A Bloch decompositionbased stochastic Galerkin method for quantum dynamics with a random external potential
Abstract
In this paper, we consider the numerical solution of the onedimensional 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 decompositionbased timesplitting 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 wellknown phenomenon of Anderson localization.
 Authors:
 Publication Date:
 OSTI Identifier:
 22572336
 Resource Type:
 Journal Article
 Resource Relation:
 Journal Name: Journal of Computational Physics; Journal Volume: 317; Other Information: Copyright (c) 2016 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; CHAOS THEORY; DISPERSIONS; NUMERICAL SOLUTION; ONEDIMENSIONAL CALCULATIONS; PERIODICITY; POLYNOMIALS; POTENTIALS; RANDOMNESS; SCHROEDINGER EQUATION; SOLID STATE PHYSICS; SOLIDS; STOCHASTIC PROCESSES
Citation Formats
Wu, Zhizhang, Email: wzz14@mails.tsinghua.edu.cn, and Huang, Zhongyi, Email: zhuang@math.tsinghua.edu.cn. A Bloch decompositionbased stochastic Galerkin method for quantum dynamics with a random external potential. United States: N. p., 2016.
Web. doi:10.1016/J.JCP.2016.04.051.
Wu, Zhizhang, Email: wzz14@mails.tsinghua.edu.cn, & Huang, Zhongyi, Email: zhuang@math.tsinghua.edu.cn. A Bloch decompositionbased stochastic Galerkin method for quantum dynamics with a random external potential. United States. doi:10.1016/J.JCP.2016.04.051.
Wu, Zhizhang, Email: wzz14@mails.tsinghua.edu.cn, and Huang, Zhongyi, Email: zhuang@math.tsinghua.edu.cn. 2016.
"A Bloch decompositionbased stochastic Galerkin method for quantum dynamics with a random external potential". United States.
doi:10.1016/J.JCP.2016.04.051.
@article{osti_22572336,
title = {A Bloch decompositionbased stochastic Galerkin method for quantum dynamics with a random external potential},
author = {Wu, Zhizhang, Email: wzz14@mails.tsinghua.edu.cn and Huang, Zhongyi, Email: zhuang@math.tsinghua.edu.cn},
abstractNote = {In this paper, we consider the numerical solution of the onedimensional 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 decompositionbased timesplitting 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 wellknown phenomenon of Anderson localization.},
doi = {10.1016/J.JCP.2016.04.051},
journal = {Journal of Computational Physics},
number = ,
volume = 317,
place = {United States},
year = 2016,
month = 7
}

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