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

Journal Article · Fri Dec 01 04:00:00 EST 2017 · Journal of Computational Physics

DOI:https://doi.org/10.1016/J.JCP.2017.08.059· OSTI ID:22701636

imec, B-3001 Leuven ^[1]; Sorée, Bart ^[2]; imec, B-3001 Leuven ^[1]; Magnus, Wim ^[2]; imec, B-3001 Leuven ^[1]

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1. Belgium
2. Department of Physics, Universiteit Antwerpen, B-2020 Antwerpen (Belgium)

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.

• OSTI ID: 22701636
• Journal Information: Journal of Computational Physics, Journal Name: Journal of Computational Physics Vol. 350; ISSN JCTPAH; ISSN 0021-9991
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
MATHEMATICAL EVOLUTION
MATHEMATICAL SOLUTIONS
ONE-DIMENSIONAL CALCULATIONS
POTENTIAL ENERGY
QUANTUM MECHANICS
TUNNEL DIODES
TUNNEL EFFECT