Approximate Analytical Solutions of the Klein–Gordon Equation with Generalized Morse Potential

In this study, the Klein–Gordon equation for the newly proposed generalized Morse potential (GMP) was solved using the newly proposed Nikiforov–Uvarov-functional analysis (NUFA) method for solving central potentials. The analytical expressions of Eigen solutions for GMP were obtained in closed form. The effects of the GMP parameters on the energy eigenvalues have been discussed. The wave function and probability density graphs with position have been illustrated, as regards the ground state, first, and second excited states for different deformation parameters. The effects of temperature and upper bound vibrational quantum number have been shown to vary for different thermodynamic functions considered for the selected deformation parameters. Hence, it was observed that the deformation parameter has a strong influence on the energy and thermodynamic function of the GMP model.