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

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 o...

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Bibliographic Details
Published in:Journal of computational physics 2014-09, Vol.272, p.386-407
Main Author: Zhou, Zhennan
Format: Article
Language:English
Subjects:
ACCURACY
Approximation
APPROXIMATIONS
ASYMPTOTIC SOLUTIONS
CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS
CORRECTIONS
ELECTROMAGNETIC FIELDS
ERRORS
Galerkin approximation
Galerkin methods
GAUSS FUNCTION
Hagedorn wave packets
HAMILTONIANS
Mathematical analysis
Mathematical models
POTENTIALS
SCHROEDINGER EQUATION
Semi-classical Schrödinger equation
Vector potential
VECTORS
WAVE PACKETS
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Summary: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(ε1/2), 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(εk/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.
ISSN:0021-9991
1090-2716
DOI:10.1016/j.jcp.2014.04.041