Modified Unequal Mixture Scalar Vector Hulthén–Yukawa Potentials Model as a Quark–Antiquark Interaction and Neutral Atoms via Relativistic Treatment Using the Improved Approximation of the Centrifugal Term and Bopp’s Shift Method

Abstract

In this work, we present approximate solutions of the modified Klein–Gordon containing an interaction of the modified unequal mixture scalar vector Hulthen–Yukawa potentials model (MUMSVHYa-PM) using the procedure of improved approximation of the centrifugal term, Bopp’s shift method and perturbation theory. This study is realized in the relativistic noncommutative three-dimensional real-space (RNC: 3D-RS) symmetries. The unequal mixture of scalar and vector Hulthen–Yukawa potentials model (UMSVHYa-PM) is extended by including new radial terms to become MUMSVHYa-PM. We show that the new energy shift depends on the global parameters characterizing the noncommutativity space-space and the confining potential parameter ( $$V_{0} ,S_{0} ,V_{_{0} }^{'} ,S_{_{0} }^{'} ,\delta )$$ in addition to the Gamma function and the discreet atomic quantum numbers $$\left( {j,l,s,m} \right) $$ . Thus, the new energy eigenvalues appear as a function of the shift energy and the energy to the UMSVHYa-PM and ordinary energy spectrum corresponds to UMSVHYa-PM. The present results are applied in calculating both the energy spectrum and the new mass of mesons such as charmonium and bottomonium that have the quark and antiquark flavor. Furthermore, we obtained new energy eigenvalues of neutral atoms such as ( $$^{{22}}\mathrm{Na},^{{12}}{\mathrm{C}}$$ and $${}^{{158}}{\mathrm{Au}})$$ atoms under MUMSVHYa-PM. We have also discussed some special cases of physical importance.