Semiclassical asymptotics of the Aharonov-Bohm interference process

TL;DR

This paper derives the semiclassical limit of a charged particle's motion around an infinitely thin solenoid with magnetic flux, clarifying the origin of discontinuities and the emergence of Aharonov-Bohm interference patterns.

Contribution

It systematically connects quantum and classical descriptions of the Aharonov-Bohm effect, revealing how interference patterns arise from semiclassical propagators and their relation to classical action.

Findings

Derived the semiclassical limit of the quantum propagator

Linked quantum angular momentum to classical angular momentum

Explained the emergence of Aharonov-Bohm interference patterns

Abstract

In order to determine the origin of discontinuities which arise when the semiclassical propagator is employed to describe an infinitely long and infinitesimally thin solenoid carrying magnetic flux, we give a systematic derivation of the semiclassical limit of the motion of an otherwise free charged particle. Our limit establishes the connection of the quantum mechanical canonical angular momentum to its classical counterpart. Moreover, we show how a picture of Aharonov-Bohm interference of two half-waves acquiring Dirac's magnetic phase when passing on either side of the solenoid emerges from the quantum propagator, and that the typical scale of the resulting interference pattern is fully determined by the ratio of the angular part of Hamilton's principal function to Planck's constant. The semiclassical propagator is recovered in the limit when this ratio diverges. We discuss the relation of our results to the whirling-wave representation of the exact propagator.