Analytical solutions of the Proca equation for modified Manning–Rosen potential with centrifugal term and direct coupling approaches by hypergeometric method

The Proca equation is one of the fundamental relativistic wave equations of massive vector (spin 1) boson fields. Proca equation is used to explain a relatively small particle that has a speed approaching the speed of light. The effect of photon imaginary mass to the quantum system that was coupled by noncentral potential consisting of radial, polar, and azimuthal modified Manning–Rosen potentials was investigated. The vector potential was coupled to the photon relativistic energy and the quadratic scalar potential was coupled directly to the quadratic photon mass. In the case of free space and when the vector potential was equal to scalar potential, therefore, the time-independent Proca equation reduced to a three-dimensional variable separable Schrodinger-like equation. By using variable separation method, the three-dimensional Schrodinger-like equation was resolved into three one-dimensional radial, polar, and azimuthal parts of Schrodinger-like equations. Each part of one-dimensional Schrodinger-like equation was solved using the hypergeometric differential equation through variables and wave function substitutions and resulted in the energy and wave function equations of massive photons. The relativistic energy equation of massive photons was a function of imaginary mass photon, the potential parameters, and the quantum numbers in radial, polar, and azimuthal variables. By using the Maxwell field approach, we got the correction factor for the energy function of this spin-1 particle.