Charged Particle with Magnetic Moment in the Aharonov-Bohm Potential

TL;DR

This paper investigates the quantum behavior of a charged particle with a magnetic moment in an Aharonov-Bohm potential, revealing that an anomalous magnetic moment induces bound states and analyzing different modeling approaches.

Contribution

It introduces a detailed analysis of bound states caused by magnetic moment anomalies in Aharonov-Bohm settings, comparing self-adjoint extension and finite flux tube models.

Findings

Anomalous magnetic moments lead to bound states in the Aharonov-Bohm effect.

Self-adjoint extension method reproduces results for flux less than one.

Finite radius flux tube models show multiple bound states depending on flux quantum number.

Abstract

We considered a charged quantum mechanical particle with spin ${1\over 2}$ and gyromagnetic ratio $g\ne 2$ in the field af a magnetic string. Whereas the interaction of the charge with the string is the well kown Aharonov-Bohm effect and the contribution of magnetic moment associated with the spin in the case $g=2$ is known to yield an additional scattering and zero modes (one for each flux quantum), an anomaly of the magnetic moment (i.e. $g>2$) leads to bound states. We considered two methods for treating the case $g>2$. \\ The first is the method of self adjoint extension of the corresponding Hamilton operator. It yields one bound state as well as additional scattering. In the second we consider three exactly solvable models for finite flux tubes and take the limit of shrinking its radius to zero. For finite radius, there are $N+1$ bound states ($N$ is the number of flux quanta in the tube).\\ For $R\to 0$ the bound state energies tend to infinity so that this limit is not physical unless $g\to 2$ along with $R\to 0$. Thereby only for fluxes less than unity the results of the method of self adjoint extension are reproduced whereas for larger fluxes $N$ bound states exist and we conclude that this method is not applicable.\\ We discuss the physically interesting case of small but finite radius whereby the natural scale is given by the anomaly of the magnetic moment of the electron $a_e=(g-2)/2\approx 10^{-3}$.