The Aharonov-Bohm scattering : the role of the incident wave

TL;DR

This paper revisits the Aharonov-Bohm scattering problem, demonstrating that using a plane wave as the incident wave yields consistent wave functions through different methods, and explores effects of finite-radius solenoids on scattering.

Contribution

It provides a new validation of the incident plane wave approach using two different methods and extends the analysis to finite-radius solenoids affecting charged particles.

Findings

Wave function matches Aharonov and Bohm's original results

Both methods confirm the validity of plane wave incident assumption

Finite-radius solenoid affects particles even at integer flux values

Abstract

The scattering problem under the influence of the Aharonov-Bohm (AB) potential is reconsidered. By solving the Lippmann-Schwinger (LS) equation we obtain the wave function of the scattering state in this system. In spite of working with a plane wave as an incident wave we obtain the same wave function as was given by Aharonov and Bohm. Another method to solve the scattering problem is given by making use of a modified version of Gordon's idea which was invented to consider the scattering by the Coulomb potential. These two methods give the same result, which guarantees the validity of taking an incident plane wave as usual to make an analysis of this scattering problem. The scattering problem by a solenoid of finite radius is also discussed, and we find that the vector potential of the solenoid affects the charged particles even when the magnitude of the flux is an odd integer as well as noninteger. It is shown that the unitarity of the $S$ matrix holds provided that a plane wave is taken to be an incident one.