Solving Dirac equation using the tridiagonal matrix representation approach

• We choose L2 basis such that the Dirac wave operator is tridiagonal matrix.
• We use the tridiagonal-matrix-representation approach.
• The wave equation becomes a symmetric three-term recursion relation.
• We solve the associated three-term recursion relation exactly.
• The energy spectrum formula is obtained.

The aim of this work is to find exact solutions of the one dimensional Dirac equation using the tridiagonal matrix representation. 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. We are honored to dedicate this work to Prof. Hashim A. Yamani on the occasion of his 70th birthday.