Exact approaches to charged particle motion in a time-dependent flux-driven ring

TL;DR

This paper explores exact quantum solutions for a charged particle in a time-dependent flux-driven ring, revealing how classical trajectories influence quantum wave packets and highlighting the limitations of Ehrenfest's theorem in topologically nontrivial systems.

Contribution

The study introduces a non-Hermitian invariant approach to obtain exact wave functions and analyzes the quantum-classical correspondence in a flux-driven ring system.

Findings

Wave packet peaks follow classical trajectories but are not expectation values.

Ehrenfest theorem does not hold in this topologically nontrivial system.

Gaussian wave packets are characterized by initial conditions and classical motion.

Abstract

We consider a charged particle which is driven by a time-dependent flux threading a circular ring system. Various approaches including classical treatment, Fourier expansion method, time-evolution method, and Lewis-Riesenfeld method are used and compared to solve the time-dependent problem. By properly managing the boundary condition of the system, a time-dependent wave function of the charged particle can be obtained by using a non-Hermitian time-dependent invariant, which is a specific linear combination of initial angular-momentum and azimuthal-angle operators. The eigenfunction of the linear invariant can be realized as a Gaussian-type wave packet with a peak moving along the classical angular trajectory, while the distribution of the wave packet is determined by the ratio of the coefficient of the initial angle to that of the initial canonical angular momentum. In this topologically nontrivial system, we find that although the classical trajectory and angular momentum can determine the motion of the wave packet; however, the peak position is no longer an expectation value of the angle operator. Therefore, in such a system, the Ehrenfest theorem is not directly applicable.