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A High-Order Integrator for the Schrödinger Equation with Time-Dependent, Homogeneous Magnetic Field

by V. Gradinaru and O. Rietmann

(Report number 2018-47)

Abstract
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.

Keywords: Pauli equation, Fourier, splitting, Lie algebra

@Techreport{GR18_801,
  author = {V. Gradinaru and O. Rietmann},
  title = {A High-Order Integrator for the Schr\"odinger Equation with Time-Dependent, Homogeneous Magnetic Field},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2018-47},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2018/2018-47.pdf },
  year = {2018}
}

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