Dirac Equation with vector and scalar potentials via Supersymmetry in Quantum Mechanics

TL;DR

This paper explores the Dirac equation with combined vector and scalar potentials using supersymmetry, providing exact solutions for energy levels and eigenfunctions in a relativistic quantum framework.

Contribution

It introduces a supersymmetric approach to solve the Dirac equation with generalized Coulomb and scalar potentials, yielding explicit energy eigenvalues and eigenfunctions.

Findings

Exact energy eigenvalues for ground and first excited states

Explicit eigenfunctions derived for specific states

Supersymmetry simplifies solving relativistic potentials

Abstract

In this work, a spin $\frac 12$ relativistic particle described by a generalized potential containing both the Coulomb potential and a Lorentz scalar potential in Dirac equation is investigated in terms of the generalized ladder operators from supersymmetry in quantum mechanics. This formalism is applied for the generalized Dirac-Coulomb problem, which is an exactly solvable potential in relativistic quantum mechanics. We obtain the energy eigenvalues and calculate explicitly the energy eigenfunctions for the ground state and the first excited state.