Alternative Method for Determining the Feynman Propagator of a Non-Relativistic Quantum Mechanical Problem

TL;DR

This paper introduces a Laplace transform-based method for directly deriving the Green function and propagator in non-relativistic quantum mechanics, simplifying calculations for simple and potentially complex systems.

Contribution

It presents a new approach using Laplace transforms to directly obtain the energy-dependent Green function and propagator, offering a potentially more straightforward alternative to Feynman's path integral method.

Findings

Method successfully applied to simple one-dimensional examples

Can be generalized to more complex potentials

Provides a direct computational pathway for propagators

Abstract

A direct procedure for determining the propagator associated with a quantum mechanical problem was given by the Path Integration Procedure of Feynman. The Green function, which is the Fourier Transform with respect to the time variable of the propagator, can be derived later. In our approach, with the help of a Laplace transform, a direct way to get the energy dependent Green function is presented, and the propagator can be obtained later with an inverse Laplace transform. The method is illustrated through simple one dimensional examples and for time independent potentials, though it can be generalized to the derivation of more complicated propagators.