Origin, Verifications, and Applications

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This is the first scientic book devoted to the Pauli exclusion principle, which is a fundamental principle of quantum mechanics and is permanently applied in chemistry, physics, and molecular biology. However, while the principle has been studied for more than 90 years, rigorous theoretical foundations still have not been established and many unsolved problems remain.

Following a historical survey in Chapter 1, the book discusses the still unresolved questions around this fundamental principle. For instance, why, according to the Pauli exclusion principle, are only symmetric and antisymmetric permutation symmetries for identical particles realized, while the Schrödinger equation is satisfied by functions with any permutation symmetry? Chapter 3 covers possible answers to this question. The construction of function with a given permutation symmetry is described in the previous Chapter 2, while Chapter 4 presents effective and elegant methods for finding the Pauli–allowed states in atomic, molecular, and nuclear spectroscopy. Chapter 5 discusses parastatistics and fractional statistics, demonstrating that the quasiparticles in a periodical lattice, including excitons and magnons, are obeying modified parafermi statistics.

With detailed appendices, *The Pauli Exclusion Principle: Origin, Verifications, and Applications* is intended as a self–sufficient guide for graduate students and academic researchers in the fields of chemistry, physics, molecular biology and applied mathematics. It will be a valuable resource for any reader interested in the foundations of quantum mechanics and its applications, including areas such as atomic and molecular spectroscopy, spintronics, theoretical chemistry, and applied fields of quantum information.

Preface xi

**1 Historical Survey 1**

1.1 Discovery of the Pauli Exclusion Principle and Early Developments 1

1.2 Further Developments and Still Existing Problems 11

References 21

**2 Construction of Functions with a Definite Permutation Symmetry 25**

2.1 Identical Particles in Quantum Mechanics and Indistinguishability Principle 25

2.2 Construction of Permutation–Symmetric Functions Using the Young Operators 29

2.3 The Total Wave Functions as a Product of Spatial and Spin Wave Functions 36

2.3.1 Two–Particle System 36

2.3.2 General Case of N–Particle System 41

References 49

**3 Can the Pauli Exclusion Principle Be Proved? 50**

3.1 Critical Analysis of the Existing Proofs of the Pauli Exclusion Principle 50

3.2 Some Contradictions with the Concept of Particle Identity and their Independence in the Case of the Multidimensional Permutation Representations 56

References 62

**4 Classification of the Pauli–Allowed States in Atoms and Molecules 64**

4.1 Electrons in a Central Field 64

4.1.1 Equivalent Electrons: L S Coupling 64

4.1.2 Additional Quantum Numbers: The Seniority Number 71

4.1.3 Equivalent Electrons: j j Coupling 72

4.2 The Connection between Molecular Terms and Nuclear Spin 74

4.2.1 Classification of Molecular Terms and the Total Nuclear Spin 74

4.2.2 The Determination of the Nuclear Statistical Weights of Spatial States 79

4.3 Determination of Electronic Molecular Multiplets 82

4.3.1 Valence Bond Method 82

4.3.2 Degenerate Orbitals and One Valence Electron on Each Atom 87

4.3.3 Several Electrons Specified on One of the Atoms 91

4.3.4 Diatomic Molecule with Identical Atoms 93

4.3.5 General Case I 98

4.3.6 General Case II 100

References 104

**5 Parastatistics, Fractional Statistics, and Statistics of Quasiparticles of Different Kind 106**

5.1 Short Account of Parastatistics 106

5.2 Statistics of Quasiparticles in a Periodical Lattice 109

5.2.1 Holes as Collective States 109

5.2.2 Statistics and Some Properties of Holon Gas 111

5.2.3 Statistics of Hole Pairs 117

5.3 Statistics of Cooper s Pairs 121

5.4 Fractional Statistics 124

5.4.1 Eigenvalues of Angular Momentum in the Three– and Two–Dimensional Space 124

5.4.2 Anyons and Fractional Statistics 128

References 133

**Appendix A: Necessary Basic Concepts and Theorems of Group Theory 135**

A.1 Properties of Group Operations 135

A.1.1 Group Postulates 135

A.1.2 Examples of Groups 137

A.1.3 Isomorphism and Homomorphism 138

A.1.4 Subgroups and Cosets 139

A.1.5 Conjugate Elements. Classes 140

A.2 Representation of Groups 141

A.2.1 Definition 141

A.2.2 Vector Spaces 142

A.2.3 Reducibility of Representations 145

A.2.4 Properties of Irreducible Representations 147

A.2.5 Characters 148

A.2.6 The Decomposition of a Reducible Representation 149

A.2.7 The Direct Product of Representations 151

A.2.8 Clebsch Gordan Coefficients 154

A.2.9 The Regular Representation 156

A.2.10 The Construction of Basis Functions for Irreducible Representation 157

References 160

**Appendix B: The Permutation Group 161**

B.1 General Information 161

B.1.1 Operations with Permutation 161

B.1.2 Classes 164

B.1.3 Young Diagrams and Irreducible Representations 165

B.2 The Standard Young Yamanouchi Orthogonal Representation 167

B.2.1 Young Tableaux 167

B.2.2 Explicit Determination of the Matrices of the Standard Representation 170

B.2.3 The Conjugate Representation 173

B.2.4 The Construction of an Antisymmetric Function from the Basis Functions for Two Conjugate Representations 175

B.2.5 Young Operators 176

B.2.6 The Construction of Basis Functions for the Standard Representation from a Product of N Orthogonal Functions 178

References 181

**Appendix C: The Interconnection between Linear Groups and Permutation Groups 182**

C.1 Continuous Groups 182

C.1.1 Definition 182

C.1.2 Examples of Linear Groups 185

C.1.3 Infinitesimal Operators 187

C.2 The Three–Dimensional Rotation Group 189

C.2.1 Rotation Operators and Angular Momentum Operators 189

C.2.2 Irreducible Representations 191

C.2.3 Reduction of the Direct Product of Two Irreducible Representations 194

C.2.4 Reduction of the Direct Product of k Irreducible Representations. 3n j Symbols 197

C.3 Tensor Representations 201

C.3.1 Construction of a Tensor Representation 201

C.3.2 Reduction of a Tensor Representation into Reducible Components 202

C.3.3 Littlewood s Theorem 207

C.3.4 The Reduction of U2j + 1 R3 209

C.4 Tables of the Reduction of the Representations U 2j+1 to the Group R3 214

References 216

**Appendix D: Irreducible Tensor Operators 217**

D.1 Definition 217

D.2 The Wigner Eckart Theorem 220

References 222

**Appendix E: Second Quantization 223**

References 227

Index 228

This is the first scientic book devoted to the Pauli exclusion principle, which is a fundamental principle of quantum mechanics and is permanently applied in chemistry, physics, and molecular biology. However, while the principle has been studied for more than 90 years, rigorous theoretical foundations still have not been established and many unsolved problems remain. Following a historical survey in Chapter 1, the book discusses the still unresolved questions around this fundamental principle. For instance, why, according to the Pauli exclusion principle, are only symmetric and antisymmetric permutation symmetries for identical particles realized, while the Schrödinger equation is satisfied by functions with any permutation symmetry? Chapter 3 covers possible answers to this question. The construction of function with a given permutation symmetry is described in the previous Chapter 2, while Chapter 4 presents effective and elegant methods for finding the Pauli-allowed states in atomic, molecular, and nuclear spectroscopy. Chapter 5 discusses parastatistics and fractional statistics, demonstrating that the quasiparticles in a periodical lattice, including excitons and magnons, are obeying modified parafermi statistics. With detailed appendices, The Pauli Exclusion Principle: Origin, Verifications, and Applications is intended as a self-sufficient guide for graduate students and academic researchers in the fields of chemistry, physics, molecular biology and applied mathematics. It will be a valuable resource for any reader interested in the foundations of quantum mechanics and its applications, including areas such as atomic and molecular spectroscopy, spintronics, theoretical chemistry, and applied fields of quantum information.

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