Exact transparent boundary condition for the multidimensional Schr\"odinger equation in hyperrectangular computational domain

TL;DR

This paper introduces an exact transparent boundary condition for the multidimensional Schrödinger equation in hyperrectangular domains, enabling precise quantum wave simulations with improved numerical stability and applicability to multi-particle systems.

Contribution

It generalizes existing boundary conditions to higher dimensions and proposes a new fully discrete 1D boundary condition derived from the finite-difference scheme.

Findings

Demonstrates accurate propagation of Gaussian wave packets in 3D

Shows particle penetration through a 3D barrier

Validates boundary condition in multi-particle quantum dynamics

Abstract

In this paper an exact transparent boundary condition (TBC) for the multidimensional Schr\"odinger equation in a hyperrectangular computational domain is proposed. It is derived as a generalization of exact transparent boundary conditions for 2D and 3D equations reported before. A new exact fully discrete (i.e. derived directly from the finite-difference scheme used) 1D transparent boundary condition is also proposed. Several numerical experiments using an improved unconditionally stable numerical implementation in the 3D space demonstrate propagation of Gaussian wave packets in free space and penetration of a particle through a 3D spherically asymmetrical barrier. The application of the multidimensional transparent boundary condition to the dynamics of the 2D system of two non-interacting particles is considered. The proposed boundary condition is simple, robust and can be useful in the field of computational quantum mechanics, when an exact solution of the multidimensional Schr\"odinger equation (including multi-particle problems) is required.