Dirac equation in the magnetic-solenoid field

TL;DR

This paper analyzes the Dirac equation in magnetic-solenoid fields, constructing self-adjoint Hamiltonian extensions, and provides new solutions for finite-radius solenoids, advancing understanding of quantum behavior in such fields.

Contribution

It introduces the first solutions of the Dirac equation with finite-radius solenoids and details the boundary conditions for magnetic-solenoid fields, extending previous theoretical frameworks.

Findings

Constructed self-adjoint extensions of the Dirac Hamiltonian in magnetic-solenoid fields.

Derived solutions of the Dirac equation for finite-radius solenoids.

Analyzed the dependence of solutions on the magnetic field inside the solenoid.

Abstract

We consider the Dirac equation in the magnetic-solenoid field (the field of a solenoid and a collinear uniform magnetic field). For the case of Aharonov-Bohm solenoid, we construct self-adjoint extensions of the Dirac Hamiltonian using von Neumann's theory of deficiency indices. We find self-adjoint extensions of the Dirac Hamiltonian in both above dimensions and boundary conditions at the AB solenoid. Besides, for the first time, solutions of the Dirac equation in the magnetic-solenoid field with a finite radius solenoid were found. We study the structure of these solutions and their dependence on the behavior of the magnetic field inside the solenoid. Then we exploit the latter solutions to specify boundary conditions for the magnetic-solenoid field with Aharonov-Bohm solenoid.