Introduction to Quantum Statistical Mechanics

The following sections are included:

• CONTENTS

• PREFACE

The following sections are included:

• INTRODUCTION: STATISTICAL APPROACH IN CLASSICAL AND QUANTUM MECHANICS

• THE CLASSICAL STATISTICAL APPROACH
  • A Transformation Operator G
  • Probability Density
  • The Liouville Theorem
  • Time Dependent Probability Density

• QUANTUM ANALOGY

• SYMMETRY PROPERTIES

• ISOLATED DYNAMICAL SYSTEMS

• A SYSTEM OF IDENTICAL MONOATOMIC MOLECULES

• PROPERTY OF REVERSIBILITY

The following sections are included:

• THE X-REPRESENTATION

• QUANTUM STATISTICAL APPROACH
  • Statistical Operators
  • The Liouville Equation
  • Operator U_{t, t0}
  • Properties of the Statistical Operators

• SYMMETRY PROPERTIES

• DISCRETE X-REPRESENTATION

• DISCRETE MOMENTUM REPRESENTATION

• COMPATIBILITY WITH THE SCHROEDINGER EQUATION

• LIMIT TRANSITION AND CYCLIC BOUNDARY CONDITION

• AN ISOLATED DYNAMICAL SYSTEM

• CONSERVATION AND NON-CONSERVATION OF PARTICLE NUMBER
  • N-Particle Wave Functions
  • X-Representation for Variable Particle Numbers
  • The Hilbert Space of Wave Functions and its Subspaces
  • A Projection Operator
  • A Combined Index

The following sections are included:

• INTEGRALS OF MOTION

• THE GIBBS CANONICAL DISTRIBUTION

• THERMODYNAMIC FUNCTIONS

• QUASI-STATIC PROCESSES
  • The concept of Quasi-Static Process
  • Construction of Quasi-Static Processes
  • Interpretation of Terms
  • Heat Capacity
  • Homogeneous Systems
  • Relation Between H and E

• PASSING TO LIMITS
  • Basic Assumptions
  • Boundary Surface
  • Limits
  • Validity of Speculations on Passing to Limits

• THE GRAND CANONICAL ENSEMBLE
  • Statistical Operators
  • Definitions of μ, Γ, and G
  • Uniqueness of μ_j
  • Thermodynamic Functions
  • Passing to the Limit V → ∞

• QUANTUM APPROACH IN THE THEORY OF CLASSICAL CANONICAL DISTRIBUTION

• ENTROPY OF DYNAMICAL SYSTEMS
  • Canonical Ensemble
  • Extremum Property of the Entropy
  • An Auxiliary Operator
  • The Grand Canonical Ensemble
  • Time Independence
  • Entropy of Non-Equilibrium States
  • Difficulties Connected with Study of Non-Equilibrium Processes

The following sections are included:

• TWO-TIME CORRELATION FUNCTIONS FOR QUANTUM SYSTEMS

• SPECTRAL INTENSITY
  • Definition
  • Basic Properties

• THE TWO-TIME GREEN FUNCTION
  • Definitions
  • The Fourier Representations
  • Passing to the Limit
  • Basic Properties

• INFINITESIMAL PERTURBATION
  • A Perturbation Hamiltonian
  • Variation of a Average Value
  • The Retarded and Advanced Green Functions
  • Some remarks

• THE GREEN FUNCTION FOR CLASSICAL SYSTEMS
  • Correlation Functions and the Spectral Intensity
  • Average Values of the Poison Brackets
  • The Retarded and Advanced Green Functions
  • Basic Properties

• CLASSICAL VARIATION OF AVERAGE VALUE

• THE LAPLACE TRANSFORMATION OF CORRELATION FUNCTIONS

The following sections are included:

• INTRODUCTION

• STATISTICAL OPERATORS OF PARTICLE COMPLEXES
  • Wave Functions and Operators
  • Statistical Operators
  • Statistical Operators of Particle Complexes
  • Fluctuations
  • Some properties of the Statistical Operators of Particle Complexes

• TIME EVOLUTION OF THE STATISTICAL OPERATORS
  • A Hierarchy of Equations for the Reduced Statistical Operators
  • Some Remarks on the Physical Meaning of the Hierarchy of Equations

• THE APPLICATION OF THE METHOD OF STATISTICAL OPERATORS TO SYSTEMS OF MONOATOMIC SPINLESS MOLECULES
  • A Model and the Evolution Equation
  • Passing to Statistical Operators F_s
  • The Limit Equation of Evolution of Statistical Operators
  • The Density of Particle Number and its Fluctuation

• REFERENCES TO PART I

In the second Part of the book, we present a consistent approach to the method of secondary quantization. The results of this method are widely used in modern quantum statistical physics, the solid state theory, the theory of magnetism, theory of lasers, and other problems of the many-body theory…

The following sections are included:

• INTRODUCTION

• SYMMETRY PROPERTIES
  • Permutations
  • Symmetry and Antisymmetry

• SYMMETRIC OPERATORS
  • Symmetric single-particle operators
  • Symmetric Two-Particle Operators
  • Symmetric s-Particle Operators

• SOME IDENTITIES

• HAMILTONIANS

The following sections are included:

• REPRESENTATION OF WAVE FUNCTIONS
  • Discrete f-Representation
  • Discrete Momentum Representation
  • Periodicity
  • Passing to the Limit. Quasi-Discrete Representation

• OCCUPATION NUMBERS
  • Definition
  • Bosons
  • Fermions

The following sections are included:

• BOSE STATISTICS
  • Operators a and a⁺
  • Quantum Bose Operators a and a⁺
  • Wave Functions

• FERMI STATISTICS
  • Operators σ and σ⁺
  • The Pauli Operators
  • The Fermi Operators a and a⁺
  • Wave Function

• COMPARISON OF WAVE FUNCTIONS OF BOSONS AND FERMIONS

• THE REPRESENTATION OF SECONDARY QUANTIZATION

• THE OPERATOR OF PARTICLE NUMBER

The following sections are included:

• INTRODUCTION

• LEMMA
  • Proof
  • Corollaries of the Lemma

• WAVE FUNCTION
  • Inverse of Representation
  • Invariance of the Scalar Product

• TRANSFORMATION OF DYNAMICAL QUANTITIES TO THE REPRESENTATION OF SECONDARY QUANTIZATION
  • Development of the Transformation
  • Additive Dynamical Quantities
  • The Particle-Number Density
  • The Particle-Number Operator
  • Binary Dynamical Quantities
  • s-Particle Operators

• THE HAMILTONIAN IN THE REPRESENTATION OF SECONDARY QUANTIZATION

• THE EVOLUTION OF OPERATOR FUNCTIONS
  • The Heisenberg Representation
  • Commutation Relations
  • Time Evolution of Operator Functions
  • Discussion

The following sections are included:

• SYSTEMS INDEPENDENT OF PARTICLE NUMBER
  • The Grand Canonical Ensemble
  • The Quasi-Discrete Representation
  • Basis and Operator Functions
  • The Representation of Secondary Quantization for Hamiltonian Γ
  • The Momentum Vector

• WAVE FUNCTIONS IN THE REPRESENTATION OF SECONDARY QUANTIZATION
  • Conditions
  • Remarks on Condition (D)
  • Derivation of the Expression for a Wave Function

• DYNAMICAL SYSTEMS CONSISTING OF BOSONS AND FERMIONS OF SEVERAL TYPES
  • Ordinary Wave Functions
  • Operator Function
  • Transformation of Dynamical Operators
  • Dynamical Quantities of Mixed type
  • Independence of the Particle Number

• COMMUTATION RELATIONS FOR OPERATOR FUNCTIONS
  • Commutativity of Arbitrary Type
  • Remark on Commutativity

• OCCUPATION NUMBERS AND THE f-REPRESENTATION (Comments on Chapter 2)
  • Bosons
  • Fermions

The following sections are included:

• THE METHOD OF SECONDARY QUANTIZATION
  • Time Evolution of the Wigner Quantum Operator
  • Passing to the Limit ħ → 0
  • Quantization and Secondary Quantization

• THE TRANSITION TO CLASSICAL MECHANICS
  • A Binary Dynamical Quantity
  • Time-Evolution of Function f
  • The Vlasov Equation

• A SYSTEM OF IDENTICAL HARD SPHERES
  • A System of Two Identical Hard Spheres
  • A System of N Identical Hard Spheres
  • The Boltzmann-Enskog Equation
  • The Diffusion Equation of a Separate Particle in a Gas
  • Binary Interaction

• REFERENCES TO PART II

The following sections are included:

• EQUILIBRIUM PROPERTIES OF IDEAL GASES
  • Ideal Gas
  • Free Energy and Average Occupation Numbers
  • Spectrum of Black-Body Radiation
  • Ideal Bose Gas
  • Ideal Fermi Gas

• DIAGONALIZATION OF QUADRATIC FORM FOR THE BOSE AND FERMI OPERATORS
  • A Method of Canonical Transformation
  • An Operator Form of Canonical Transformation

• DIAGONALIZATION OF QUADRATIC HAMILTONIANS IN THE THEORY OF SUPERFLUIDITY AND SUPERCONDUCTIVITY
  • The Case of Bose-Einstein Statistics
  • The Case of Fermi-Dirac Statistics
  • An Operator Form of Canonical Transformation for the Bose-Einstein Statistic

• THE BLOCH-DE DOMINICIS STATISTICAL THEOREM
  • The Bloch-de Dominicis Theorem
  • The Proof of the Bloch-de Dominicis Theorem

The following sections are included:

• DIAGONALIZATION OF A QUADRATIC FORM OF THE BOSE OPERATORS
  • Canonical Transformation
  • A Set of Equations for Parameters of Canonical Transformation
  • Matrix Form
  • Eigenvalues
  • Condition on Parameters of the Canonical Transformation
  • Diagonalization
  • Hamiltonian with Linear Operator Term

• DIAGONALIZATION OF THE QUADRATIC FORM OF THE FERMI OPERATORS
  • Canonical Transformation
  • A Set of Equations for Parameters of Canonical Transformation
  • Matrix Form
  • Eigenvalue
  • Condition on Parameters of Canonical Transformation
  • A Lemma on Eigenvectors and Eigenvalues
  • Diagonalization

• REFERENCES TO PART III

The following sections are included:

• AN IDEAL BOSE GAS

• CONDENSATION IN MOMENTUM SPACE AND THE PHENOMENON OF SUPERFLUIDITY

• SUPERFLUIDITY
  • Quasi-Particles
  • Occurrence of Superfluidity
  • Equations of Motion
  • Elementary Excitations and Superfluidity
  • Remarks
  • Momentum Distribution
  • Correlation Function
  • Discussion

The following sections are included:

• GREEN FUNCTIONS CONSTRUCTED FROM USUAL AVERAGES. ADDITIVE CONSERVATION LAWS AND SELECTED RULES.

• DEGENERATION OF THE STATE OF STATISTICAL EQUILIBRIUM. INTRODUCTION OF QUASI-AVERAGES.
  • Degeneration of States of Statistical Equilibrium
  • Quasi-Averages
  • Degeneration in the Theory of the Crystalline State
  • An Ideal Bose Gas
  • Quasi-Averages in the Theory of Superconductivity
  • The Dicke Model
  • Quasi-Averages for an Arbitrary System

The following sections are included:

• LIMIT PROPERTIES OF FREE ENERGIES

• QUASI-AVERAGES FOR A HAMILTONIAN WITH A NEGATIVE FOUR-FERMION INTERACTION

The following sections are included:

• PRINCIPLE OF CORRELATION WEAKENING

• THEOREMS ON SINGULARITIES OF TYPE 1/q² AND SOME APPLICATIONS OF THE METHOD OF QUASI-AVERAGES

The following sections are included:

• SOME REMARKS ON THE ERGODIC THEORY

• A DYNAMICAL SYSTEM INTERACTING WITH BOSON FIELD

• REFERENCES TO PART IV

The following sections are included:

• Photos

• INDEX