# Stable and efficient momentum-space solutions of the time-dependent Schrödinger equation for one-dimensional atoms in strong laser fields

## Abstract

One-dimensional model systems have a particular role in strong-field physics when gaining physical insight by computing data over a large range of parameters, or when performing numerous time propagations within, e.g., optimal control theory. Here we derive a scheme that removes a singularity in the one-dimensional Schrödinger equation in momentum space for a particle in the commonly used soft-core Coulomb potential. By using this scheme we develop two numerical approaches to the time-dependent Schrödinger equation in momentum space. The first approach employs the expansion of the momentum-space wave function over the eigenstates of the field-free Hamiltonian, and it is shown to be more efficient for laser parameters usual in strong field physics. The second approach employs the Crank–Nicolson scheme or the method of lines for time-propagation. The both methods are readily applicable for large-scale numerical simulations in one-dimensional model systems.

- Authors:

- Publication Date:

- OSTI Identifier:
- 22382151

- Resource Type:
- Journal Article

- Journal Name:
- Journal of Computational Physics

- Additional Journal Information:
- Journal Volume: 279; Other Information: Copyright (c) 2014 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0021-9991

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ATOMS; COMPUTERIZED SIMULATION; COULOMB FIELD; EIGENSTATES; HAMILTONIANS; LASER RADIATION; MATHEMATICAL SOLUTIONS; ONE-DIMENSIONAL CALCULATIONS; OPTIMAL CONTROL; SCHROEDINGER EQUATION; SINGULARITY; SPACE; TIME DEPENDENCE; WAVE FUNCTIONS

### Citation Formats

```
Shvetsov-Shilovski, N.I., E-mail: nikolay.shvetsov@tut.fi, and Räsänen, E., E-mail: erasanen@tut.fi.
```*Stable and efficient momentum-space solutions of the time-dependent Schrödinger equation for one-dimensional atoms in strong laser fields*. United States: N. p., 2014.
Web. doi:10.1016/J.JCP.2014.09.006.

```
Shvetsov-Shilovski, N.I., E-mail: nikolay.shvetsov@tut.fi, & Räsänen, E., E-mail: erasanen@tut.fi.
```*Stable and efficient momentum-space solutions of the time-dependent Schrödinger equation for one-dimensional atoms in strong laser fields*. United States. doi:10.1016/J.JCP.2014.09.006.

```
Shvetsov-Shilovski, N.I., E-mail: nikolay.shvetsov@tut.fi, and Räsänen, E., E-mail: erasanen@tut.fi. Mon .
"Stable and efficient momentum-space solutions of the time-dependent Schrödinger equation for one-dimensional atoms in strong laser fields". United States. doi:10.1016/J.JCP.2014.09.006.
```

```
@article{osti_22382151,
```

title = {Stable and efficient momentum-space solutions of the time-dependent Schrödinger equation for one-dimensional atoms in strong laser fields},

author = {Shvetsov-Shilovski, N.I., E-mail: nikolay.shvetsov@tut.fi and Räsänen, E., E-mail: erasanen@tut.fi},

abstractNote = {One-dimensional model systems have a particular role in strong-field physics when gaining physical insight by computing data over a large range of parameters, or when performing numerous time propagations within, e.g., optimal control theory. Here we derive a scheme that removes a singularity in the one-dimensional Schrödinger equation in momentum space for a particle in the commonly used soft-core Coulomb potential. By using this scheme we develop two numerical approaches to the time-dependent Schrödinger equation in momentum space. The first approach employs the expansion of the momentum-space wave function over the eigenstates of the field-free Hamiltonian, and it is shown to be more efficient for laser parameters usual in strong field physics. The second approach employs the Crank–Nicolson scheme or the method of lines for time-propagation. The both methods are readily applicable for large-scale numerical simulations in one-dimensional model systems.},

doi = {10.1016/J.JCP.2014.09.006},

journal = {Journal of Computational Physics},

issn = {0021-9991},

number = ,

volume = 279,

place = {United States},

year = {2014},

month = {12}

}