Author(s)

Ibsal A. Assi^1*, Hocine Bahlouli^2,3

¹Department of Physics and Physical Oceanography, Memorial University of Newfoundland, St. John’s A1B3X7, NL, Canada.
²Physics Department, King Fahd University of Petroleum & Minerals, Dhahran, KSA.
³Saudi Center for Theoretical Physics, Jeddah, KSA.

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.

Dirac Equation, Tridiagonal Representation, Three-Term Recursion Relation, Orthogonal Polynomials, Energy Spectrum, Isospectral Potentials, Spin-Symmetric Coupling, Pseudo-Spin-Symmetric Coupling, Graphene

Assi, I. and Bahlouli, H. (2017) Analytical Solutions of the 1D Dirac Equation Using the Tridiagonal Representation Approach. Journal of Applied Mathematics and Physics, 5, 2072-2092. doi: 10.4236/jamp.2017.510172.