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Time-integration of Gaussian variational approximation for the magnetic Schrödinger equation

Malik Scheifinger, Kurt Busch, Marlis Hochbruck, Caroline Lasser

Abstract

In the present paper we consider the semiclassical magnetic Schrödinger equation, which describes the dynamics of charged particles under the influence of a electro-magnetic field. The solution of the time-dependent Schrödinger 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.

Time-integration of Gaussian variational approximation for the magnetic Schrödinger equation

Abstract

In the present paper we consider the semiclassical magnetic Schrödinger equation, which describes the dynamics of charged particles under the influence of a electro-magnetic field. The solution of the time-dependent Schrödinger 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.

Paper Structure

This paper contains 24 sections, 7 theorems, 66 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Contributions of the paper
  3. Outline of the paper
  4. Notation
  5. Penning trap
  6. Variational approximation
  7. Variational equations of motion
  8. Asymptotic accuracy
  9. Classical versus variational approximation
  10. Hagedorn parametrization
  11. Transformation of equations of motion
  12. Averaged magnetic momenta
  13. Equations for the center and the width
  14. Equations for the phase
  15. Time integration for the equations of motion
  16. ...and 9 more sections

Key Result

Lemma 4.1

The variational equations of motion eq:eom:var-all for the parameters of a Gaussian wave packet are equivalent to where $\langle a \rangle_{{u^{}}} = \langle \mathrm{op}_\mathrm{Weyl}(a) \rangle_{{u^{}}}$ for any smooth $a:\mathbb{R}^{2d}\to\mathbb{R}$, $(x,\xi)\mapsto a(x,\xi)$. In Hagedorn's parametrisation, the matrix evolution eq:eqmo_C-l31 satisfies

Figures (6)

  • Figure 1: Exact trajectory of a proton in a hyperbolic Penning trap with data from \ref{['tab:trap-frequencies']} and initial condition specified in \ref{['subsec:Penning-trap']} on the dimensionless time interval $[0,2\pi]$.
  • Figure 2: Simulation of the motion of a particle in a magnetic field in dimension two with potentials \ref{['eq:sublinear-pot-expls']} and initial values \ref{['eq:sublinear-IV']}. Errors of the numerical solution to \ref{['eq:euler-lagrange-q-v']} approximated by the Boris-type algorithm (left) and the mRK4 method (right) measured in the Frobenius norm scaled with the inverse number of entries.
  • Figure 3: $L^2$-error of a Gaussian wave packet approximated by the Boris-type method \ref{['Algo:Boris']} (left) and the mRK4 method (right) against a reference Gaussian wave packet with coefficients approximated by the classical RK4 method with time stepsize $\tau=10^{-4}$. The potentials are given by \ref{['eq:sublinear-pot-expls']} and the initial values by \ref{['eq:sublinear-IV']}. Different values for $\varepsilon$ are considered.
  • Figure 4: Energy error against the initial energy (top) and $L^2$-norm error (bottom) using the potentials and initial values in Section \ref{['subsec:convg-plots']}. The endtime is chosen as $T=200$ and $\varepsilon=10^{-3}$. On the left, the maximal energy error is illustrated over all time stamps for different time stepsizes $\tau$. On the right, the energy error over the time interval $[0, 200]$ is plotted for a fixed time stepsize $\tau = 10^{-1}$.
  • Figure 5: Maximum errors of the parameters approximated by the Boris-type method \ref{['Algo:Boris']} (left), the RK4 method (middle) and the mRK4 method (right) against the exact solution using the potentials \ref{['eq:trap-potentials']} with data given in \ref{['tab:trap-frequencies']} and initial values \ref{['eq:penning-IV']}. Measured in the Frobenius norm scaled with the inverse number of components.
  • ...and 1 more figures

Theorems & Definitions (15)

  • Lemma 4.1
  • proof
  • Proposition 4.2
  • proof
  • Theorem 4.3
  • proof
  • Remark 4.4: Linear potential ${A^{}}$
  • Lemma 4.5
  • proof
  • Lemma 5.1
  • ...and 5 more