Angular-Radial Integrability of Coulomb-like Potentials in Dirac Equations

TL;DR

This paper investigates the integrability of the Dirac equation with Coulomb-like potentials in polar coordinates, showing that angular parts are always integrable and radial parts reduce to solving a Riccati equation, demonstrating the method's versatility.

Contribution

It introduces a general method for analyzing the integrability of Dirac equations with Coulomb-like potentials, reducing the radial problem to a Riccati equation and illustrating it with known and new examples.

Findings

Angular dependence always integrable

Radial dependence reduces to Riccati equation

Method applicable to various Coulomb-like potentials

Abstract

We consider the Dirac equation, written in polar formalism, in presence of general Coulomb-like potentials, that is potentials arising from the time component of the vector potential and depending only on the radial coordinate, in order to study the conditions of integrability, given as some specific form for the solution: we find that the angular dependence can always be integrated, while the radial dependence is reduced to finding the solution of a Riccati equation so that it is always possible at least in principle. We exhibit the known case of the Coulomb potential and one special generalization as examples to show the versatility of the method.