Sharp and Infinite Boundaries in the Path Integral Formalism

TL;DR

This paper analyzes the path integral formalism for sharp infinite potentials, validating the image method through energy eigenstates, supersymmetric quantum mechanics, and applications to the half-harmonic oscillator, revealing its effectiveness in boundary problems.

Contribution

It demonstrates the validity of the image method in the path integral formalism for sharp boundaries using eigenstates and supersymmetric techniques, and applies it to complex potentials.

Findings

The propagator for an infinite wall is generated by a complete set of boundary-satisfying eigenstates.

Supersymmetric quantum mechanics relates potentials with and without sharp boundaries, aiding propagator derivation.

The method successfully derives the propagator and eigenfunctions for the half-harmonic oscillator.

Abstract

We revisit the analysis of sharp infinite potentials within the path integral formalism using the image method [1]. We show that the use of a complete set of energy eigenstates that satisfy the boundary conditions of an infinite wall precisely generates the propagator proposed in Ref. [1]. We then show the validity of the image method by using supersymmetric quantum mechanics to relate a potential without a sharp boundary to the infinite square well and derive its propagator with an infinite number of image charges. Finally, we show that the image method readily generates the propagator for the half-harmonic oscillator, a potential that has a sharp infinite boundary at the origin and a quadratic potential in the allowed region, and leads to the well known eigenvalues and eigenfunctions.