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European Journal of Physics

An alternative method to calculate propagators

ALVARO DE SOUZA DUTRA, Antonio S. de Castro

Published Jul 1, 1989• 2 citations
Publisher

Abstract

An alternative method to calculate propagators that can be applied in a simple way to polynomial potentials is presented. The method does not use path integral or eigenfunction techniques, but may require the solution of a non-linear ordinary differential equation. As an illustration the propagators for a particle moving in a homogeneous time-varying field and for the harmonic oscillator are calculated. Other simple cases are suggested.

DOI
10.1088/0143-0807/10/3/006

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An alternative method to calculate propagators that can be applied in a simple way to polynomial potentials is presented. The method does not use path integral or eigenfunction techniques, but may require the solution of a non-linear ordinary differential equation. As an illustration the propagators for a particle moving in a homogeneous time-varying field and for the harmonic oscillator are calculated. Other simple cases are suggested.
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.
As an alternative formulation of quantum mechanics, path integral is based on the notion of transition amplitude which gives the wave function of a quantum system at a time t f by acting on the wave function at an earlier time ti.We show that for a general quadratic form for the Lagrangian of the system, transition amplitude has the form f (t f -ti)e i h S class., where S class. is the classical action.We then present an algebraic method to evaluate the function f (t f -ti) without refereing to the path integral calculations.We examine the presented method to the cases of free particle and harmonic oscillator and obtain their propagators.