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

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.

References in corpus (6)

• Electric Field Effect in Atomically Thin Carbon Films
• Relating pseudospin and spin symmetries through charge conjugation and chiral transformations: the case of the relativistic harmonic oscillator
• An extended class of L2-series solutions of the wave equation
• Scattering and bound states for a class of non-central potentials
• Extending the class of solutions of the Dirac equation using tridiagonal matrix representations
• Spurious Modes in Dirac Calculations and How to Avoid Them