Quantum Statistical Mechanics

Leo P. Kadanoff, Gordon Baym, David Pines

• Publisher: CRC Press
• Year: 2018
• Print ISBN: 9780367320102
• eBook ISBN: 9780429972843
• Edition: 1
• Topic: Scienze fisiche
• Subtopic: Fisica

This book is a very early systematic treatment of the application of the field-theoretical methods developed after the Second World War to the quantum mechanical many-body problem at finite temperature. It describes various techniques that remain basic tools of modern condensed matter physicists.

1	Mathematical Introduction

1-1 BASIC DEFINITION

The properties of a quantum mechanical system composed of many identical particles are most conveniently described in terms of the second-quantized, Heisenberg representation, particle-creation, and annihilation operators. The creation operator, ψ†(r, t), when acting to the right on a state of the system, adds a particle to the state at the space-time point r, t; the annihilation operator ψ(r, t), the adjoint of the creation operator, acting to the right, removes a particle from the state at the point r, t.

The macroscopic operators of direct physical interest can all be expressed in terms of products of a few ψ’s and ψ†’s. For example, the density of particles at the point r, t is

$\text{n}(r,\text{t})=\psi †(r,\text{t})\psi (r,\text{t})$

(1-1a)

since the act of removing and then immediately replacing a particle at r, t measures the density of particles at that point. The operator for the total number of particles is

$\text{N}(\text{t})={\displaystyle \int \text{d}r\psi †(r,\text{t})}\psi (r,\text{t})$

(1-1b)

Similarly, the total energy of a system of particles of mass m interacting through an instantaneous two-body potential v(r) is given by

$\begin{array}{lll}\text{H}(\text{t})\hfill & =\hfill & {\displaystyle \int \text{d}r}\frac{\nabla \psi †(r,\text{t})•\nabla \psi (r,\text{t})}{2\text{m}}\hfill \\ \hfill & \hfill & +1/2{\displaystyle \int \text{d}r\text{d}{r}^{\prime }\psi †(r,\text{t})\psi †(r^{\prime },\text{t})\text{v}(|r-r^{\prime }|)\psi (r^{\prime },\text{t})\psi (r,\text{t})}\hfill \end{array}$

(1-2)

In general we shall take $\text{ℏ}=...$

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Table of Contents
• 1 Mathematical Introduction
• 2 Information Contained in G> and G<
• 3 The Hartree and Hartree-Fock Approximations
• 4 Effect of Collisions on G
• 5 A Technique for Deriving Green’s Function Approximations
• 6 Transport Phenomena
• 7 The Hartree Approximation, the Collisionless Boltzmann Equation, and the Random Phase Approximation
• 8 Relation between Real and Imaginary Time Response Functions
• 9 Slowly Varying Disturbances and the Boltzmann Equation
• 10 Quasi-Equilibrium Behavior: Sound Propagation
• 11 The Landau Theory of the Normal Fermi Liquid
• 12 The Shielded Potential
• 13 The T Approximation
• Appendix
• References and Supplementary Reading