Exact solution of the relativistic finite-difference equation for the Coulomb plus a ring-shaped-like potential

Abstract

In this paper, a three-dimensional problem of the motion of a charged relativistic particle in a noncentral Coulomb plus ring-shaped potential is studied. Our investigation is based on a finite-difference version of relativistic quantum mechanics. The energy eigenvalues and the corresponding wave functions are obtained analytically. It is shown that radial part and the angular part of the wave functions are expressed through the Meixner–Pollaczek polynomials and Jacobi polynomials, respectively. All relativistic expressions, for example, radial wave functions and energy spectrum, have the correct nonrelativistic limit. We also build a dynamical symmetry group for the radial part of the equation of motion, which allows us to find the energy spectrum purely algebraically.