Solutions of the Dirac Equation in a Magnetic Field and Intertwining Operators

TL;DR

This paper applies the intertwining technique to solve the Dirac equation for a relativistic particle in a cylindrically symmetric magnetic field, revealing the Hamiltonian's shape invariance and enabling spectrum analysis.

Contribution

It adapts the intertwining method to Dirac Hamiltonians in a specific magnetic field, demonstrating shape invariance and solution generation.

Findings

Hamiltonian is shape invariant

Bound states can be obtained explicitly

Method extends to relativistic quantum systems

Abstract

The intertwining technique has been widely used to study the Schr\"odinger equation and to generate new Hamiltonians with known spectra. This technique can be adapted to find the bound states of certain Dirac Hamiltonians. In this paper the system to be solved is a relativistic particle placed in a magnetic field with cylindrical symmetry whose intensity decreases as the distance to the symmetry axis grows and its field lines are parallel to the $x-y$ plane. It will be shown that the Hamiltonian under study turns out to be shape invariant.