Solution of the spin and pseudo-spin symmetric Dirac equation in 1+1 space-time using the tridiagonal representation approach

TL;DR

This paper derives exact solutions for the Dirac equation in 1+1 dimensions with spin and pseudo-spin symmetry, using a tridiagonal matrix approach and orthogonal polynomials to find energy spectra and wavefunctions.

Contribution

It introduces a novel tridiagonal representation method to solve the Dirac equation with additional potentials, extending known solutions beyond previous classes.

Findings

Exact energy spectra obtained for specific potentials.

Wavefunctions expressed in terms of orthogonal polynomials.

Method applicable to a broader class of Dirac equations.

Abstract

The aim of this work is to find exact solutions of the Dirac equation in 1+1 space-time beyond the already known class. We consider exact spin (and pseudo-spin) symmetric Dirac equations where the scalar potential is equal to plus (and minus) the vector potential. We also include pseudo-scalar potentials in the interaction. The spinor wavefunction is written as a bounded sum in a complete set of square integrable basis, which is chosen such that the matrix representation of the Dirac wave operator is tridiagonal and symmetric. This makes the matrix wave equation a symmetric three-term recursion relation for the expansion coefficients of the wavefunction. We solve the recursion relation exactly in terms of orthogonal polynomials and obtain the state functions and corresponding relativistic energy spectrum and phase shift.