Free and bound spin-polarized fermions in the fields of Aharonov--Bohm kind

TL;DR

This paper constructs a self-adjoint Hamiltonian for spin-polarized electrons in an Aharonov--Bohm field, analyzes its domain, and derives scattering properties, revealing bound states and extending to neutral fermions with anomalous magnetic moments.

Contribution

It introduces a self-adjoint extension of the Pauli Hamiltonian in the Aharonov--Bohm field, explicitly incorporating electron spin and singular functions, and applies this to scattering and bound state analysis.

Findings

Existence of bound states for certain extension parameters.

Derived scattering amplitude and cross section for spin-polarized electrons.

Extended analysis to neutral fermions with anomalous magnetic moments.

Abstract

The scattering of electrons by an Aharonov--Bohm field is considered from the viewpoint of quantum-mechanical problem of constructing a self-adjoint Hamiltonian for the Pauli equation. The correct domain for the self-adjoint Hamiltonian, which takes into account explicitly the electron spin is found. A one-parameter self-adjoint extension of the Hamiltonian for spin-polarized electrons in the Aharonov--Bohm field is selected. The correct domain of the self-adjoint Hamiltonian can contain regular and singular (at the point ${\bf r}=0$) square-integrable functions on the half-line with measure $rdr$. We argue that the physical reason of the existence of singular functions is the additional attractive potential, which appear due to the interaction between the spin magnetic moment of fermion and Aharonov--Bohm magnetic field. The scattering amplitude and cross section are obtained for spin-polarized electrons scattered by the Aharonov--Bohm field. It is shown that in some range of the extension parameter there appears a bound state. Since the Hamiltonian of the nonrelativistic Dirac--Pauli equation for a massive neutral fermion with the anomalous magnetic moment (AMM) in the electric field of a linear charge aligned perpendicularly to the fermion motion has the form of the Hamiltonian for the Pauli equation in the Aharonov--Bohm flux tube, we also calculate the scattering amplitude and cross section for the neutral fermion.