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 ΔErmqytot(n,δ,η,b,A,V0′,Θ,σ,χ,j,l,s,m) and the perturbative nonrelativistic corrections ΔEmqynr(n,δ,η,b,A,V0′,Θ,σ,χ,j,l,s,m) appeared as a function of the parameters (δ,η,b,A,V0′), 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 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≠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 mn(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.

中文翻译：

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在 RNCQM 和 NRNCQM 对称性中使用改进的离心项近似和 Bopp 位移法对变形的 Klein-Gordon 和薛定谔方程内的修正等标量和向量 Manning-Rosen 加二次 Yukawa 势进行新的理论研究

我们使用了 Bopp 位移和标准微扰理论方法的优雅工具，使用改进的等向量标量 Manning-Rosen 加上二次 Yukawa 势 (MVSMQY -Ps) 模型。此外，我们在扩展量子力学的对称性中对一些选定的双原子分子如N ₂、I _{2 、HCl、CH和LiH采用了离心项的改进近似来获得近似解。}相对论位移能量Δ乙r米q是的总(n,δ,η,b,一种,五0',θ,σ,χ,j,l,s,米)和微扰非相对论修正Δ乙米q是的nr(n,δ,η,b,一种,五0',θ,σ,χ,j,l,s,米)作为参数的函数出现（δ,η,b,一种,五0'), 不可交换性的参数 (θ,σ,χ)，除了原子量子数 (n,j,l,s,米). 在相对论和非相对论问题中，我们都证明了在 RQM 和 NRQM 的普通情况下，谱能量的修正小于主能量。我们的结果对普通量子力学的直接限制表明，MVSMQAY-Ps 下的当前结果与文献中获得的结果一致。在 NCQM 的新对称性中，不可能得到精确的解析解l=0和l≠0, 只能近似求解。我们观察到 MVSMQY-Ps 模型下的 DKGE 具有类似于具有自旋 1 的介子的 Duffin-Kemmer 方程的物理行为；它可以描述具有 RNCQM 对称性的自旋为 1 的粒子的动态状态。此外，我们还处理了复合系统，例如由ñ=2质量粒子米n(n=1,2)在非交换代数框架中。Bopp位移和标准微扰理论方法得到的NRNCQM和RNCQM结果与普通NCQM得到的结果完全重叠，表明本研究的理论研究是优秀的。