Eur. Phys. J. C (2013) 73: 2402
https://doi.org/10.1140/epjc/s10052-013-2402-1

Regular Article - Theoretical Physics

On Aharonov–Casher bound states

¹ Departamento de Física, Universidade Federal do Maranhão, Campus Universitário do Bacanga, 65085-580, São Luís, MA, Brazil
² Departamento de Matemática e Estatística, Universidade Estadual de Ponta Grossa, 84030-900, Ponta Grossa, PR, Brazil
³ Departamento de Física, Universidade Federal de Campina Grande, Caixa Postal 10071, 58109-970, Campina Grande, PB, Brazil
⁴ Departamento de Física e Química, Universidade Federal do Espírito Santo, 29060-900, Vitória, ES, Brazil

^* e-mail: fmandrade@uepg.br

Received: 13 November 2012
Published online: 24 April 2013

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 ∇⋅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.