The Aharonov--Bohm effect in scattering theory

TL;DR

This paper analyzes the Aharonov--Bohm effect as a scattering process using the WKB method, revealing classical reflection and quantum diffraction phenomena, with implications for controlling particle propagation via magnetic flux.

Contribution

It demonstrates the effectiveness of the WKB method in scattering problems involving the Aharonov--Bohm effect and clarifies the independence of phenomena from boundary conditions.

Findings

Scattering cross section has classical and quantum components.

Aharonov--Bohm effect appears as a fringe shift in diffraction.

Particle propagation can be controlled by magnetic flux.

Abstract

The Aharonov--Bohm effect is considered as a scattering event with nonrelativistic charged particles of the wavelength which is less than the transverse size of an impenetrable magnetic vortex. The quasiclassical WKB method is shown to be efficient in solving this scattering problem. We find that the scattering cross section consists of two terms, one describing the classical phenomenon of elastic reflection and another one describing the quantum phenomenon of diffraction; the Aharonov--Bohm effect is manifested as a fringe shift in the diffraction pattern. Both the classical and the quantum phenomena are independent of the choice of a boundary condition at the vortex edge, providing that probability is conserved. We show that a propagation of charged particles can be controlled by altering the flux of a magnetic vortex placed on their way.