Techniques and Applications of Path Integration

L. S. Schulman

• Publisher: Dover Publications
• Year: 2012
• Print ISBN: 9780486445281
• eBook ISBN: 9780486137025
• Topic: Physical Sciences
• Subtopic: Physics
• Index: Physics

A book of techniques and applications, this text defines the path integral and illustrates its uses by example. It is suitable for advanced undergraduates and graduate students in physics; its sole prerequisite is a first course in quantum mechanics. For applications requiring specialized knowledge, the author supplies background material.
The first part of the book develops the techniques of path integration. Topics include probability amplitudes for paths and the correspondence limit for the path integral; vector potentials; the Ito integral and gauge transformations; free particle and quadratic Lagrangians; properties of Green's functions and the Feynman-Kac formula; functional derivatives and commutation relations; Brownian motion and the Wiener integral; and perturbation theory and Feynman diagrams.
The second part, dealing with applications, covers asymptotic analysis and the calculus of variations; the WKB approximation and near caustics; the phase of the semiclassical amplitude; scattering theory; and geometrical optics. Additional topics include the polaron; path integrals for multiply connected spaces; quantum mechanics on curved spaces; relativistic propagators and black holes; applications to statistical mechanics; systems with random impurities; instantons and metastability; renormalization and scaling for critical phenomena; and the phase space path integral.

PART ONE

Introduction

ONE

Introducing and Defining the Path Integral

The best place to find out about path integrals is in Feynman’s paper.^a Our approach is not to use path integrals as a way of arriving at quantum mechanics, although Feynman has used this point of view in his book with Hibbs. Rather we assume knowledge of quantum mechanics and deduce the path integral formalism from it. This gets us into the subject quickly.

The wave function of a nonrelativistic spinless particle in one dimension evolves according to Schrödinger’s equation

(1.1)

(1.2)

Our interest is in the propagator or Green’s function G which satisfies the equation

(1.3)

in operator notation. In coordinate space this is written

(1.4)

The G’s are related by

(1.5)

Knowing G means having a solution to the time dependent Schrödinger equation in the sense that if ψ(t₀) is the state of the system at t₀, ψ(t), given by

(1.6)

is the state at t. For time independent H an operator solution of (1.3) can immediately be written down:

(1.7)

where θ is the step function. Since H is assumed to be time independent we can, without loss of generality, take t₀=0. Then for t>0 we have

(1.8)

where the argument 0 has been deleted.

The path integral arises from the fact that

(1.9)

Letting λ = it/

yields

(1.10)

with the product in the brackets taken N times. Now we make use of a fundamental fact about the exponential of two operators, namely

(1.11)

This is proved easily enough,^b and in a power series expansion the coefficient of the λ²/N² term is

In subsequent manipulations we assume that the O(1/N²) term is well behaved, that it stays bounded when applied to states, and so on. For reasonable potentials this assumption is justified; more is said on this topic in the appendix.

What we are now aiming for is to replace the term

(1.12)

by the term

(1.13)

For real numbers (rather than operators) this replacement is a reflection of a fundamental fact about the exponential. The expression

converges to e^x despite the presence of y_n so long as y_n→0 as n→∞. (A proof of this assertion can be had by taking large enough n that |y_n| <δ and by using the bound

and assuming n large enough that |(x-δ)/n| < 1.)

For operators a bit of care is required, and the trick is to express the diff...

Table of contents

• Title Page
• Copyright Page
• Dedication
• Preface
• Table of Contents
• PART ONE - Introduction
• PART TWO - Selected Applications of the Path Integral
• Author Index
• Subject Index
• Supplements