Summary: Numerical Simulation of Quantum Waveguides
Anton Arnold1
, Matthias Ehrhardt2
and Maike Schulte3
1
Institut für Analysis und Scientific Computing, TU Wien, Wiedner Hauptstr. 8, 1040 Wien, Austria
2
Weierstraß­Institut für Angewandte Analysis und Stochastik, Mohrenstr. 39, 10117 Berlin, Germany
3
Institut für Numerische und Angewandte Mathematik, Universität Münster, Einsteinstr. 62,
48149 Münster, Germany
Abstract. This chapter is a review of the research of the authors from the last decade and focuses
on the mathematical analysis of the Schrödinger model for nano­scale semiconductor devices. We
discuss transparent boundary conditions (TBCs) for the time­dependent Schrödinger equation on a
two dimensional domain.
First we derive the two dimensional discrete TBCs in conjunction with a conservative Crank­
Nicolson­type finite difference scheme and a compact nine­point scheme. For this difference equa-
tions we derive discrete transparent boundary conditions (DTBCs) in order to get highly accurate
solutions for open boundary problems. The presented discrete boundary­valued problem is uncon-
ditionally stable and completely reflection­free at the boundary.

Source: Arnold, Anton - Institut für Analysis und Scientific Computing, Technische Universität Wien

Collections: Mathematics