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Chapter 1 Fundamental Principles1

1.1 Introduction： The Characters of Thermodynamics and Sta-tistical Physics and Their Relationship1

1.2 Basic Thermodynamic Identities3

1.3 Fundamental Principles and Conclusions of Classical Sta-tistics7

1.3.1 Microscopic and Macroscopic Descriptions， Statistical Distribution Functions8

1.3.2 Liouville Theorem10

1.3.3 Statistical Independence13

1.3.4 Microscopical Canonical ， Canonical and Grand Ca-nonical Ensembles14

1.4 Boltzmann Gas16

1.5 Density Matrix19

1.5.1Density Matrix20

1.5.2 Some General Properties of the Density Matrix22

1.6 Liouville Theorem in Quantum Statistics25

1.7 Canonical Ensemble29

1.8 Grand Canonical Ensemble33

1.8.1 Fundamental Expression of the Grand Canonical En-semble33

1.8.2 Derivation of the Fundamental Thermodynamic Identity34

1.9 Probability Distribution and Slater Sum36

1.9.1Meaning of the Diagonal Elements of the Density Ma-trix36

1.9.2 Slater Summation37

1.9.3 Example： Probability of the Harmonic Ensemble39

1.10 Theory of the Reduced Density Matrix44

Chapter 2 The Perfect Gas in Quantum Statistics55

2.1Indistinguishability Principle for Identical Particles55

2.2 Bose Distribution and Fermi Distribution60

2.2.1Perfect Gases in Quantum Statistics60

2.2.2 Bose Distribution61

2.2.3 Fermi Distribution62

2.2.4 Comparison of Three Distributions； Gibbs Paradox Again62

2.3 Density of States， Chemical Potential and Equation of State65

2.3.1Density of States65

2.3.2 Virial Equation for Quantum Ideal Gases66

2.4 Black-body Radiation69

2.4.1Thermodynamic Quantities for the Black-body Radia-tion Field71

2.4.2 Exitance and Variety of Displacement Laws72

2.4.3 Waveband Radiant Exitance and Waveband Photon Exitance75

2.5 Bose-Einstein Condensation in Bulk79

2.5.1Bose Condensation， Dynamical Quantities with Tem-perature Lower Than the λ Point79

2.5.2 Discontinuity of the Derivatives of Specific Heat and λ Phenomena82

2.5.3 Two-Fluid Theory85

2.5.4 2-D Case86

2.6 Degenerate Fermi Gases and Fermi Sphere87

2.6.1 Properties of Fermi Gases at Absolute Zero87

2.6.2 Specific Heat of Free Electron Gases89

2.6.3 State Equation， Heat Capacity at Constant Pressure，and Heavy Fermions93

2.7 Fermi Integrals and their Low Temperature Expansion95

2.8 Magnetism of Fermi Gases97

2.8.1 Spin Magnetism： Paramagnetism98

2.8.2 Energy Spectra and Stationary States of Electrons in a Homogeneous Magnetic Field100

2.8.3 Diamagnetism of Orbital Motion of Free Electrons103

2.9 Peierls Perturbation Expansion of Free Energy106

2.9.1 Classical Case107

2.9.2 Quantum Case108

2.9.3 Expansion of Free Energy of an Ideal Gas in an Exter-nal Field110

2.10 Appendix114

Chapter 3 Second Quantization and Model Hamiltonians117

3.1 Necessity of Second Quantization117

3.2 Second Quantization for Bose System119

3.3 Second Quantization ·Fermi System128

3.4 Some Conservation Laws135

3.5 Some Model Hamiltonians140

3.6 Electron Gases with Coulomb Interaction145

3.6.1 Completely Ionized Gases — the High Temperature Plasma146

3.6.2 The Degenerate Electron Gas with Coulomb Interac-tion （Metal Plasma）150

3.7 Anderson Model167

Chapter 4 Least Action Principle， Field Quantization and the Electron-Phonon System169

4.1 Classical Description of Lattice Vibrations169

4.2 Continuous Media Model of Lattice Vibration （Classical）174

4.3 The Least Action Principle， Euler-Lagrange Equation and Hamilton Equation177

4.4 Lagrangian and Hamiltonian of Continuous Media188

4.5 Quantization of the Lattice Vibration Field191

4.6 Debye Theory of Specific Heat of Solids197

4.7 The Electron-Phonon System and the Frohlich Hamiltonian203

Chapter 5 Bose-Einstein Condensation208

5.1 Spatial and Momentum Distributions of Bose-Einstein Con-densation in Harmonic Traps and Bloch Summation208

5.1.1 Introduction208

5.1.2 Generalized Expression for Particle Density209

5.1.3 Distributions for Ideal Systems211

5.1.4 New Expression with Clear Physical Picture212

5.1.5 Momentum Distributions215

5.1.6 Results of Numerical Calculations216

5.1.7 Discussion and Concluding Remarks216

5.1.8 Momentum Distribution of BEC219

5.2 BEC in Confined Geometry and Thermodynamic Mapping220

5.2.1 Introduction220

5.2.2 Confinement Effects222

5.2.3 Thermodynamic Mapping222

5.2.4 Mapping Relation for Confined BEC225

5.2.5 Determination of the Critical Temperature229

5.2.6 Discussion233

Chapter 6 Some Inverse Problems in Quantum Statistics235

6.1 Introduction235

6.2 Specific Heat-Phonon Spectrum Inversion237

6.2.1 Technique for Eliminating Divergences239

6.2.2 Unique Existence Theorem and Exact SPIE Solution241

6.2.3 Summary242

6.3 Concrete Realization of Inversion244

6.3.1 The Specific Heat-Phonon Spectrum Inversion Problem244

6.3.2 Results and Concluding Remarks250

6.4 Mobius Inversion Formula252

6.4.1 Riemann ξ Function and Mobius Function252

6.4.2 Mobius Inversion Formula253

6.4.3 The Modified Mobius Inversion Formula255

6.4.4 Applications in Physics256

6.5 Unification of the Theories258

6.5.1 Introduction258

6.5.2 Deriving Chen’s Formula from Dai’s Exact Solution259

6.5.3 Concluding Remarks262

6.6 Appendix263

Chapter 7 An Introduction to Theory of Green’s Func-tions265

7.1 Temperature -Time Green’s Functions265

7.1.1 Definition of Temperature -Time Green’s Functions265

7.1.2 The Equation of Motion of Double-Time Green’s Functions269

7.1.3 Time Correlation Functions270

7.2 Spectral Theorem272

7.2.1 Spectral Representation of Time Correlation Functions272

7.2.2 Spectral Representations of Retarded and Advanced Green’s Functions274

7.2.3 Spectral Representation of Causal Green’s Functions278

7.3 Example： Ideal Quantum Gases279

7.4 Theory of Superconductivity with Double-Time Green’s Functions283

7.5 Higher-Order Spectral Theorem， Sum Rules and Unique-ness290

Chapter 8 A Unified Diagonalization Theorem for Quad-ratic Hamiltonian291

8.1 A Model Hamiltonian293

8.2 Diagonalization Theorem for Fermi Quadratic Forms296

8.3 Conclusion： A Unified Diagonalization Theorem305

Chapter 9 Functional Integral Approach： A Third Formu-lation of Quantum Statistical Mechanics308

9.1 Introduction308

9.1.1 Hubbard’s Method309

9.1.2 Difficulties310

9.2 An Operator Identity311

9.3 Functional Integral Formulation of Quantum Statistical Me-chanics311

9.4 Reality and Method of Steepest Descents314

9.5 Discussion and Concluding Remarks318

9.6 Some Recent Developments318

9.7 Application： An Exact Solution319

References324

Index330