On Aharonov-Casher bound states

TL;DR

This paper investigates bound states in the Aharonov-Casher problem, deriving new energy expressions considering singular potentials and boundary conditions, and applies the findings to a combined Aharonov-Casher and harmonic oscillator system.

Contribution

It introduces a novel method to derive bound state energies for the Aharonov-Casher problem using self-adjoint extensions and boundary conditions at the origin.

Findings

Derived an explicit expression for the Aharonov-Casher bound state energy.

Compared energies with and without the singular potential.

Provided a physical approach to determine the self-adjoint extension parameter.

Abstract

In this work bound states for the Aharonov-Casher problem are considered. According to Hagen's work on the exact equivalence between spin-1/2 Aharonov-Bohm and Aharonov-Casher effects, is known that the $\boldsymbol{\nabla}\cdot\mathbf{E}$ term cannot be neglected in the Hamiltonian if the spin of particle is considered. This term leads to the existence of a singular potential at the origin. By modeling the problem by boundary conditions at the origin which arises by the self-adjoint extension of the Hamiltonian, we derive for the first time an expression for the bound state energy of the Aharonov-Casher problem. As an application, we consider the Aharonov-Casher plus a two-dimensional harmonic oscillator. We derive the expression for the harmonic oscillator energies and compare it with the expression obtained in the case without singularity. At the end, an approach for determination of the self-adjoint extension parameter is given. In our approach, the parameter is obtained essentially in terms of physics of the problem.