A High-Order Integrator for the Schrödinger Equation with Time-Dependent, Homogeneous Magnetic Field
The SMAI Journal of computational mathematics, Volume 6 (2020), pp. 253-271.

We construct a family of numerical methods for the Pauli equation of charged particles in a time-dependent, homogeneous magnetic field. These methods are described in a general setting comprising systems of multiple particles and extend the usual splitting and Fourier grid approach. The issue is that the magnetic field causes charged particles to rotate. The corresponding rotations of the wave function are highly incompatible with the Fourier grid approach used for the standard Schrödinger equation. Motivated by the theory of Lie algebras and their representations, our new approach approximates the exact flow map in terms of rotated potentials and rotated initial data, and thereby avoids this issue. Finally, we provide numerical examples to examine convergence and preservation of norm and energy.

Published online:
DOI: 10.5802/smai-jcm.69
Classification: 81Q05, 65M12, 65M22
Keywords: quantum mechanics, time-dependent Schrödinger equation, magnetic field, discrete Fourier transform, numerical method, splitting, convergence rate, Lie group, Lie algebra
Vasile Gradinaru 1; Oliver Rietmann 1

1 Seminar for Applied Mathematics, ETH Zurich, Switzerland
License: CC-BY-NC-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Vasile Gradinaru; Oliver Rietmann. A High-Order Integrator for the Schrödinger Equation with Time-Dependent, Homogeneous Magnetic Field. The SMAI Journal of computational mathematics, Volume 6 (2020), pp. 253-271. doi : 10.5802/smai-jcm.69. https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.69/

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