Boundary conditions for the Schr\"odinger equation in the numerical simulation of quantum systems

TL;DR

This paper investigates boundary conditions in numerical simulations of quantum systems, revealing that local boundary conditions are suitable for closed systems but not for open systems, and proposes a method to address these challenges.

Contribution

The paper clarifies the nature of boundary conditions in quantum simulations and introduces a novel approach to simulate open systems without requiring infinite domains.

Findings

Closed systems can be modeled with local boundary conditions.

Open systems cannot be described by local boundary conditions due to the uncertainty principle.

A new method allows finite lattice simulations that preserve physical wave behavior.

Abstract

We study the problem of the boundary conditions in the numerical simulation of closed and open quantum systems, described by a Schr\"odinger equation. On one hand, we show that a closed quantum system is defined by local boundary conditions. On the other hand, we argue that, because of the uncertainty principle, no local boundary condition can be defined for open quantum systems. For this reason plane waves or wave packet trains cannot be simulated on a finite numerical lattice with the usual procedures. We suggest a method that avoids these difficulties by using only a small numerical lattice and maintains the correspondence with the physical picture, in which the incident and scattered waves may be infinitely extended.