Effects of Two-Dimensional Noncommutative Theories on Bound States Schrödinger Diatomic Molecules under New Modified Kratzer-Type Interactions

. In this work, an analytical expression for the nonrelativistic energy spectrum of some diatomic molecules was obtained through the Bopp’s shift method in the noncommutative (NC) two-dimensional real space-phase symmetries (NC: 2D-RSP) with a new modified Kratzer-type potential (NMKP) in the framework of two infinitesimal parameters θ and θ due to (space-phase) noncommutativity, by means of the solution of the noncommutative Schrödinger equation. The perturbation property of the spin-orbital Hamiltonian operator and new Zeeman effect of two-dimensional system are investigated. We have shown that, the new energy of diatomic molecule is the sum of ordinary energy of modified Kratzer-type potential, in commutative space, and new additive terms due to the contribution of the additive part of the NMKP. We have shown also that, the group symmetry of (NC: 2D-RSP) reduce to new sub-group symmetry of NC two-dimensional real space (NC: 2D-RSP) under new modified Kratzer-type interactions.


Introduction
In some area of physics, non-relativistic quantum mechanics play important roles, finding the exact solutions of the Schrödinger equation (SE) by various methods for a class of central potentials in different fields of sciences, like nuclear, optics, etc. [1][2][3]. In particular, the modified Kratzer-type potential (MKP) have the general features of the true interaction energy, inter atomic and dynamical properties in solid-state physics and play an important role in the history of molecular structures molecular physics ( 2 N , CO , NO , CH ,… ) and interactions [4][5][6], in addition, this potential offered one of the most important exactly models of atomic and molecular physics and quantum chemistry. It may be apply to energy spectrum for the CO diatomic molecule with different quantum numbers and, on another hand, the MKP can be describe the interaction between two atoms and have attracted a great of interest for some decades in the history of quantum chemistry [7][8]. The noncommutativity of space-time, which known firstly by Heisenberg and was formalized by Snyder at 1947, suggest by the physical recent results in string theory [9]. Motivated by these, over the past few years, theoretical physicists have shown a great deal of interest in solving Schrödinger equation, Klein-Gordon and Dirac equation for various potentials in NC space-phase to obtaining profound interpretations at microscopic scale [10][11][12][13][14][15][16][17][18][19][20][21][22][23] and in particularly, our previously works . The notions of noncommutativity of space and phase based essentially on Seiberg-Witten map, the Bopp's shift method and the star product, which modified the ordinary product ( )( ) The two variables ( ) µ µ p x , satisfy the usual canonical commutation relations in quantum mechanics. The present paper consists of five sections, and the rest of this work is arranged as follows: In the second and the third sections, we have briefly review the SE with 2D MKP and we shall briefly give the fundamental concepts of the Bopp's shift method and then we derive the deformed potential ( )

Review the Spectrum of (MKP) in Ordinary Quantum Mechanics
It is necessary to review the ordinary energy eigenvalues for MKP in order to understand the parallels between this and noncommutative theories and to gives a guides us to our new energy eigenvalues, thus, the content of the present section is devoted to review the wave function  for two-dimensional SE satisfied the following equation [4]:

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where the MKP is given by: . Nevertheless, the above potential can be consider as a particular case from the general form of the following Mie-type potential ( ) r V mt [4][5][6]: where ( ) (11) Here µ to denotes the reduced mass of the diatomic molecules. It is important to notice that the modified Kratzer-type was treated in the case of three dimensions by Cuneyt Berkdemir et al. in the reference [5].

Theoretical overview of Bopp's shift method
In order to construct a 2D model of NMKP, the essential step is to rewrite the ordinary SE to the new following modified Schrödinger equation (MSE) which play a major role in (NC: 2D-RSP) symmetries [23][24][25][26][27][28][29][30][31][32][33][34]: where ( ) International Letters of Chemistry, Physics and Astronomy Vol. 76 3 In recently work, we are interest with the first variety in eq. (13). We may go a step further and consider the Bopp's method (modified by a shift), which allows us to reducing the above MSE to new ordinary form, in addition two fundamental translations of space and phase which are presenting in eq. (5): The new modified Hamiltonian that appears above is given by: Based on our references [34][35][36][37][38][39][40][41][42][43], we can write the two operators 2 r and 2 p in (NC: 2D-RSP) as follows: It is clear that, the first three terms in above equation represent the ordinary MKP while the rest terms are produced by the deformations of space-phase noncommutativity. Now simultaneously transforming ( ) r V kpˆ and (20) The above operator can be considering of the sum of ( )

Two-dimensional spin-orbital Hamiltonian operators for NMKP
In this sub-section we apply the same strategy, which we have seen in our previously works [33][34][35][36][37][38][39][40][41][42][43][44][45][46][47][48] Here S denote the spin of diatomic molecules and α is real constant, thus, the spin-orbital interactions appear automatically because of the new properties of space-phase. Now, it is possible to rewrite the above equation as follows: We have replaced the coupling S , which allow us to obtaining the eigenvalues ( )

The exact spin-orbital spectrum for NMKP in (NC: 2D-RSP) symmetries
In order to find the differences in the energy spectrum ( ) , , kp , we use perturbation theory up to first order in θ and θ and through the structure constants which specified the dimensionality of NMKP of diatomic molecular, which is sufficient to obtain differences in the energy, thus, we have the following results:

International Letters of Chemistry, Physics and Astronomy Vol. 76
Then, the nonrelativistic energy levels ( ) s l j n E r , , , kp at first order of two parameters θ and θ for diatomic molecular will expressed as a function of the previously factors as: , to achieve this goal; we apply the following special integral of hypergeometric function [49]:    . Therefore, substituting, equation (28) into equation (26) Thus, the global group symmetry (NC: 2D-RSP) reduce to new sub-group symmetry (NC: 2D-RS) for NMKP.

The exact magnetic spectrum for NMKP in (NC: 2D-RSP) symmetries
On other hand, it's possible to found another automatically symmetry for NMKP related to the influence of an external uniform magnetic field ℵ , if we perform the mapping we avoid repetition in calculations: Here χ and σ are two infinitesimal real proportional's constants and further insight can be gained when we choose the magnetic field k ℵ = ℵ , then we can make the following translation: However, very little has been achieved in the solution of MSE for studied potential NMKP.

Results and Discussion of Global Spectrum for NMKP in (NC: 2D-RSP) Symmetries
We have solved the modified radial Schrödinger equation and obtained the differences in the energy eigenvalues where ( ) is the new part of energy due to the noncommutativity of space-phase: It is clear the obtained result is proportional to the infinitesimal parameters (θ , χ ). It is also noteworthy that, the quantum number m can be takes (  (41)

Conclusion
In this conclusion, we briefly summarize what has been achieved in this reach work and comment on the outlook on future work that can follow from this paper: . v) If we consider the case when the two infinitesimal parameters' θ and θ become zero we recover all results of standard ordinary quantum mechanics for nonrelativistic MKP.
We hope to get some interesting applications to this new potential in the study of different fields of matter sciences, because our results are not only interesting for the pure theoretical physicists but also for experimental physicists (solid-state physics, the history of molecular structures molecular physics ( 2 N , CO , NO , CH ,… ) and interactions). Finally, we hope that our obtained results may serve as benchmarks for these potentials.