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arXiv:quant-ph/0205085 · American Journal of Physics

Three methods for calculating the Feynman propagator

F. A. Barone, Henrique Boschi-Filho, Carlos Farina

Published Apr 10, 2003• 40 citations
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Abstract

We present three methods for calculating the Feynman propagator for the nonrelativistic harmonic oscillator. The first method was employed by Schwinger a half a century ago, but has rarely been used in nonrelativistic problems since. Also discussed is an algebraic method and a path integral method so that the reader can compare the advantages and disadvantages of each method.

DOI
10.1119/1.1538573

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We present three methods for calculating the Feynman propagator for the nonrelativistic harmonic oscillator. The first method was employed by Schwinger a half a century ago, but has rarely been used in nonrelativistic problems since. Also discussed is an algebraic method and a path integral method so that the reader can compare the advantages and disadvantages of each method.
We present a purely analytical method to calculate the propagator for the quantum harmonic oscillator using Feynman’s path integral. Though the details of the calculation are involved, the general approach uses only matrix diagonalization and well-known integrals, techniques which an advanced undergraduate should understand. The full propagator, including both the prefactor and the classical action, is obtained from a single calculation which involves the exact diagonalization of the discretized action for the system.
In this paper, the propagators for a particle moving in a time-dependent linear potential and a free particle with linear damping are calculated by the application of the integrals of the motion of a quantum system. The propagator for a charged harmonic oscillator is derived from the Feynman path integrals and the propagator for a damped harmonic oscillator is evaluated by the Schwinger method. The relation between the integrals of the motion, Feynman path integrals, and Schwinger method are also described.