Energy spectra with the Dirac equation of the q-deformed generalized Pöschl-Teller potential via the Feynman approach for \(^{39}K_{2}\left( a^{3}\sum _{u}^{+}\right) \)

Abstract

Context

The diatomic molecules of potassium \(^{\varvec{39}}\varvec{K}_{\varvec{2}}\left( \varvec{a}^{\varvec{3}}\varvec{\sum }_{\varvec{u}}^{\varvec{+}}\right) \) is widely used in industrial chemicals and alternative energy. Besides that, \(^{\varvec{39}}\varvec{K}_{\varvec{2}}\left( \varvec{a}^{\varvec{3}}\varvec{\sum }_{\varvec{u}}^{\varvec{+}}\right) \) 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.

Methods

The Dirac equation has been solved for an arbitrary spin-orbit quantum number \(\varvec{\kappa }\) using the path integral technique with the \(\varvec{q}\)-deformed generalized Pöschl-Teller potential \(\varvec{(DGPT)}\). 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 \(\varvec{n}\) and \(\varvec{\kappa }\). With the constraint \(\varvec{\tilde{\lambda }}\varvec{>}\varvec{\tilde{\eta }+1}\), 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 \(^{\varvec{39}}\varvec{K}_{\varvec{2}}\left( \varvec{a}^{\varvec{3}}\varvec{\sum }_{\varvec{u}}^{\varvec{+}}\right) \) molecule were computed and compared with the \(\varvec{RKR}\) data. The average absolute percentage deviations from the \(\varvec{RKR}\) data obtained for the potassium molecule are \(\varvec{0.5018\%}\). This demonstrates that the \(\varvec{(DGPT)}\) model is a very consistent model to study and characterize diatomic molecules.