Summary: NANOSYSTEMS: PHYSICS, CHEMISTRY, MATHEMATICS, 2011, 1 (1), P. 122
WKB-based schemes for two-band Schr¨odinger
equations in the highly oscillatory regime
, Anton Arnold1
Institute for Analysis and Scientific Computing, Vienna University of Technology,
Wiedner Hauptstr. 8, A-1040 Wien, Austria.
PACS 02.60.Lj, 02.60.Lh, 85.30.De
An efficient and accurate numerical method is presented for the solution of highly oscillatory differential
equations in one spatial dimension. While standard methods would require a very fine grid to resolve the
oscillations, the presented approach uses first an analytic WKB-type transformation, which filters out the
dominant oscillations. The resulting ODE-system is much smoother and can hence be discretized on a much
coarser grid, with significantly reduced numerical costs.
Here we are concerned with stationary two-band Schr¨odinger equations employed in quantum trans-
port applications. We focus on the Kanemodel and the two band k·pmodel. The accuracy of the presented
method is illustrated on a numerical example.
Keywords: Schr¨odinger equation, Kanemodel, two-band k · pmodel, highly oscillating wave functions,