Analytical solutions of the Dirac equation using the Tridiagonal Representation Approach: General study, limitations, and possible applications

TL;DR

This paper extends the Tridiagonal Representation Approach to solve the one-dimensional Dirac equation, transforming it into an algebraic recursion relation, and explores its applications and limitations, including potential use in graphene.

Contribution

It introduces a generalized method for solving the Dirac equation using TRA, including new classes of solutions and analysis of solvable potentials with symmetry considerations.

Findings

Solutions relate to known orthogonal polynomials in some cases

New classes of polynomials emerge in other cases

Potential applications in graphene are discussed

Abstract

This paper aims at extending our previous work on the solution of the one-dimensional Dirac equation using the Tridiagonal Representation Approach (TRA). In the approach, we expand the spinor wavefunction in terms of suitable square integrable basis functions that support a tridiagonal matrix representation of the wave operator. This will transform the problem from solving a system of coupled first order differential equations to solving an algebraic three-term recursion relation for the expansion coefficients of the wavefunction. In some cases, solutions to this recursion relation can be related to well-known classes of orthogonal polynomials whereas in other situations solutions represent new class of polynomials. In this work, we will discuss various solvable potentials that obey the tridiagonal representation requirement with special emphasis on simple cases with spin-symmetric and pseudospin-symmetric potential couplings. We conclude by mentioning some potential applications in graphene.