Classical and Quantum Mechanical Motion in Magnetic Fields

TL;DR

This paper investigates the classical and quantum dynamics of particles in specific magnetic fields, revealing how classical trajectories relate to quantum expectations and introducing a numerical method for quantum simulations.

Contribution

It establishes the classical dependence of escape speeds on gauge-fixed potentials and extends a numerical method to quantum problems with magnetic fields.

Findings

Classical escape speeds depend on gauge choice.

Quantum trajectories show similarities and differences with classical paths.

A numerical method for quantum magnetic problems is demonstrated.

Abstract

We study the motion of a particle in a particular magnetic field configuration both classically and quantum mechanically. For flux-free radially symmetric magnetic fields defined on circular regions, we establish that particle escape speeds depend, classically, on a gauge-fixed magnetic vector potential, and demonstrate some trajectories associated with this special type of magnetic field. Then we show that some of the geometric features of the classical trajectory (perpendicular exit from the field region, trapped and escape behavior) are reproduced quantum mechanically using a numerical method that extends the norm-preserving Crank-Nicolson method to problems involving magnetic fields. While there are similarities between the classical trajectory and the position expectation value of the quantum mechanical solution, there are also differences, and we demonstrate some of these.