Finite Difference-Time Domain solution of the Dirac equation and the dynamics of the Aharonov-Bohm effect

TL;DR

This paper presents a 3D FDTD method to solve the time-dependent Dirac equation, revealing electron wave packet dynamics related to the Aharonov-Bohm effect and confirming classical predictions about velocity changes without forces.

Contribution

It introduces a finite difference-time domain approach for the Dirac equation and explores quantum-classical correspondence in the Aharonov-Bohm effect.

Findings

Electron wave packet velocity changes without force influence.

Solution aligns with classical dynamics assuming canonical momentum conservation.

Magnetic field presence links momentum conservation to Lorentz force action.

Abstract

The time-dependent Dirac equation is solved using the three-dimensional Finite Difference-Time Domain (FDTD) method. The dynamics of the electron wave packet in a vector potential is studied in the arrangements associated with the Aharonov-Bohm effect. The solution of the Dirac equation showed a change in the velocity of the electron wave packet even in a region where no forces acted on the electron. The solution of the Dirac equation agreed with the prediction of classical dynamics under the assumption that the dynamics was defined by the conservation of generalized or canonical momentum. It was also shown that in the case when the magnetic field was not zero, the conservation of generalized or canonical momentum was equivalent to the action of the Lorentz force.