INTRODUCTION TO QUANTUM STATISTICAL MECHANICS (2ND EDITION)

Introduction to Quantum Statistical Mechanics (Second Edition) may be used as an advanced textbook by graduate students, even ambitious undergraduates in physics. It is also suitable for non experts in physics who wish to have an overview of some of the classic and fundamental quantum models in the subject. The explanation in the book is detailed enough to capture the interest of the reader, and complete enough to provide the necessary background material needed to dwell further into the subject and explore the research literature. CONTENTS 6 Preface 16 PART I QUANTUM STATISTICAL MECHANICS 18 CHAPTER 1 THE LIOUVILLE EQUATION IN CLASSICAL MECHANICS 20 1. Introduction: Statistical approach in classical and quantum mechanics 20 2. The classical statistical approach 22 a) A transformation operator G 22 b) Probability density D 23 c) The Liouville theorem 24 d) Time dependent probability density Dt, t0 26 3. Quantum analogy 28 4. Symmetry properties 29 5. Isolated dynamical systems 31 6. A system of identical monoatomic molecules 32 7. Property of reversibility 34 CHAPTER 2 THE LIOUVILLE EQUATION IN QUANTUM MECHANICS 35 1. The X-representation 35 2. Quantum statistical approach 38 a) Statistical operators 38 b) The Liouville equation 40 c) Operator Ut, t0 41 d) Properties of the statistical operators 44 3. Symmetry properties 45 4. Discrete X-representation 50 5. Discrete momentum representation 52 6. Compatibility with the Schroedinger equation 53 7. Limit transition and cyclic boundary condition 55 8. An isolated dynamical system 58 9. Conservation and non-conservation of particle number 60 a) N-particle wave functions 60 b) X-representation for variable particle numbers 62 c) The Hilbert space of wave functions and its subspaces 64 d) A projection operator 66 e) A combined index 67 CHAPTER 3 CANONICAL DISTRIBUTION AND THERMODYNAMIC FUNCTIONS 68 1. Integrals of motion 68 2. The Gibbs canonical distribution 69 3. Thermodynamic functions 71 4. Quasi-static processes 73 a) The concept of quasi-static process 73 b) Construction of quasi-static processes 75 c) Interpretation of terms 77 d) Heat capacity 78 e) Homogeneous systems 79 f) Relation between H and E 79 5. Passing to limits 81 a) Basic assumptions 81 b) Boundary surface 82 c) Limits 83 d) Validity of speculations on passing to limits 84 6. The grand canonical ensemble 85 a) Statistical operators 85 b) De nitions of , , and G 86 c) Uniqueness of j 87 d) Thermodynamic functions 90 e) Passing to the limit V 92 7. Quantum approach in the theory of classical canonical distribution 94 8. Entropy of dynamical systems 96 a) Canonical ensemble 96 b) Extremum property of the entropy 97 c) An auxiliary operator 97 d) The grand canonical ensemble 99 e) Time independence 101 f) Entropy of non-equilibrium states 101 g) Difficulties connected with study of non-equilibrium processes 103 CHAPTER 4 TWO-TIME CORRELATION FUNCTIONS AND THE GREEN FUNCTIONS IN THE THEORY OF STATISTICAL EQUILIBRIUM 105 1. Two-time correlation functions for quantum systems 105 2. Spectral Intensity 108 a) Definition 108 b) Basic properties 109 3. The two-time Green function 111 a) Definitions 111 b) The Fourier representations 112 c) Passing to the limit 112 d) Basic properties 115 4. Infinitesimal perturbation 116 a) A perturbation Hamiltonian 116 b) Variation of a average value 117 c) The retarded and advanced Green Functions 119 d) Some remarks 120 5. The Green function for classical systems 121 a) Correlation functions and the spectral intensity 122 b) Average values of the poison brackets 125 c) The retarded and advanced Green functions 126 d) Basic properties 128 6. Classical variation of average value 130 7. The Laplace transformation of correlation functions 133 CHAPTER 5 STATISTICAL OPERATORS 136 1. Introduction 136 2. Statistical operators of particle complexes 137 a) Wave functions and operators 137 b) Statistical operators 137 c) Statistical operators of particle complexes 138 d) Fluctuations 141 e) Some properties of the statistical operators of particle complexes 143 3. Time evolution of the statistical operators 144 a) A hierarchy of equations for the reduced statistical operators 144 b) Some remarks on the physical meaning of the hierarchy of equations 147 4. The application of the method of statistical operators to systems of monoatomic spinless molecules 149 a) A model and the evolution equation 149 b) Passing to statistical operators Fs 149 c) The limit equation of evolution of statistical operators 151 d) The density of particle number and its fluctuation 153 References to Part I 156 PART II SOME ASPECTS OF THE METHOD OF SECONDARY QUANTIZATION 160 CHAPTER 1 MATRIX REPRESENTATION OF SYMMETRIC DYNAMICAL OPERATORS 163 1. Introduction 163 2. Symmetry properties 164 a) Permutations 164 b) Symmetry and antisymmetry 166 3. Symmetric operators 166 a) Symmetric single-particle operators 167 b) Symmetric two-particle operators 168 c) Symmetric s-particle operators 169 4. Some identities 170 5. Hamiltonians 172 CHAPTER 2 PASSING FROM CONTINUOUS TO DISCRETE REPRESENTATION. INTRODUCTION OF THE OCCUPATION NUMBERS 174 1. Representation of wave functions 174 a) Discrete f-representation 174 b) Discrete momentum representation 176 c) Periodicity 177 d) Passing to the limit. Quasi-discrete representation 178 2. Occupation numbers 179 a) Definition 179 b) Bosons 181 c) Fermions 182 CHAPTER 3 REPRESENTATION OF SECONDARY QUANTIZATION FOR WAVE FUNCTIONS OF BOSONS AND FERMIONS 184 1. Bose statistics 184 a) Operators a and a+ 184 b) Quantum Bose operators a and a+ 186 c) Wave functions 186 2. Fermi statistics 189 a) Operators and + 189 b) The Pauli operators 190 c) The Fermi operators a and a+ 192 d) Wave function 194 3. Comparison of wave functions of bosons and fermions 196 4. The representation of secondary quantization 197 5. The operator of particle number 199 CHAPTER 4 REPRESENTATION OF SECONDARY QUANTIZATION FOR DYNAMICAL OPERATORS 201 1. Introduction 201 2. Lemma 202 a) Proof 203 b) Corollaries of the Lemma 206 3. Wave function 207 a) Inverse of representation 207 b) Invariance of the scalar product 208 4. Transformation of dynamical quantities to the representation of secondary quantization 209 a) Development of the transformation 209 b) Additive dynamical quantities 210 c) The particle-number density 211 d) The particle-number operator 212 e) Binary dynamical quantities 213 f) s-particle operators 215 5. The Hamiltonian in the representation of secondary quantization 216 6. The evolution of operator functions 217 a) The Heisenberg representation 217 b) Commutation relations 218 c) Time evolution of operator functions 219 d) Discussion 221 CHAPTER 5 GENERAL REMARKS ON THE METHOD OF SECONDARY QUANTIZATION 224 1. Systems independent of particle number 224 a) The grand canonical ensemble 224 b) The quasi-discrete representation 225 c) Basis and operator functions 226 d) The representation of secondary quantization for Hamiltonian 227 e) The momentum vector 228 2. Wave functions in the representation of secondary quantization 229 a) Conditions 229 b) Remarks on condition (D) 230 c) Derivation of the expression for a wave function 231 3. Dynamical systems consisting of bosons and fermions of several types 235 a) Ordinary wave functions 235 b) Operator function 236 c) Transformation of dynamical operators 237 d) Dynamical quantities of mixed type 239 e) Independence of the particle number 240 4. Commutation relations for operator functions 241 a) Commutativity of arbitrary type 241 b) Remark on commutativity 243 5. Occupation numbers and the f-representation (Comments on Chapter 2) 243 CHAPTER 6 SOME ANALOGUES OF THE METHOD OF SECONDARY QUANTIZATION IN CLASSICAL MECHANICS 247 1. The method of secondary quantization 247 a) Time evolution of the Wigner quantum operator 248 b) Passing to the limit 0 251 c) Quantization and secondary quantization 252 2. The transition to classical mechanics 252 a) A binary dynamical quantity 253 b) Time-evolution of function f 254 c) The Vlasov equation 256 3. A system of identical hard spheres 258 a) A system of two identical hard spheres 258 b) A system of N identical hard spheres 264 c) The Boltzmann-Enskog equation 267 d) The diffusion equation of a separate particle in a gas 270 e) Binary interaction 272 References to Part II 274 PART III QUADRATIC HAMILTONIANS AND THEIR APPLICATION 276 CHAPTER 1 QUADRATIC HAMILTONIANS IN STATISTICAL MECHANICS 278 1. Equilibrium properties of ideal gases 278 a) Ideal gas 278 b) Free energy and average occupation numbers 279 c) Spectrum of black-body radiation 282 d) Ideal Bose gas 283 e) Ideal Fermi gas 284 2. Diagonalization of quadratic form for the Bose and Fermi operators 287 a) A method of canonical transformation 288 b) An operator form of canonical transformation 290 3. Diagonalization of quadratic Hamiltonians in the theory of superuidity and superconductivity 292 a) The case of Bose-Einstein statistics 293 b) The case of Fermi-Dirac statistics 295 c) An operator form of canonical transformation for the Bose-Einstein statistic 297 4. The Bloch-de Dominicis statistical theorem 299 a) The Bloch-de Dominicis theorem 300 b) The proof of the Bloch-de Dominicis theorem 302 CHAPTER 2 DIAGONALIZATION OF ARBITRARY QUADRATIC FORMS 304 1. Diagonalization of a quadratic form of the Bose operators 304 a) Canonical transformation 304 b) A set of equations for parameters of canonical transformation 306 c) Matrix form 307 d) Eigenvalues 309 e) Condition on parameters of the canonical transformation 311 f) Diagonalization 314 g) Hamiltonian with linear operator term 316 2. Diagonalization of the quadratic form of the Fermi operators 317 a) Canonical transformation 318 b) A set of equations for parameters of canonical transformation 319 c) Matrix form 320 d) Eigenvalue 321 e) Condition on parameters of canonical transformation 322 f) A lemma on eigenvectors and eigenvalues 328 g) Diagonalization 330 References to Part III 332 PART IV SUPERFLUIDITY AND QUASI-AVERAGES IN PROBLEMS OF STATISTICAL MECHANICS 334 CHAPTER 1 SUPERFLUIDITY AND A NON-IDEAL BOSE GAS 336 1. An ideal Bose gas 336 2. Condensation in momentum space and the phenomenon of superuidity 339 3. Superuidity 349 a) Quasi-particles 350 b) Occurrence of superuidity 353 c) Equations of motion 355 d) Elementary excitations and superuidity 360 e) Remarks 363 f) Momentum distribution 364 g) Correlation function 368 h) Discussion 371 CHAPTER 2 QUASI-AVERAGES IN PROBLEMS OF STATISTICAL MECHANICS 373 1. Green functions constructed from usual averages. Additive conservation laws and selected rules 373 2. Degeneration of the state of statistical equilibrium. Introduction of quasi-averages 376 a) Degeneration of states of statistical equilibrium 377 b) Quasi-averages 379 c) Degeneration in the theory of the crystalline state 380 d) An ideal Bose gas 383 e) Quasi-averages in the theory of superconductivity 387 f) The Dicke model 394 g) Quasi-averages for an arbitrary system 397 CHAPTER 3 ON THE DEFINITION OF QUASI-AVERAGES FOR A HAMILTONIAN WITH NEGATIVE FOUR-FERMION INTERACTION 400 1. Limit properties of free energies 400 2. Quasi-averages for a Hamiltonian with a negative four-fermion interaction 404 CHAPTER 4 THE PRINCIPLE OF CORRELATION WEAKENING AND THEOREMS ON SINGULARITIES OF TYPE 1=q2 414 1. Principle of correlation weakening 414 2. Theorems on singularities of type 1=q2 and some applications of the method of quasi-averages 419 CHAPTER 5 ON SOME QUESTIONS CONNECTED WITH PROBLEMS OF THE FOUNDATION OF STATISTICAL MECHANICS 421 1. Some remarks on the ergodic theory 421 2. A dynamical system interacting with boson field 425 References to Part IV 430 Photos 434 Index 436 Problem Non Ideal Bose Gas, Superfluidity and Fundamental Aspects of Quasiaverages Application Quasiaverages in the Theory of Superconductivity Correlations Weakening and Theorems on Singularities of the Type 1/Q2 Some Remarks on the Ergodic Theory (and Real Example From Statistical Mechanics Where the Usual Ergodic Conditions Not Satisfed) Presented Revised (More Simplified Version) Method of Second Quantized Representation for Wave Functions of Boson and Fermion Particles and Dynamical Operators.