Volume 26, Issue 2
The Gaussian Wave Packet Transform for the Semi-Classical Schrödinger Equation with Vector Potentials

Zhennan Zhou & Giovanni Russo

Commun. Comput. Phys., 26 (2019), pp. 469-505.

Published online: 2019-04

[An open-access article; the PDF is free to any online user.]

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  • Abstract

In this paper, we reformulate the semi-classical Schrödinger equation in the presence of electromagnetic field by the Gaussian wave packet transform. With this approach, the highly oscillatory Schrödinger equation is equivalently transformed into another Schrödinger type wave equation, the w equation, which is essentially not oscillatory and thus requires much less computational effort. We propose two numerical methods to solve the w equation, where the Hamiltonian is either divided into the kinetic, the potential and the convection part, or into the kinetic and the potential-convection part. The convection, or the potential-convection part is treated by a semi-Lagrangian method, while the kinetic part is solved by the Fourier spectral method. The numerical methods are proved to be unconditionally stable, spectrally accurate in space and second order accurate in time, and in principle they can be extended to higher order schemes in time. Various one dimensional and multidimensional numerical tests are provided to justify the properties of the proposed methods.

  • Keywords

Semi-classical Schrödinger equation, Gaussian wave packets, splitting methods, Fourier-spectral methods.

  • AMS Subject Headings

65M70, 35Q41, 74Q10, 65Z05

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COPYRIGHT: © Global Science Press

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@Article{CiCP-26-469, author = {}, title = {The Gaussian Wave Packet Transform for the Semi-Classical Schrödinger Equation with Vector Potentials}, journal = {Communications in Computational Physics}, year = {2019}, volume = {26}, number = {2}, pages = {469--505}, abstract = {

In this paper, we reformulate the semi-classical Schrödinger equation in the presence of electromagnetic field by the Gaussian wave packet transform. With this approach, the highly oscillatory Schrödinger equation is equivalently transformed into another Schrödinger type wave equation, the w equation, which is essentially not oscillatory and thus requires much less computational effort. We propose two numerical methods to solve the w equation, where the Hamiltonian is either divided into the kinetic, the potential and the convection part, or into the kinetic and the potential-convection part. The convection, or the potential-convection part is treated by a semi-Lagrangian method, while the kinetic part is solved by the Fourier spectral method. The numerical methods are proved to be unconditionally stable, spectrally accurate in space and second order accurate in time, and in principle they can be extended to higher order schemes in time. Various one dimensional and multidimensional numerical tests are provided to justify the properties of the proposed methods.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2018-0131}, url = {http://global-sci.org/intro/article_detail/cicp/13099.html} }
TY - JOUR T1 - The Gaussian Wave Packet Transform for the Semi-Classical Schrödinger Equation with Vector Potentials JO - Communications in Computational Physics VL - 2 SP - 469 EP - 505 PY - 2019 DA - 2019/04 SN - 26 DO - http://doi.org/10.4208/cicp.OA-2018-0131 UR - https://global-sci.org/intro/article_detail/cicp/13099.html KW - Semi-classical Schrödinger equation, Gaussian wave packets, splitting methods, Fourier-spectral methods. AB -

In this paper, we reformulate the semi-classical Schrödinger equation in the presence of electromagnetic field by the Gaussian wave packet transform. With this approach, the highly oscillatory Schrödinger equation is equivalently transformed into another Schrödinger type wave equation, the w equation, which is essentially not oscillatory and thus requires much less computational effort. We propose two numerical methods to solve the w equation, where the Hamiltonian is either divided into the kinetic, the potential and the convection part, or into the kinetic and the potential-convection part. The convection, or the potential-convection part is treated by a semi-Lagrangian method, while the kinetic part is solved by the Fourier spectral method. The numerical methods are proved to be unconditionally stable, spectrally accurate in space and second order accurate in time, and in principle they can be extended to higher order schemes in time. Various one dimensional and multidimensional numerical tests are provided to justify the properties of the proposed methods.

Zhennan Zhou & Giovanni Russo. (2019). The Gaussian Wave Packet Transform for the Semi-Classical Schrödinger Equation with Vector Potentials. Communications in Computational Physics. 26 (2). 469-505. doi:10.4208/cicp.OA-2018-0131
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