Young Diagrams and Irreducible Representations |
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165 | (2) |
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B.2 The Standard Young-Yamanouchi Orthogonal Representation |
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167 | (15) |
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167 | (3) |
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B.2.2 Explicit Determination of the Matrices of the Standard Representation |
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170 | (3) |
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B.2.3 The Conjugate Representation |
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173 | (2) |
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B.2.4 The Construction of an Antisymmetric Function from the Basis Functions for Two Conjugate Representations |
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175 | (1) |
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176 | (2) |
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B.2.6 The Construction of Basis Functions for the Standard Representation from a Product of N Orthogonal Functions |
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178 | (3) |
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181 | (1) |
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Appendix C The Interconnection between Linear Groups and Permutation Groups |
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182 | (35) |
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182 | (7) |
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182 | (3) |
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C.1.2 Examples of Linear Groups |
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185 | (2) |
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C.1.3 Infinitesimal Operators |
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187 | (2) |
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C.2 The Three-Dimensional Rotation Group |
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189 | (12) |
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C.2.1 Rotation Operators and Angular Momentum Operators |
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189 | (2) |
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C.2.2 Irreducible Representations |
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191 | (3) |
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C.2.3 Reduction of the Direct Product of Two Irreducible Representations |
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194 | (3) |
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C.2.4 Reduction of the Direct Product of k Irreducible Representations. 3n - j Symbols |
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197 | (4) |
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C.3 Tensor Representations |
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201 | (13) |
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C.3.1 Construction of a Tensor Representation |
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201 | (1) |
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C.3.2 Reduction of a Tensor Representation into Reducible Components |
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202 | (5) |
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C.3.3 Littlewood's Theorem |
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207 | (2) |
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C.3.4 The Reduction of U2j +1 → R3 |
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209 | (5) |
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C.4 Tables of the Reduction of the Representations Uγ2j+,1 to the Group R3 |
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214 | (3) |
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216 | (1) |
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Appendix D Irreducible Tensor Operators |
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217 | (6) |
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217 | (3) |
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D.2 The Wigner-Eckart Theorem |
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220 | (3) |
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222 | (1) |
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Appendix E Second Quantization |
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223 | (5) |
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227 | (1) |
| Index |
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228 | |