Boundary conditions: The path integral approach

TL;DR

This paper extends the path integral approach to quantum mechanics to include systems with boundaries, using boundary amplitude distributions and complex phases to incorporate various boundary conditions.

Contribution

It introduces a generalization of the path integral method to handle bounded domains with non-local boundary conditions, analyzing one-dimensional particle dynamics.

Findings

Different boundary conditions correspond to distinct boundary amplitude prescriptions.

The approach effectively describes quantum dynamics confined to bounded regions.

Analysis of one-dimensional systems demonstrates the method's applicability.

Abstract

The path integral approach to quantum mechanics requires a substantial generalisation to describe the dynamics of systems confined to bounded domains. Non-local boundary conditions can be introduced in Feynman's approach by means of boundary amplitude distributions and complex phases to describe the quantum dynamics in terms of the classical trajectories. The different prescriptions involve only trajectories reaching the boundary and correspond to different choices of boundary conditions of selfadjoint extensions of the Hamiltonian. One dimensional particle dynamics is analysed in detail.