# Numerical approximation of the Schrödinger equation with the electromagnetic field by the Hagedorn wave packets

## Abstract

In this paper, we approximate the semi-classical Schrödinger equation in the presence of electromagnetic field by the Hagedorn wave packets approach. By operator splitting, the Hamiltonian is divided into the modified part and the residual part. The modified Hamiltonian, which is the main new idea of this paper, is chosen by the fact that Hagedorn wave packets are localized both in space and momentum so that a crucial correction term is added to the truncated Hamiltonian, and is treated by evolving the parameters associated with the Hagedorn wave packets. The residual part is treated by a Galerkin approximation. We prove that, with the modified Hamiltonian only, the Hagedorn wave packets dynamics give the asymptotic solution with error O(ε{sup 1/2}), where ε is the scaled Planck constant. We also prove that, the Galerkin approximation for the residual Hamiltonian can reduce the approximation error to O(ε{sup k/2}), where k depends on the number of Hagedorn wave packets added to the dynamics. This approach is easy to implement, and can be naturally extended to the multidimensional cases. Unlike the high order Gaussian beam method, in which the non-constant cut-off function is necessary and some extra error is introduced, the Hagedorn wave packets approachmore »

- Authors:

- Publication Date:

- OSTI Identifier:
- 22314898

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Computational Physics; Journal Volume: 272; Other Information: Copyright (c) 2014 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; ACCURACY; APPROXIMATIONS; ASYMPTOTIC SOLUTIONS; CORRECTIONS; ELECTROMAGNETIC FIELDS; ERRORS; GAUSS FUNCTION; HAMILTONIANS; POTENTIALS; SCHROEDINGER EQUATION; VECTORS; WAVE PACKETS

### Citation Formats

```
Zhou, Zhennan, E-mail: zhou@math.wisc.edu.
```*Numerical approximation of the Schrödinger equation with the electromagnetic field by the Hagedorn wave packets*. United States: N. p., 2014.
Web. doi:10.1016/J.JCP.2014.04.041.

```
Zhou, Zhennan, E-mail: zhou@math.wisc.edu.
```*Numerical approximation of the Schrödinger equation with the electromagnetic field by the Hagedorn wave packets*. United States. doi:10.1016/J.JCP.2014.04.041.

```
Zhou, Zhennan, E-mail: zhou@math.wisc.edu. Mon .
"Numerical approximation of the Schrödinger equation with the electromagnetic field by the Hagedorn wave packets". United States.
doi:10.1016/J.JCP.2014.04.041.
```

```
@article{osti_22314898,
```

title = {Numerical approximation of the Schrödinger equation with the electromagnetic field by the Hagedorn wave packets},

author = {Zhou, Zhennan, E-mail: zhou@math.wisc.edu},

abstractNote = {In this paper, we approximate the semi-classical Schrödinger equation in the presence of electromagnetic field by the Hagedorn wave packets approach. By operator splitting, the Hamiltonian is divided into the modified part and the residual part. The modified Hamiltonian, which is the main new idea of this paper, is chosen by the fact that Hagedorn wave packets are localized both in space and momentum so that a crucial correction term is added to the truncated Hamiltonian, and is treated by evolving the parameters associated with the Hagedorn wave packets. The residual part is treated by a Galerkin approximation. We prove that, with the modified Hamiltonian only, the Hagedorn wave packets dynamics give the asymptotic solution with error O(ε{sup 1/2}), where ε is the scaled Planck constant. We also prove that, the Galerkin approximation for the residual Hamiltonian can reduce the approximation error to O(ε{sup k/2}), where k depends on the number of Hagedorn wave packets added to the dynamics. This approach is easy to implement, and can be naturally extended to the multidimensional cases. Unlike the high order Gaussian beam method, in which the non-constant cut-off function is necessary and some extra error is introduced, the Hagedorn wave packets approach gives a practical way to improve accuracy even when ε is not very small.},

doi = {10.1016/J.JCP.2014.04.041},

journal = {Journal of Computational Physics},

number = ,

volume = 272,

place = {United States},

year = {Mon Sep 01 00:00:00 EDT 2014},

month = {Mon Sep 01 00:00:00 EDT 2014}

}