$δ$-Function Perturbations and Boundary Problems by Path Integration

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 $δ$-function perturbations is outlined, which includes the discussion of multiple $δ$-function perturbations, $δ$-function perturbations along perpendicular lines and planes, and moving $δ$-function perturbations. The limiting process, where the strength of the $δ$-function perturbations gets infinite repulsive, has the effect of producing impenetrable walls at the locations of the $δ$-function perturbations, i.e.\ a consistent description for boundary problems with Dirichlet boundary-condition emerges. Several examples illustrate the formalism.