Quantum Statistical Mechanics Book

Preface xi

1 Foundations of quantum statistical mechanics 1

1.1 The density operator and probability 1

1.2 The Gleason theorem and consequences 6

1.3 Calculation of averages of observables 9

Appendix 1A: Gleason theorem 12

References 18

2 Elementary examples 19

2.1 Introduction 19

2.2 Harmonic oscillator 19

2.3 Spin one-half and two-level atoms 27

Appendix 2A: the Fokker-Planck equation 34

References 35

3 Quantum statistical master equation 37

3.1 Reduced observables 37

3.2 The Pauli equation 39

3.3 The weak coupling master equation for open systems 42

3.4 Pauli equation: time scaling 46

3.5 Reservoir states: rigorous results and models 53

3.6 The completely positive evolution 54

Appendix 3A: Chapman-Kolmogorov master equation 57

References 59

4 Quantum kinetic equations 61

4.1 Introduction 61

4.2 Reduced density matrices and the B.B.G.Y.K. hierarchy 61

4.3 Derivation of the quantum Boltzmann equation 63

4.4 Phase space quantum Boltzmann equation 66

4.5 Memory of initial correlations 76

4.6 Quantum Vlasov equation 79

Appendix 4A: Phase space distribution functions 80

References 83

5 Quantum irreversibility 85

5.1 Quantum reversibility 85

5.2 Master equation and irreversibility 87

5.3 Time irreversibility of the generalized master and Pauli equations 87

5.4 Irreversibility of the quantum operator Boltzmann equation 89

5.5 Reversibility of the quantum Vlasov equation 90

5.6 Completely positive dynamical semigroup: a model 92

Appendix 5A: the quantum time reversal operator 94

References 96

6 Entropy and dissipation: the microscopic theory 98

6.1 Introduction 98

6.2Macroscopic non-equilibrium thermodynamics 98

6.3 Dissipation and the quantum Boltzmann equation 105

6.4 Negative probability and the quantum $$ theorem 111

6.5 Entropy and master equations 113

Appendix 6A: quantum recurrence 120

References 121

7 Global equilibrium: thermostatics and the microcanonical ensemble 123

7.1 Boltzmann's thermostatic entropy 124

7.2 Thermostatics 125

7.3 Canonical and grand canonical distribution of Gibbs 126

7.4 Equilibrium fluctuations 129

7.5 Negative probability in equilibrium 131

7.6 Non-interacting fermions and bosons 132

7.7 Equilibrium limit theorems 136

References 139

8 Bose-Einstein ideal gas Condensation 141

8.1 Introduction 141

8.2 Continuum box model of condensation 142

8.3 Harmonic oscillator trap and condensation 145

8.4 4He: the λ transition 148

8.5 Fluctuations: comparison of the grand canonical and canonical ensemble 150

8.6 A master equation view of Bose condensation 152

Appendix 8A: exact treatment of condensate traps 155

References 158

9 Scaling, renormalization and the Ising model 159

9.1 Introduction 159

9.2 Mean field theory and critical indices 160

9.3 Scaling 167

9.4 Renormalization 169

9.5 Renormalization and scaling 172

9.6 Two-dimensional Ising model renormalization 174

References 177

10 Relativistic covariant statistical mechanics of many particles 178

10.1 Introduction 178

10.2 Quantum many-particle dynamics: the event picture 180

10.3 Two-event Boltzmann equation 183

10.4 Some results of the quantum event Boltzmann equation 187

10.5 Relativistic quantum equilibrium event ensembles 191

References 197

11 Quantum optics and damping 199

11.1 Introduction 199

11.2 Atomic damping: atomic master equation 199

11.3 Cavity damping: the micromaser: detection 206

11.4 Detection master equation for the cavity field 207

Appendix 11A: the field Quantization and interaction 214

References 219

12 Entanglements 221

12.1 Introduction 221

12.2 Entanglements: foundations 221

12.3 Entanglements: Q bits 224

12.4 Entanglement consequences: quantum teleportation, the Bob and Alice story 226

12.5 Entanglement consequences: dense coding 228

12.6 Entanglement consequences: quantum computation 228

12.7 Decoherence: entanglement destruction 231

12.8 Decoherence correction (error correction) 235

Appendix 12A: entanglement and the Schmidt decomposition 236

References 238

13 Quantum measurement and irreversibility 240

13.1 Introduction 240

13.2 Ideal quantum measurement 241

13.3 Irreversibility: measurement master equations 243

13.4 An Open system master equation model for measurement 246

13.5 Stochastic energy based collapse 248

References 251

14 Quantum Langevin equation and quantum Brownian motion 253

14.1 Introduction 253

14.2 Quantum Langevin equation 254

14.3 Quantum Langevin equation with measurement 260

References 262

15 Linear response: fluctuation and dissipation theorems 264

15.1 Introduction 264

15.2 Quantum linear response in the steady state 266

15.3 Linear response, time dependent 269

15.4 Fluctuation and dissipative theorems 272

15.5 Comments and comparisons 277

References 279

16 Time-dependent quantum Green's functions 281

16.1 Introduction 281

16.2 One- and two-time quantum Green's functions and their properties 282

16.3 Analytic properties of Green's functions 284

16.4 Connection to linear response theory 288

16.5 Green's function hierarchy truncation 289

16.6 Keldysh time-loop path perturbation theory 297

References 302

17 Decay scattering 303

17.1 Basic notions and the Wigner-Weisskopf theory 303

17.2 Wigner-Weisskopf method: pole approximation 306

17.3 Wigner-Weisskopf method and Lee-Friedrichs model with a single channel 312

17.4 Wigner-Weisskopf and multichannel decay 318

17.5 Wigner-Weisskopf method with many-channel decay: the Lee-Friedrichs model 321

17.6 Gel'fand triple 332

17.7 Lax-Phillips theory 335

17.8 Application to the Stark model 354

References 362

18 Quantum statistical mechanics, extended 365

18.1 Intrinsic theory of irreversibility 365

18.2 Complex Liouvillian eigenvalue method: introduction 366

18.3 Operators and states with diagnoal singularity 367

18.4 Super operators and time evolution 369

18.5 Subdynamics and analytic continuation 371

8.6 The Pauli equation revisited 375

References 378

19 Quantum transport with tunneling and reservoir ballistic transport 379

19.1 Introduction 379

19.2 Pauli equation and boundary interaction 380

19.3 Ballistic transport 383

19.4 Green's function closed-time path theory to transport 385

References 389

20 Black hole thermodynamics 390

20.1 Introduction to black holes 390

20.2 Equilibrium Thermodynamic analogies: the first law 394

20.3 The second law of thermodynamics and black holes 397

20.4 Extended entropy principle for black holes 399

20.5 Acausal evolution: extended irreversible dynamics in black holes 401

Reference 401

A Problems 404

A.1 Comments on the problems 404

A.2 "Foundations" problems 404

A.3 Kinetic dynamics problems 407

A.4 Equilibrium and phase transition problems 409

References 410

Index 411