L2 series solution of the relativistic Dirac-Morse problem for all energies

TL;DR

This paper presents an analytic, series-based solution to the relativistic Dirac equation with Morse potential for all energies, utilizing hypergeometric functions and orthogonal polynomials to handle bound and scattering states.

Contribution

It introduces a novel series solution approach for the Dirac-Morse problem applicable to all energy levels, including bound and scattering states, with explicit wavefunction representations.

Findings

Wavefunctions expressed in terms of confluent hypergeometric functions.

Matrix representation of the operator is tridiagonal for scattering states.

Wavefunction coefficients follow a three-term recursion relation solved by orthogonal polynomials.

Abstract

We obtain analytic solutions for the one-dimensional Dirac equation with the Morse potential as an infinite series of square integrable functions. These solutions are for all energies, the discrete as well as the continuous. The elements of the spinor basis are written in terms of the confluent hypergeometric functions. They are chosen such that the matrix representation of the Dirac-Morse operator for continuous spectrum (i.e., for scattering energies larger than the rest mass) is tridiagonal. Consequently, the wave equation results in a three-term recursion relation for the expansion coefficients of the wavefunction. The solution of this recursion relation is obtained in terms of the continuous dual Hahn orthogonal polynomials. On the other hand, for the discrete spectrum (i.e., for bound states with energies less than the rest mass) the spinor wave functions result in a diagonal matrix representation for the Dirac-Morse Hamiltonian.