Extending the class of solutions of the Dirac equation using tridiagonal matrix representations

TL;DR

This paper develops a method to find new exact solutions to the one-dimensional Dirac equation by using tridiagonal matrix representations, extending beyond conventional solutions, and applies it to a specific potential case.

Contribution

It introduces a novel approach using tridiagonal matrix representations to derive new exact solutions of the Dirac equation beyond known classes.

Findings

Derived relativistic energy spectrum for the model

Obtained wavefunctions from recursion relations

Confirmed nonrelativistic limit matches known results

Abstract

The aim of this work is to find exact solutions of the one-dimensional Dirac equation that do not belong to the already known conventional class. We write the spinor wavefunction as a bounded infinite sum in a complete basis set, which is chosen such that the matrix representation of the Dirac wave operator becomes tridiagonal and symmetric. This makes the wave equation equivalent to a symmetric three-term recursion relation for the expansion coefficients of the wavefunction. We solve the recursion relation and obtain the relativistic energy spectrum and corresponding state functions. Restriction to the diagonal representation results in the conventional class of solutions. As illustration, we consider the square well with half-sine potential bottom in the vector and scalar coupling under spin symmetry. We obtain the relativistic energy spectrum and show that the nonrelativistic limit agrees with results found elsewhere.