Exact and Approximate Solutions of Dirac–Morse Problem in Curved Space-Time

Abstract

In this work, we analyze the Dirac–Morse problem with spin and pseudo-spin symmetries in deformed nuclei. So, we consider the Dirac equation with the scalar U(r) and vector V(r) Morse-type potentials and tensor Hellmann-type potential in curved space-time whose line element is of type $ds^2 = (1+\alpha ^2 U(r))^2(dt^2- dr^2) - r^2d\theta ^2-r^2\sin ^2\theta d\phi ^2$. From the effective tensor potential $A_{eff}(r) = \lambda /r + \alpha ^2\lambda U(r)/r + A(r)$, that contain terms of spin-orbit coupling, line element and electromagnetic field, we analyze dirac’s spinor in two ways: (i) in the first, we solve the problem approximately considering $A_{eff}(r)$ not null; (ii) in the second analysis, we obtain exact solutions of radial spinor and eigenenergies considering $A_{eff}(r) = 0$. In both cases, we consider two types of coupling of vector and scalar potentials, with spin symmetry for $V(r) = U(r)$ and pseudo-spin symmetry for $V(r) = -\,U(r)$. We analyzed the effect of coupling the electromagnetic field with the curvature of space in eigenenergies and radial spinor.