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

## Abstract

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.

- 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; ONE-DIMENSIONAL CALCULATIONS; PERIODICITY; POLYNOMIALS; POTENTIALS; RANDOMNESS; SCHROEDINGER EQUATION; SOLID STATE PHYSICS; SOLIDS; STOCHASTIC PROCESSES

### Citation Formats

```
Wu, Zhizhang, E-mail: wzz14@mails.tsinghua.edu.cn, and Huang, Zhongyi, E-mail: zhuang@math.tsinghua.edu.cn.
```*A Bloch decomposition-based 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, E-mail: wzz14@mails.tsinghua.edu.cn, & Huang, Zhongyi, E-mail: zhuang@math.tsinghua.edu.cn.
```*A Bloch decomposition-based stochastic Galerkin method for quantum dynamics with a random external potential*. United States. doi:10.1016/J.JCP.2016.04.051.

```
Wu, Zhizhang, E-mail: wzz14@mails.tsinghua.edu.cn, and Huang, Zhongyi, E-mail: zhuang@math.tsinghua.edu.cn. Fri .
"A Bloch decomposition-based 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 decomposition-based stochastic Galerkin method for quantum dynamics with a random external potential},

author = {Wu, Zhizhang, E-mail: wzz14@mails.tsinghua.edu.cn and Huang, Zhongyi, E-mail: zhuang@math.tsinghua.edu.cn},

abstractNote = {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.},

doi = {10.1016/J.JCP.2016.04.051},

journal = {Journal of Computational Physics},

number = ,

volume = 317,

place = {United States},

year = {Fri Jul 15 00:00:00 EDT 2016},

month = {Fri Jul 15 00:00:00 EDT 2016}

}