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VLSI Design
Volume 6 (1998), Issue 1-4, Pages 313-319
http://dx.doi.org/10.1155/1998/38298

Numerically Absorbing Boundary Conditions for Quantum Evolution Equations

1Fachbereich Mathematik, TU-Berlin, Berlin D-10623, Germany
2Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA

Copyright © 1998 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Transparent boundary conditions for the transient Schrödinger equation on a domain Ω can be derived explicitly under the assumption that the given potential V is constant outside of this domain. In 1D these boundary conditions are non-local in time (of memory type). For the Crank-Nicolson finite difference scheme, discrete transparent boundary conditions are derived, and the resulting scheme is proved to be unconditionally stable. A numerical example illustrates the superiority of discrete transparent boundary conditions over existing ad-hoc discretizations of the differential transparent boundary conditions. As an application of these boundary conditions to the modeling of quantum devices, a transient 1D scattering model for mixed quantum states is presented.