In this work, we present approximate and analytical solutions of the deformed Klein–Gordon containing an interaction of the equal vector and scalar potential newly generalized modified screened Coulomb plus inversely quadratic Yukawa potential. To overcome the centrifugal barrier, we employed the well-known Greene and Aldrich approximation scheme. This study is realized in the relativistic noncommutative 3-dimensional real space symmetries. This potential is suggested to compute bound-state normalizations and the energy levels of neutral atoms. Furthermore, it is considered a good potential in studying Hydrogen-like atoms. Both ordinary Bopp’s shift method, perturbation theory, and the Greene–Aldrich approximation method of handling centrifugal barriers are surveyed to get generalized excited states energy \(E_{r-nc}^{sip} \left( {V_{0} ,\,V_{1} ,\,k\left( {j,l,s} \right) ,\,\alpha ,\,n,\,j,\,l,\,m,\,s} \right) \), as a function of the shift energy, the energy of the modified screened Coulomb plus inversely quadratic Yukawa potential, the discreet atomic quantum numbers \((j,\,l,\,s,\,m)\), the potential parameters (\(V_{0} ,\,V_{1} ,\,\alpha )\) and the infinitesimal noncommutativity parameters (\(\theta \) and \(\sigma \)). We have shown that the degeneracy of the initial spectral under the potential in the relativistic commutative quantum mechanics RCQM is broken and replaced by newly degeneracy of energy levels; this gives more precisions in measurement and better off compared to results of ordinary RCQM.

在这项工作中，我们介绍了变形的Klein-Gordon的近似和解析解，其中包含等矢量和标量势的相互作用，以及新近广义修改过的屏蔽库仑加反二次余川势。为了克服离心障碍，我们采用了著名的Greene和Aldrich近似方案。这项研究是在相对论的非可交换3维实空间对称中实现的。建议使用此电势来计算束缚态归一化和中性原子的能级。此外，它被认为是研究类氢原子的良好潜力。研究了普通的Bopp位移法，扰动理论和处理离心屏障的Greene-Aldrich近似方法，以获得广义的激发态能量\（E_ {r-nc} ^ {sip} \ left（{V_ {0}，\，V_ {1}，\，k \ left（{j，l，s} \ right），\，\ alpha， \，n，\，j，\，l，\，m，\，s} \ right）\），作为位移能量的函数，经过修改的屏蔽库仑能量加上逆二次汤川势，即离散原子量子数\（（j，\，l，\，s，\，m）\），潜在参数（\（V_ {0}，\，V_ {1}，\，\ alpha）\）和无穷小非交换性参数（\（\ theta \）和\（\ sigma \））。我们已经表明，相对论可交换量子力学RCQM中的势能下的初始光谱的简并性被打破，并被新的能级简并性所取代；与普通的RCQM结果相比，这提供了更高的测量精度和更好的性能。