Journal of Physical Studies 25(4), Article 4301 [17 pages] (2021)
DOI: https://doi.org/10.30970/jps.25.4301

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

Abdelmadjid Maireche 

Laboratory of Physics and Material Chemistry, Physics department, Sciences Faculty, University of M'sila
BP 239 Chebilia-M'sila, Algeria
abdelmadjid.maireche@univ-msila.dz

Received 30 June 2021; in final form 14 July 2021; accepted 19 August 2021; published online 19 November 2021

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-Gordon and deformed Schrödinger equations using the modified equal vector scalar Manning-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 $Δ E_{\rm mcy}^{\rm tot}\left( n,δ ,η ,b,A,V_{0},V_{0}',Θ ,σ ,χ ,j,l,s,m\right) $ and the perturbative nonrelativistic corrections $Δ E_{\rm mcy}^{\rm nr}\left( n,δ ,η ,b,A,V_{0},V_{0}',Θ ,σ ,χ ,j,l,s,m\right) $ appeared as a function of the parameters $\left( δ ,η ,b,A,V_{0},V_{0}'\right) $, the parameters of noncommutativity $\left( Θ ,σ ,χ \right) $, in addition to the atomic quantum numbers $\left( n,j,l,s,m\right) $. 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 $l\neq 0$ can only be solved approximately. We have observed that the DKGE under the DVSMCY-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 relativistic noncommutative quantume mechanics.

Key words: Klein–Gordon equation, Schrödinger equation, Manning–Rosen potential, class of Yukawa potentials, the diatomic molecules, noncommutative geometry, Bopp's shift method and star products.

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