Annotations to Quantum Statistical Mechanics

275 Pages·2018·2.61 MB·English

by  In-Gee Kim

Annotations to Quantum
Statistical Mechanics
Pan Stanford Series on Renewable Energy — Volume 2
Annotations to Quantum
Statistical Mechanics
In-Gee Kim
editors
Preben Maegaard
Anna Krenz
Wolfgang Palz
The Rise of Modern Wind Energy
Wind Power
for the World
February 6, 2018 11:30 PSP Book - 9in x 6in 00-Annotations2QSM-Prelims
Published by
Pan Stanford Publishing Pte. Ltd.
Penthouse Level, Suntec Tower 3
8 Temasek Boulevard
Singapore 038988
Email:
February 6, 2018 11:30 PSP Book - 9in x 6in 00-Annotations2QSM-Prelims
To my wife Gajean
and
my daughter Eugenie
February 6, 2018 11:30 PSP Book - 9in x 6in 00-Annotations2QSM-Prelims
February 6, 2018 11:30 PSP Book - 9in x 6in 00-Annotations2QSM-Prelims
Contents
Preface xi
Preface of Quantum Statistical Mechanics: Green’s Function
Methods in Equilibrium and Nonequilibrium Problems xv
1 Physical Prerequisites 1
1.1 Basic QuantumMechanics 1
1.2 Representations and Equations of Motion 12
1.3 Second Quantization 23
2 Mathematical Introduction 33
2.1 Basic Definitions 33
2.2 The Boundary Condition 37
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3 Information Contained in G and G 43
3.1 Dynamical Information 43
3.2 Statistical Mechanical Information Contained in G 46
4 The Hartree and Hartree–Fock Approximations 51
4.1 Equations of Motion 51
4.2 Free Particles 54
4.3 Hartree Approximation 55
4.4 Hartree–Fock Approximation 60
5 Effects of Collisions on G 63
5.1 Lifetime of Single-Particle States 63
5.2 Born Approximation Collisions 65
5.3 Structure of c and A 69
5.4 Interpretation of the Born Collision Approximation 72
5.5 Boltzmann Equation Interpretation 75
February 6, 2018 11:30 PSP Book - 9in x 6in 00-Annotations2QSM-Prelims
viii Contents
6 A Technique for Deriving Green’s Function Approximations 77
6.1 Ordinary Perturbation Theory 81
6.2 Expansion of in V and G0 85
6.3 Expansion of in V and G 87
7 Transport Phenomena 89
7.1 Boltzmann Equation Approach to Transport 90
7.2 Green’s Function Description of Transport 98
7.3 Conservation Laws for g(U) 102
7.4 Relation of g(U) to the Distribution Function f (p, R, T) 105
8 Hartree Approximation, Collision-Less Boltzmann Equation,
and Random Phase Approximation 107
8.1 Collision-Less Boltzmann Equation 110
8.2 Linearization of the Hartree Approximation: The
Random Phase Approximation 111
8.3 Coulomb Interaction 114
8.4 Low-Temperature Fermion System and Zero Sound 120
8.5 Breakdown of the Random Phase Approximation 124
9 Relation between Real and Imaginary Time Response
Functions 129
9.1 Linear Response 129
9.2 Continuation of Imaginary Time Response to Real
Times 135
9.3 Equations of Motion in the Real-Time Domain 139
10 Slowly Varying Disturbances and the Boltzmann Equation 145
10.1 Derivation of the Boltzmann Equation 146
10.2 Generalization of the Boltzmann Equation 153
11 Quasi-Equilibrium Behavior: Sound Propagation 165
11.1 Complete Equilibrium Solutions 165
11.2 Local Equilibrium Solutions 169
11.3 Conservation Laws 172
11.4 Application of Conservation Laws to the
Quasi-Equilibrium Situation 174
11.5 Sound Propagation 180
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Contents ix
12 The Landau Theory of the Normal Fermi Liquid 185
12.1 The Boltzmann Equation 185
12.2 Conservation Laws 190
12.3 Thermodynamic Properties 195
13 Shielded Potential 201
13.1 Green’s Function Approximation for Coulomb Gas 201
13.2 Calculation of the Equation of State of a Coulomb
Gas 209
14 T Approximation 225
14.1 Structure of the T Matrix 225
14.2 Breakdown of the T Approximation in Metals 235
Appendix A: Finite-Temperature Perturbation Theory 241
References and Supplementary Reading 251
Index 255