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American Journal of Physics

Path integrals and propagators for quadratic Lagrangians in three dimensions

Kai-Man Poon, Gerardo Muñoz

Published Jun 1, 1999• 7 citations
Publisher

Abstract

Functional methods are employed to derive the propagator for a three-dimensional system described by a Lagrangian at most quadratic in the velocity and position variables. Two physically relevant examples are presented to illustrate the concrete application of the general formula.
DOI
10.1119/1.19321

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A class of time-dependent classical Lagrangians possessing an invariant quadratic in momentum is considered from a quantal point of view. Quantum mechanics is introduced through the Feynman propagator defined as a path integral involving the classical action. It is shown, without carrying out an explicit path integration, that the propagator for such a time-dependent system is related to the propagator of an associated time-independent problem. The expansion of the propagator in terms of the eigenfunctions of the invariant operator is derived and the equivalence of the present theory to that of Lewis and Reisenfeld (1969) is discussed. Explicit analytic forms of propagators are obtained for some cases to illustrate the application of the present approach.
These lectures are intended for graduate students who want to acquire a working knowledge of path integral methods in a wide variety of fields in physics. In general the presentation is elementary and path integrals are developed in the usual heuristic, non-mathematical way for application in many diverse problems in quantum physics. Three main parts deal with path integrals in non-relativistic quantum mechanics, many-body physics and field theory and contain standard examples (quadratic Lagrangians, tunneling, description of bosons and fermions, quantization of gauge theories etc.) as well as specialized topics (scattering, dissipative systems, spin \& color in the path integral, lattice methods etc.). In each part simple Fortran programs which can be run on a PC, illustrate the numerical evaluation of (Euclidean) path integrals by Monte-Carlo or variational methods. Also included are the set of problems which accompanied the lectures and their solutions.