Abstract
In this study, the deformed Klein–Gordon equation, Dirac equation, and Schrödinger equations were solved with the improved the modified Mobius square potential model (IMSPM, in short) using Bopp’s shift and standard perturbation theory methods in the symmetries of extended quantum mechanics. By employing the improved approximation to the centrifugal term, the relativistic and nonrelativistic bound state energies were obtained for some selected diatomic molecules (H2, I2, CO, NO, and HCl). The relativistic shift energy ΔEmsptot(n,α,A,B,V0,Θ,σ,χ,j,l,s,m) and the perturbative nonrelativistic corrections ΔEmspnr(n,α,A,B,V0,Θ,σ,χ,j,l,s,m) appeared as a function of the parameters (α,A,B,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 the improved modified Mobius square potential model is consistent with what is obtained in the literature. In the new symmetries of NCQM, is not possible to get the exact analytical solutions for l=0 and l≠0 can only be solved approximately. Through this research, we came to two noteworthy results, the first is related to the deformed Klein-Gordon equation under the influence of the improved modified Mobius square potential model becomes equivalent to the Duffin-Kemmer equation for a meson with spin-s, while the second result concerns the deformed Schrödinger equation, which can now describe the state of high-energy fermionic particles similar to the Dirac equation in the literature.
Highlights
Despite the passage of nearly a century on the three fundamental equations are known in the literature Schrödinger, Klien-Gordon, and Dirac equations, they are still attracting the interest of many researchers
Before we start constructing the new solutions of the deformed Klein-Gordon theory, the deformed Dirac theory, and the deformed Schrödinger theory under the improved modified Mobius squared potential model, we give a summary of the corresponding usual solutions in ordinary relativistic quantum mechanics RQM and nonrelativistic quantum mechanics NRQM
We consider a special case of interest of the improved modified Mobius square potential, known by the improved Deng-Fan potential is given in Eq (63) in RNCQM symmetries
Summary
Despite the passage of nearly a century on the three fundamental equations are known in the literature Schrödinger, Klien-Gordon, and Dirac equations, they are still attracting the interest of many researchers. Before we start constructing the new solutions of the deformed Klein-Gordon theory, the deformed Dirac theory, and the deformed Schrödinger theory under the improved modified Mobius squared potential model, we give a summary of the corresponding usual solutions in ordinary relativistic quantum mechanics RQM and nonrelativistic quantum mechanics NRQM. Okorie et al.,[1] Onyenegecha et al.,[2] and Ikot et al.[3] using both Nikiforov-Uvarov method and the Greene and Aldrich approximation for the centrifugal term[48] to obtain the expressions for the wave function as hypergeometric polynomials and the corresponding energy values for the modified Mobius squared potential, in RQM and NRQM symmetries as, With and α is the Yukawa tensor interaction, in meson physics, and are strength and range of the nucleon force, respectively. The CNCCRs with star product in Eqs. (3) and (4) become new CNCCRs without the notion of star product as follows (see, e.g., 57–60,65,66,71): μν lows: We insert Eqs. (43) into Eq (42), this allows it to be rewritten in the following form: The generalized positions and momentum coordinates μ μ μ and μ μ in the symmetries of DRKGT are defined in terms of the corresponding coordinates in the symmetries of RQM and spectively[60,65]: via, reμν
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