Solutions of Klein–Gordon and Dirac Equations for Non-pure Dipole Potential in 2D Systems

Abstract

We study the quantum relativistic wave equations (Klein–Gordon and Dirac) for the non-pure dipole potential $$V(r)=-Ze/r+D\cos \theta /r^{2}$$ , in the case of two-dimensional systems. We consider either spin symmetry or anti-spin symmetry cases in our computations. We give the analytical expressions of the eigenfunctions, compute the exact values of the energies and study their dependence according to the dipole moment D. Our study generalizes the energies of the Kratzer potential as well as the magnetic quantum number m, which is replaced with the Mathieu characteristic values obtained during the resolution of the angular equations. For each magnetic quantum number, we demonstrate the existence of a critical value for the dipole moment, beyond which the corresponding bound state can no longer exist. We find that the critical value is null when $$m=0$$ ; this means that these s-states cannot exist for this system and this is in agreement with non-relativistic studies.