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A Coherent Semiclassical Transport Model for Pure-state Quantum Scattering



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A Coherent Semiclassical Transport Model for Pure state Quantum Scattering Shi Jin and Kyle A Novak January 28 2009 Abstract We present a time dependent semiclassical transport model for coherent pure state scattering with quantum barriers The model is based on a complex valued Liouville equation with interface conditions at quantum barriers computed from the steady state Schro dinger equation By retaining the phase information at the barrier this coherent model adequately describes quantum scattering and interference at quantum barriers with a computational cost comparable to that of the classical mechanics We construct both Eulerian and Lagrangian numerical methods for this model and validate it using several numerical examples including multiple quantum barriers 1 Introduction The motion of electrons in a plasma or a semiconductor can be modeled with classical mechanics when the change in the applied potential is moderate But in a region where the potential changes rapidly over the length on the order of a de Broglie wavelength quantum mechanics is required to accurately capture wave phenomena such as tunneling resonance and partial transmission and reflections Because quantum scale parameters often control the accuracy and consistency of the solution one often must resolve the dynamics entirely at the quantum scale But for large scale problems such an approach is numerically infeasible If the quantum region is sufficiently localized a viable approach is to solve the problem using a multiscale method that combines the large scale classical model with the small scale quantum model This research was supported by NSF grant No DMS 0608720 and NSF FRG grant DMS 0757285 Shi Jin was also supported by a Van Vleck Research Prize for University of Wisconsin Department of Mathematics University of Wisconsin 480 Lincoln Drive Madison Wisconsin 53706 1338 USA jin math wisc edu Department of Mathematics and Statistics Air Force Institute of Technology 2950 Hobson Way Wright



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