The development of potential theory heightens the understanding of fundamental interactions in quantum systems. In this paper, the bound state solution of the modified radial Klein–Gordon equation is presented for generalised tanh-shaped hyperbolic potential from the Nikiforov–Uvarov method. The resulting energy eigenvalues and corresponding radial wave functions are expressed in terms of the Jacobi polynomials for arbitrary l states. It is also demonstrated that energy eigenvalues strongly correlate with potential parameters for quantum states. Considering particular cases, the generalised tanh-shaped hyperbolic potential and its derived energy eigenvalues exhibit good agreement with the reported findings. Furthermore, the rovibrational energies are calculated for three representative diatomic molecules, namely H₂, HCl and O₂. The lowest excitation energies are in perfect agreement with experimental results. Overall, the potential model is displayed to be a viable candidate for concurrently prescribing numerous quantum systems.

In this study, we present an analytical solution of the Dirac equation in a generalized tanh-shape hyperbolic potential, which allows us to unify various well-known quantum potentials under a single theoretical framework. This versatile potential model is used to compute the mass spectra for charmonium and bottomonium, with excellent agreement with experimental measurements, and does better than some potential models in predicting the several orbital states. Our results not only validate the GTHP as a powerful tool for describing heavy quarkonium systems but also suggest its broader applicability in exploring quantum systems where similar potentials are effective. This work is a stepping stone for new research into fermionic systems with complex interactions, by jointly providing insights into foundational aspects of quantum mechanics as well as applications in particle physics.