Feynman integrals with absorbing boundaries

V. Janiš

Avi Marchewka, Z. Schuss

We propose a formulation of an absorbing boundary for a quantum particle. The formulation is based on a Feynman-type integral over trajectories that are confined by the absorbing boundary. Trajectories that reach the absorbing wall are instantaneously terminated and their probability is discounted from the population of the surviving trajectories. This gives rise to a unidirectional absorption current at the boundary. We calculate the survival probability as a function of time. Several modes of absorption are derived from our formalism: total absorption, absorption that depends on energy levels, and absorption of non-interacting particles. Several applications are given: the slit experiment with an absorbing screen and with absorbing lateral walls, and one dimensional particle between two absorbing walls. The survival probability of a particle between absorbing walls exhibits decay with beats.

Miroslav Pardy

Avi Marchewka, Z. Schuss

We propose a formulation of an absorbing boundary for a quantum particle. The formulation is based on a Feynman-type integral over trajectories that are confined by an absorbing boundary. Trajectories that reach the absorbing wall are instantaneously terminated and their probability is discounted from the population of the surviving trajectories. This gives rise to a unidirectional absorption current at the boundary. We calculate the survival probability as a function of time. Several applications are given: the calculation of the absorption current in the slit experiment with a totally absorbing screen, the calculation of the survival probability of a particle between two totally absorbing walls, and the calculation of the survival probability and of the reflection coefficient of a Gaussian wave packet incident on a totally absorbing wall. The survival probability of a one-dimensional particle between two totally absorbing walls exhibits decay with beats.

Rashi Kaimal, Roderich Tumulka

In quantum mechanics, among the proposals for how to compute the probability distribution of detection time and place along a given spatial surface or in a given volume, two leading candidates involve absorbing boundary conditions (ABCs) and complex absorbing potentials (CAPs) for the Schrödinger equation. Here, we extend earlier results on how the two are related. It was previously shown that ABCs arise as a limiting case of CAPs, but for that a (reflecting) Neumann boundary condition was assumed at the end of the region with the CAP. Here, we obtain ABCs in a different limit using a Dirichlet boundary condition at the end of the CAP region, which seems more practically relevant and realizable.