Solutions of Schrödinger and Klein-Gordon Equations with Hulthen Plus Inversely Quadratic Exponential Mie-Type Potential

We proposed a novel potential called Hulthen plus Inversely Quadratic Exponential Mie-Type potential (HIQEMP). We use parametric Nikiforov-Uvarov method to study approximate solutions of Schrödinger and Klein-Gordon equations with the novel potential. We obtain bound state energies and the normalized wave function expressed in terms of Jacobi polynomial. The proposed potential is applicable in the field of vibrational and rotational spectroscopy. To ascertain the accuracy of our results, we apply the nonrelativistic limit to the Klein-Gordon equation to obtain the energy equation which is exactly the same as nonrelativistic Schrodinger energy equation. This is a proof that relativistic equation can be converted to nonrelativistic equation using a nonrelativistic limit with the help of Greene-Aldrich approximation to the centrifugal term. The wave functions were normalized analytically using two infinite series of confluent hypergeometric functions. We implement MATLAB algorithm to obtain the numerical bound state energy eigen-values for both Schrödinger and Klein-Gordon equations. Our potential reduce to many existing potentials and the result is in agreement with existing literature. The energy spectral diagrams were plotted using origin software. The bound state energy from Schrodinger equation decreases with increase in quantum state while that of Klein-Gordon equation increases with increase in quantum state.