A Bloch decomposition-based stochastic Galerkin method for quantum dynamics with a random external potential

In this paper, we consider the numerical solution of the one-dimensional Schrödinger equation with a periodic lattice potential and a random external potential. This is an important model in solid state physics where the randomness results from complicated phenomena that are not exactly known. Here...

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Bibliographic Details
Published in:Journal of computational physics 2016-07, Vol.317, p.257-275
Main Authors: Wu, Zhizhang, Huang, Zhongyi
Format: Article
Language:English
Subjects:
Bloch decomposition
CHAOS THEORY
CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS
Computation
Decomposition
DISPERSIONS
Galerkin methods
Generalized polynomial chaos
Mathematical models
NUMERICAL SOLUTION
ONE-DIMENSIONAL CALCULATIONS
PERIODICITY
POLYNOMIALS
POTENTIALS
RANDOMNESS
SCHROEDINGER EQUATION
Schrödinger equation
SOLID STATE PHYSICS
SOLIDS
STOCHASTIC PROCESSES
Stochasticity
Time-splitting
Uncertainty quantification
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Summary:In this paper, we consider the numerical solution of the one-dimensional Schrödinger equation with a periodic lattice potential and a random external potential. This is an important model in solid state physics where the randomness results from complicated phenomena that are not exactly known. Here we generalize the Bloch decomposition-based time-splitting pseudospectral method to the stochastic setting using the generalized polynomial chaos with a Galerkin procedure so that the main effects of dispersion and periodic potential are still computed together. We prove that our method is unconditionally stable and numerical examples show that it has other nice properties and is more efficient than the traditional method. Finally, we give some numerical evidence for the well-known phenomenon of Anderson localization.
ISSN:0021-9991
1090-2716
DOI:10.1016/j.jcp.2016.04.051