SUSYQM and SWKB Approaches to the Relativistic Equations with Hyperbolic PotentialV₀tanh²(r/d)

We apply supersymmetric quantum mechanics (SUSYQM) and shape invariance to investigate the Klein-Gordon and Dirac equations with the hyperbolic potential V 0 tanh 2 ( r / d ). We find that the master equations for the Klein-Gordon and Dirac equations (radial wave function F ( r )) with this potential for the S wave are unified as a same second order differential equation. The energy levels are obtained by SUSYQM and supersymmetric WKB (SWKB) approaches. We also apply the traditional hypergeometric differential equation approach to solve the master equation and obtain the exact solutions. We show that the vibrational quantum number n for the energy levels has to be an odd integer only through studying the properties of the eigenfunctions expressed by the Gegenbauer polynomials. Finally, we study the harmonic limit of this system.