Narrow Results

In this work, on the condition that scalar potential is equal to vector potential, the bound state solutions of the Klein–Fock–Gordon equation of the Manning–Rosen plus ring-shaped like potential are obtained by Nikiforov–Uvarov method. The energy levels are worked out and the corresponding normalized eigenfunctions are obtained in terms of orthogonal polynomials for arbitrary l states. The conclusion also contain central Manning–Rosen, central and noncentral Hulthén potential.

We employed the elegant tool of Bopp’s shift and standard perturbation theory methods to obtain a new relativistic and nonrelativistic approximate bound state solution of deformed Klein–Gordon DKG and deformed Schrödinger DSE equations using the modified equal vector scalar Manning–Rosen plus quadratic Yukawa potential (MVSMQY-Ps) model. Furthermore, we have employed the improved approximation to the centrifugal term for some selected diatomic molecules such as N₂, I₂, HCl, CH and LiH in the symmetries of extended quantum mechanics to obtain the approximate solutions. The relativistic shift energy $\mathrm{\Delta }{E}_{rmqy}^{\mathrm{\text{tot}}}(n,\delta ,\eta ,b,A,V_0^{\prime },\mathrm{\Theta },\sigma ,\chi ,j,l,s,m)$ and the perturbative nonrelativistic corrections $\mathrm{\Delta }{E}_{mqy}^{nr}(n,\delta ,\eta ,b,A,V_0^{\prime },\mathrm{\Theta },\sigma ,\chi ,j,l,s,m)$ appeared as a function of the parameters ($\delta ,\eta ,b,A,V_0^{\prime }$), the parameters of noncommutativity ($\mathrm{\Theta },\sigma ,\chi$), in addition to the atomic quantum numbers ($n,j,l,s,m)$. In both relativistic and nonrelativistic problems, we show that the corrections on the spectrum energy are smaller than the main energy in the ordinary cases of RQM and NRQM. A straightforward limit of our results to ordinary quantum mechanics shows that the present result under MVSMQAY-Ps is consistent with what is obtained in the literature. In the new symmetries of NCQM, it is not possible to get the exact analytical solutions for $l=0$ and $l\ne 0$, and can only be solved approximately. We have observed that the DKGE under the MVSMQY-Ps model has a physical behavior similar to the Duffin–Kemmer equation for meson with spin-1; it can describe a dynamic state of a particle with spin-1 in the symmetries of RNCQM. Moreover, we have treated composite systems such as molecules made of $N=2$ particles of masses $m_n(n=1,2)$ in the frame of noncommutative algebra. The NRNCQM and RNCQM results obtained within Bopp’s shift and standard perturbation theory methods overlap entirely with the results obtained by ordinary NCQM, and it displays that the theoretical investigation in this study is excellent.