New bound-state solutions of the deformed Klein–gordon and Schrödinger equations for arbitrary l-state with the modified equal vector and scalar Manning–Rosen plus a class of Yukawa potentials in RNCQM and NRNCQM symmetries

In this work, 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 the deformed Klein(cid:21)Gordon and deformed Schr(cid:4)odinger equations using the modi-(cid:28)ed equal vector scalar Manning(cid:21)Rosen plus a class of Yukawa potentials (DVSMCY-Ps, in short) model. Furthermore, we have employed the improved approximation to the centrifugal term for some selected diatomic molecules, such as N 2 , I 2 , HCl, CH, LiH, and CO, in the symmetries of extended quantum mechanics to obtain the approximate solutions. The relativistic shift energy ∆ E totmcy ( n, δ, η, b, A, V 0 , V (cid:48) 0 , Θ , σ, χ, j, l, s, m ) and the perturbative nonrelativistic corrections ∆ E nrmcy ( n, δ, η, b, A, V 0 , V (cid:48) 0 , Θ , σ, χ, j, l, s, m ) appeared as a function of the parameters ( δ, η, b, A, V 0 , V (cid:48) 0 ) , the parameters of noncommutativity (Θ , σ, χ ) , in addition to the atomic quantum numbers ( n, j, l, s, m ) . In both relativistic and nonrelativistic problems, we show that the corrections to the spectrum energy are smaller than the main energy in the ordinary cases of relativistic quantum mechanics and nonrelativistic quantum mechanics. A straightforward limit of our results to ordinary quantum mechanics shows that the present result under DVSMCY-Ps is consistent with what is obtained in the literature. In the new symmetries of noncommutative quantum mechanics, it is not possible to get exact analytical solutions for l = 0 , and l (cid:54) = 0 can only be solved approximately. We have observed that the DKGE under the DVSMCY-Ps model has a physical behavior similar to the Du(cid:30)n(cid:21)Kemmer equation for meson with spin-1, it can describe a dynamic state of a particle with spin-1 in the symmetries of relativistic noncommutative quantume mechanics.