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arXiv · 2504.03407

Time-integration of Gaussian variational approximation for the magnetic Schr\"odinger equation

Malik Scheifinger·Kurt Busch·Marlis Hochbruck·Caroline Lasser
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Abstract

In the present paper we consider the semiclassical magnetic Schr\"odinger equation, which describes the dynamics of charged particles under the influence of a electro-magnetic field. The solution of the time-dependent Schr\"odinger equation is approximated by a single Gaussian wave packet via the time-dependent Dirac--Frenkel variational principle. For the approximation we use ordinary differential equations of motion for the parameters of the variational solution and extend the second-order Boris algorithm for classical mechanics to the quantum mechanical case. In addition, we propose a modified version of the classical fourth order Runge--Kutta method. Numerical experiments explore parameter convergence and geometric properties. Moreover, we benchmark against the analytical solution of the Penning trap.

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Malik Scheifinger, Kurt Busch, Marlis Hochbruck, Caroline Lasser. 2025-04-04. Time-integration of Gaussian variational approximation for the magnetic Schr\"odinger equation. https://arxiv.org/abs/2504.03407

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