Bound state solutions of the hyper-radial Klein-Gordon equation under the Deng-Fan potential by WKB and SWKB methods

The energy levels of the Klein–Gordon equation in hyper-radial space under the Deng-Fan potential energy function are studied by the SWKB and WKB approximation methods. We obtained the analytic solution of the energy spectra and the ground state wave function in closed form. Furthermore, we obtained the energy equation corresponding to the Schrodinger equation by invoking the non-relativistic limit. The variations of the non-relativistic N -dimensional energy spectra with the potential parameters and radial quantum number are investigated. The energy levels are degenerate for N = 2 , N = 4 and increase with the dimensionality number. The ground state wave function and its gradient are continuous at the boundary r = 0 , r = ∞ . Our results for the energy spectra are in excellent agreement with the ones obtained by other analytical methods where similar centrifugal approximations were applied.