Solution of the Dirac Equation for Potential Interaction

TL;DR

This paper reviews a method for solving the three-dimensional Dirac equation with spherically symmetric potentials, enabling the derivation of Schrödinger-like equations for spinor components that connect to well-known solvable problems.

Contribution

It consolidates a recently introduced approach that simplifies solving the Dirac equation and extends exactly solvable nonrelativistic potentials to the relativistic domain.

Findings

Successfully applied to shape invariant potentials

Reproduces nonrelativistic limits accurately

Establishes relativistic extensions of classic potentials

Abstract

An effective approach for solving the three-dimensional Dirac equation for spherically symmetric local interactions, which we have introduced recently, is reviewed and consolidated. The merit of the approach is in producing Schrodinger-like equation for the spinor components that could simply be solved by correspondence with well-known exactly solvable non-relativistic problems. Taking the nonrelativistic limit will reproduce the nonrelativistic problem. The approach has been used successfully in establishing the relativistic extension of all classes of shape invariant potentials as well as other exactly solvable nonrelativistic problems. These include the Coulomb, Oscillator, Scarf, Poschl-Teller, Woods-Saxon, etc.