Path integrals for boundaries in diffusion and quantum mechanics

Abstract

Diffusion and quantum mechanics with boundary conditions arise naturally in various situations in physics. While reflecting and absorbing boundaries are well mathematically described, intermediate scenarios are not that clear. Consider a reflective boundary which is removed for a time δ and subsequently reinstated. First, we place a Brownian particle at one side of this barrier and study its path-integral propagator. We obtained a closed expression for the particle’s probability of being before or after the barrier at a certain time. We then consider the same barrier, but when the removing time is unforeseen, so the particle could cross to the other side at some time within a known interval [0,T]. A path-integral propagator is computed, and it is shown numerically that the limit to the exact path integral is convergent. Finally, we consider a quantum particle in the presence of a reflective barrier removed at t = ∆t1 for a time δ and then reinstated. We propose a path-integral propagator for this process and show that the corresponding wave function satisfies the Schrodinger equation.