Green's Functions in Quantum Physics (eBook)

Preface to the Third Edition 9
Preface to the Second Edition 10
Preface to the First Edition 12
Contents 14
Part I Green’s Functions in Mathematical Physics 19
1 Time-Independent Green’s Functions 20
1.1 Formalism 20
1.2 Examples 26
1.3 Summary 31
Further Reading 34
Problems7 34
2 Time-Dependent Green’s Functions 37
2.1 First-Order Case 37
2.2 Second-Order Case 42
2.3 Summary 49
Further Reading 52
Problems 52
Part II Green’s Functions in One-Body Quantum Problems 54
3 Physical Signi.cance of Application to the Free-Particle Case 55
3.1 General Relations 55
3.2 The Free-Particle Case 57
3.3 The Free-Particle Klein–Gordon Case 61
3.4 Summary 64
Further Reading 65
Problems 65
4 Green’s Functions and Perturbation Theory 68
4.1 Formalism 68
4.2 Applications 77
4.3 Summary 84
Further Reading 87
Problems 87
5 Green’s Functions for Tight-Binding Hamiltonians 89
5.1 Introductory Remarks 89
5.2 The Tight-Binding Hamiltonian (TBH) 92
5.3 Green’s Functions 99
5.4 Summary 113
Further Reading 114
Problems 114
6 Single Impurity Scattering 123
6.1 Formalism 123
6.2 Explicit Results for a Single Band 130
6.3 Applications 137
6.4 Summary 149
Further Reading 151
Problems 152
7 Two or More Impurities Disordered Systems
7.1 Two Impurities 153
7.2 In.nite Number of Impurities 162
7.3 Summary 180
Further Reading 181
Problems 181
8 Electrical Conductivity and Green’s Functions 184
8.1 Electrical Conductivity and Related Quantities 184
8.2 Various Methods of Calculation 187
8.3 Conductivity in Terms of Green’s Functions 194
8.4 Summary 206
Further Reading 208
Problems 208
9 Localization, Transport, and Green’s Functions 210
9.1 An Overview 210
9.2 Disorder, Di.usion, and Interference 214
9.3 Localization 219
9.4 Conductance and Transmission 227
9.5 Scaling Approach 230
9.6 Other Calculational Techniques 235
9.7 Localization and Green’s Functions 237
9.8 Applications 249
9.9 Summary 251
Further Reading 254
Problems 254
Part III Green’s Functions in Many-Body Systems 258
10 Definitions 259
10.1 Single-Particle Green’s Functions in Terms 259
of Field Operators 259
10.2 Green’s Functions for Interacting Particles 263
10.3 Green’s Functions for Noninteracting Particles 267
10.4 Summary 270
Further Reading 271
Problems 271
11 Properties and Use of the Green’s Functions 273
11.1 Analytical Properties of 273
and 273
11.2 Physical Signi.cance and Use of 278
and 278
11.3 Quasiparticles 285
11.4 Summary 291
Further Reading 293
Problems 293
12 Calculational Methods for 294
12.1 Equation of Motion Method 294
12.2 Diagrammatic Method for Fermions at 298
= 0 298
12.3 Diagrammatic Method for 307
0 307
12.4 Partial Summations. Dyson’s Equation 309
12.5 Other Methods of Calculation 315
12.6 Summary 315
Further Reading 316
Problems 316
13 Applications 318
13.1 Normal Fermi Systems. Landau Theory 318
13.2 High-Density Electron Gas 321
13.3 Dilute Fermi Gas 328
13.4 Superconductivity 331
13.5 The Hubbard Model 336
13.6 Summary 341
Further Reading 342
Problems 343
Appendices 345
A Dirac’s delta Function 345
B Dirac’s bra and ket Notation 348
C Solutions of Laplace and Helmholtz Equations in Various Coordinate Systems1 352
C.1 Helmholtz Equation (. 352
+ 352
( 352
= 0 352
C.2 Vector Derivatives 354
C.3 Schr¨ odinger Equation in Centrally Symmetric 355
3- and 2-Dimensional Potential 355
D Analytic Behavior of G(z) Near a Band Edge 357
E Wannier Functions 360
F Renormalized Perturbation Expansion (RPE) 361
G Boltzmann’s Equation 366
H Transfer Matrix, S-Matrix, etc. 371
Further Reading 380
I Second Quantization 381
Solutions of Selected Problems 392
Chapter 1 392
Chapter 2 407
Chapter 3 408
Chapter 4 413
Chapter 5 420
Chapter 6 426
Chapter 7 430
References 447
Index 461