$\delta$-Function Perturbations and Boundary Problems by Path Integration

TL;DR

This paper develops a path integral approach to boundary problems in quantum mechanics, incorporating delta-function perturbations to model boundaries and obstacles, including moving and multiple perturbations, leading to a unified formalism.

Contribution

It introduces a formalism for including delta-function perturbations in path integrals, enabling the modeling of complex boundary conditions and moving boundaries in quantum systems.

Findings

Formalism for delta-function perturbations in path integrals.

Modeling of boundary conditions via infinite-strength delta perturbations.

Application to various boundary problems with examples.

Abstract

A wide class of boundary problems in quantum mechanics is discussed by using path integrals. This includes motion in half-spaces, radial boxes, rings, and moving boundaries. As a preparation the formalism for the incorporation of $\delta$-function perturbations is outlined, which includes the discussion of multiple $\delta$-function perturbations, $\delta$-function perturbations along perpendicular lines and planes, and moving $\delta$-function perturbations. The limiting process, where the strength of the $\delta$-function perturbations gets infinite repulsive, has the effect of producing impenetrable walls at the locations of the $\delta$-function perturbations, i.e.\ a consistent description for boundary problems with Dirichlet boundary-condition emerges. Several examples illustrate the formalism.