Solution of the relativistic Dirac-Hulthen problem

TL;DR

This paper analytically solves the relativistic Dirac equation with the Hulthen potential, deriving energy spectra and wavefunctions, and explores their properties and nonrelativistic limits.

Contribution

It provides an exact analytical solution for the Dirac-Hulthen problem, including energy spectra and wavefunctions, and analyzes their properties and limits.

Findings

Energy spectrum splits into two subspaces based on parameter sign

Wavefunctions expressed in Jacobi polynomials

Nonrelativistic limit reproduces known spectra

Abstract

The one-particle three-dimensional Dirac equation with spherical symmetry is solved for the Hulthen potential. The s-wave relativistic energy spectrum and two-component spinor wavefunctions are obtained analytically. Conforming to the standard feature of the relativistic problem, the solution space splits into two distinct subspaces depending on the sign of a fundamental parameter in the problem. Unique and interesting properties of the energy spectrum are pointed out and illustrated graphically for several values of the physical parameters. The square integrable two-component wavefunctions are written in terms of the Jacobi polynomials. The nonrelativistic limit reproduces the well-known nonrelativistic energy spectrum and results in Schrodinger equation with a "generalized" three-parameter Hulthen potential, which is the sum of the original Hulthen potential and its square.