Introduction to Quantum Statistical Mechanics

The following sections are included:

• PREFACE

• CONTENTS

The following sections are included:

• Introduction: Classical and Quantum Mechanics in the Statistical Context

• The Classical Statistical Scheme
  • Transformation operator G
  • Probability distribution density
  • The Liouville Theorem
  • Time dependent density t, t0

• Quantum Analogy

• Symmetry Properties

• Time-Independent Hamiltonians

• Example

• Property of Reversibility

In this chapter, we shall generalise the concept of probability distribution in phase space and the corresponding Liouville equation for dynamical systems to quantum mechanics.

The following sections are included:

• Integrals of Motion

• Gibbs Canonical Distribution

• Method for Thermodynamic Functions: Assumptions and Definitions

• Quasi-Static Processes
  • Discussion
  • Construction for quasi-static processes
  • Interpretation of terms
  • Heat capacity C_v
  • Homogeneous systems
  • Relation between H and E

• Limiting Transitions
  • Basic assumptions
  • Boundary surface
  • Limits
  • Validity of approaches to limiting transitions

• Grand Canonical Ensemble
  • Statistical operators
  • Definitions: μ, Γ, G
  • Uniqueness of μ_j
  • Thermodynamic functions
  • Limiting transitions as V → ∞

• Quantum Ideas in Classical Canonical Distributions

• Entropy in Dynamical Systems
  • In canonical ensembles
  • Limiting form of entropy
  • Auxiliary parameter τ and function R(τ)
  • Limiting property
  • Time independence
  • Entropy in non-equilibrium states: Example
  • Approximations for non-equilibrium states

Let us continue with the study of average values of dynamical variables, obtained from statistical operators in a grand canonical ensemble.

In part 2, we present a self-contained account of the second quantization method which has results that are widely used in modern quantum statistical physics, solid state physics, the theory of magnetism, theory of lasers and many-particle problems…

In this chapter and subsequent ones, certain developments, conventions and discussions parallel closely those of earlier chapters in the first half of the volume. Rather than economise on words, the repetitions have been made, no doubt abbreviated, to provide a summary and review of basic ideas. This creates a wholly self-contained unit in the second half that allows those already acquainted with the basics to start immediately on the elements of Second Quantization.

The following sections are included:

• Representations
  • Discrete f-representation
  • Discrete momentum representation
  • Periodicity: Fourier integral representations
  • Limiting transitions: Quasi-discrete representation

• Occupation Numbers
  • Representation and definitions
  • Bosons
  • Fermions

In order to transform dynamical quantities represented by operators operating on the usual wavefunctions into operators operating on wavefunctions C ( . . ,n_f, . . ) , it would be interesting to consider the so-called Bose and Fermi quantum operators.

The following sections are included:

• Bases for Transformations to Second Quantized Representations

• Lemma
  • Proof
  • Corollary

• Wavefunctions C, φ
  • Transformation and correspondence
  • Invariance of scalar product

• Transformation of Dynamical Quantities to Second-Quantized Representation
  • Procedure for transformation
  • One-point operator U
  • Space density ρ
  • Particle number operator
  • Two-point operator B
  • s-point operator σ ^(s)

• Second Quantized Representation for Hamiltonian H

• Time Evolution of Operator Functions
  • Heisenberg representation
  • Transformation relations
  • Equation for time evolution
  • Discussion

In this chapter we shall make additional remarks about the second quanitzation method.

We shall now consider some analogues of the second quantization method in classical mechanics. For this purpose, we shall deal with a system of N identical spinless particles with a Hamiltonian consisting of a sum of the kinetic energies of particles and a sum of potential energies of 2-particle interactions, the latter being described by a radially symmetric function Φ(r) (cf. (1.38) in Chapter 1).

The following sections are included:

• INDEX