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arXiv · 1801.08847

Efficient solution of the Wigner-Liouville equation using a spectral decomposition of the force field

Maarten L. Van de Put·Bart Sorée·Wim Magnus
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Abstract

The Wigner-Liouville equation is reformulated using a spectral decomposition of the classical force field instead of the potential energy. The latter is shown to simplify the Wigner-Liouville kernel both conceptually and numerically as the spectral force Wigner-Liouville equation avoids the numerical evaluation of the highly oscillatory Wigner kernel which is nonlocal in both position and momentum. The quantum mechanical evolution is instead governed by a term local in space and non-local in momentum, where the non-locality in momentum has only a limited range. An interpretation of the time evolution in terms of two processes is presented; a classical evolution under the influence of the averaged driving field, and a probability-preserving quantum-mechanical generation and annihilation term. Using the inherent stability and reduced complexity, a direct deterministic numerical implementation using Chebyshev and Fourier pseudo-spectral methods is detailed. For the purpose of illustration, we present results for the time-evolution of a one-dimensional resonant tunneling diode driven out of equilibrium.

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Maarten L. Van de Put, Bart Sorée, Wim Magnus. 2018-01-26. Efficient solution of the Wigner-Liouville equation using a spectral decomposition of the force field. https://doi.org/10.1016/j.jcp.2017.08.059

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