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

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.

16 pages, 2 tables, 3 figures

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Cited by in corpus (3)

• Solution of the spin and pseudo-spin symmetric Dirac equation in 1+1 space-time using the tridiagonal representation approach
• Exact solvability of two new 3D and 1D nonrelativistic potentials within the TRA framework
• Analytical solutions of the Dirac equation using the Tridiagonal Representation Approach: General study, limitations, and possible applications