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Preprint 6-2008

Numerical Simulation of Quantum Waveguides

Source file is available as :   Portable Document Format (PDF)

Author(s) : Anton Arnold , Matthias Ehrhardt , Maike Schulte

Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 6-2008

MSC 2000

65M12 Stability and convergence of numerical methods
35Q40 Equations from quantum mechanics
45K05 Integro-partial differential equations

Abstract :
In this chapter we propose transparent boundary conditions (TBCs) for the time-dependent Schrödinger equation on a two-dimensional domain. First we derive the two-dimensional discrete TBCs in conjunction with a conservative Crank-Nicolson-type finite difference scheme and a novel compact nine-point scheme. For this difference equations we derive discrete transparent boundary conditions (DTBCs) in order to get highly accurate solutions for open boundary problems. The presented discrete boundary-valued problem is unconditionally stable and completely reflection-free at the boundary. Several numerical experiments illustrate the perfect absorption of outgoing waves independent of their impact angle at the boundary. Finally, we apply inhomogeneous DTBCs to the transient simulation of quantum waveguides with a prescribed electron inflow.

Keywords : Quantum waveguide, Schrödinger equation, transparent boundary condition, finite difference scheme

Notes :
Chapter in: F.Columbus (ed.), Very-Large-Scale Integration (VLSI): Architecture, Performance and Nano Applications, to appear fall 2008.

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