Energy spectra with the Dirac equation of the q-deformed generalized Pöschl-Teller potential via the Feynman approach...

The diatomic molecules of potassium $_{}^{39}{K}_2\left(a^3,{\sum }_u^{+}\right)$ is widely used in industrial chemicals and alternative energy. Besides that, $_{}^{39}{K}_2\left(a^3,{\sum }_u^{+}\right)$ is very useful for researching molecular interactions and energy states, especially in the context of quantum chemistry and spectroscopy. In the present work, a newly proposed diatomic potential model within relativistic and non-relativistic quantum mechanics has been considered, to obtain corresponding energy eigenvalues and related normalized eigenfunctions.

The Dirac equation has been solved for an arbitrary spin-orbit quantum number $\kappa$ using the path integral technique with the $q$ -deformed generalized Pöschl-Teller potential $\left(DGPT\right)$ . By including a Pekeris-type approximation to handle the centrifugal factor, it was possible to obtain the spin and pseudospin-symmetric solution of the relativistic energy eigenvalues and wave equation. To assess the correctness of this work, Maple software was used to present some numerical findings for various values of $n$ and $\kappa$ . With the constraint $\tilde{\lambda }>\tilde{\eta }+1$ , it was shown that in the situation of pseudospin symmetry, only bound states exist with negative energy. In the non-relativistic limits, the non-relativistic ro-vibrational energy expression of the diatomic molecule is derived from the relativistic energy equation under spin symmetry. Under Varshni conditions, both vibrational and ro-vibrational energies of the $_{}^{39}{K}_2\left(a^3,{\sum }_u^{+}\right)$ molecule were computed and compared with the $\mathrm{RKR}$ data. The average absolute percentage deviations from the $\mathrm{RKR}$ data obtained for the potassium molecule are $0.5018\%$ . This demonstrates that the $\left(DGPT\right)$ model is a very consistent model to study and characterize diatomic molecules.