The Investigation of Approximate Solutions of Deformed Klein–Gordon and Schrödinger Equations Under Modified More General Exponential Screened Coulomb Potential Plus Yukawa Potential in NCQM Symmetries

Abstract

By employing the improved approximation to deal with the centrifugal term, we solve the deformed Klein–Gordon and deformed Schrödinger equations with the modified more general exponential screened Coulomb potential plus Yukawa potential for all values of l (orbital angular momentum quantum number). Using Bopp’s shift and standard perturbation theory methods, the new approximate analytical energy shift and the corrections of bound state energy eigenvalues in relativistic noncommutative quantum mechanics RNCQM and nonrelativistic noncommutative quantum mechanics NRNCQM are obtained for some selected diatomic molecules such as (N $$_{\\mathrm {2}}$$ 2 , CO, and CO). It is seen that relativistic shift energy and the perturbative nonrelativistic corrections are sensitive to the quantum numbers $$\\left( {j,s,m} \\right) $$ j , s , m in addition to the usual states numbers $$\\left( {n,l} \\right) $$ n , l , the potential depth of the more general exponential screened Coulomb potential, and Yukawa potential ( $$V_{0} $$ V 0 , $$V_{1} )$$ V 1 ) , the adjustable positive parameter $$\\alpha $$ α and the infinitesimal parameters of noncommutativity $$\\left( {\\Theta ,\\sigma ,\\chi } \\right) $$ Θ , σ , χ . We have highlighted three physical phenomena that automatically generate a result of the topological properties of noncommutativity, the first physical phenomena are the perturbative spin-orbit coupling, the second the magnetic induction while the third corresponds to the rotational proper phenomena. 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 quantum field theory and quantum mechanics. A straightforward limit of our results to ordinary quantum mechanics shows that the present result under the modified more general exponential screened Coulomb potential plus Yukawa potential 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 $$l=0$$ l = 0 and $$l\\ne 0$$ l ≠ 0 , the approximate solutions are available. We have observed that the deformed Klein–Gordon equation under the modified more general exponential screened Coulomb potential plus Yukawa potential has a physical behavior similar to the Duffin–Kemmer equation for a meson with spin-1, it can describe the dynamic state of a particle with spin-1 in the symmetries of RNCQM. 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.