 
Summary: Discrete Transparent Boundary Conditions
for the Schr¨odinger Equation
Matthias Ehrhardt, Anton Arnold
Fakult¨at f¨ur Mathematik und Informatik,
Universit¨at des Saarlandes,
Postfach 15 11 50,
D66041 Saarbr¨ucken, Germany
ehrhardt@num.unisb.de, arnold@num.unisb.de
Abstract
This paper is concerned with transparent boundary conditions for the one dimensional
timedependent Schr¨odinger equation. They are used to restrict the original PDE prob
lem that is posed on an unbounded domain onto a finite interval in order to make this
problem feasible for numerical simulations. The main focus of this article is on the appro
priate discretization of such transparent boundary conditions in conjunction with some
chosen discretization of the PDE (usually CrankNicolson finite differences in the case
of the Schr¨odinger equation). The presented discrete transparent boundary conditions
yield an unconditionally stable numerical scheme and are completely reflectionfree at
the boundary.
1. Introduction
Many physical problems are described mathematically by a partial differential equation
