THE INFLUENCE OF DEFORMATION PHASE-SPACE ON SPECTRA OF HEAVY QUARKONIA IN IMPROVED ENERGY POTENTIAL AT FINITE TEMPERATURE MODEL OF SHRODINGER EQUATION VIA THE GENERALIZED BOPP’S SHIFT METHOD AND STANDARD PERTURBATION THEORY

In this work, we obtain solutions of the deformed Schrödinger equation (DSE) with improved internal energy potential at a finite temperature model in a 3-dimensional nonrelativistic noncommutative phase-space (3D-NRNCPS) symmetries framework, using the generalized Bopp’s shift method in the case of perturbed nonrelativistic quantum chromodynamics (pNRQCD). The modified bound state energy spectra are obtained for the heavy quarkonium system such as charmonium 𝑐𝑐 and bottomonium 𝑏𝑏 at ﬁnite temperature. It is found that the perturbative solutions of the discrete spectrum are sensible to the discreet atomic quantum numbers (𝑗, 𝑙, 𝑠, 𝑚) of the 𝑄𝑄 ( 𝑄 = 𝑐, 𝑏 ) state, the parameters of internal energy potential (𝑇, 𝛼 (cid:3046) (𝑇), 𝑚 (cid:3005) (𝑇), 𝛽, 𝑐) , which are the Debye screening mass 𝑚 (cid:3005) (𝑇) , the running coupling constant 𝛼 (cid:3046) (𝑇), the critical temperature 𝛽 , the free parameter 𝑐 in addition to noncommutativity parameters ( 𝛩 , 𝜃 ). The new Hamiltonian operator in 3D-NRNCPS symmetries is composed of the corresponding operator in commutative phase-space and three additive parts for spin-orbit interaction, the new magnetic interaction, and the rotational Fermi-term. The obtained energy eigenvalues are applied to obtain the mass spectra of heavy quarkonium systems ( 𝑐𝑐 and 𝑏𝑏 ). The total complete degeneracy of the new energy levels of the improved internal energy potential changed to become equal to the new value 3𝑛 (cid:2870) in 3D-NRNCPS symmetries instead of the value 𝑛 (cid:2870) in the symmetries of 3D-NRQM. Our non-relativistic results obtained from DSE will possibly be compared with the Dirac equation in high-energy physics.


INTRODUCTION
It is well known that the ordinary Schrödinger equation (SE) describes the dynamics of quantum systems at low energy without considering the temperature effect.Recently, the finite temperature SE allows us to study quantum systems such as superconductivity mechanisms and Bose-Einstein condensates at an arbitrary temperature, and when the temperature is equal to zero, it becomes identical to the SE [1].Very recently, many authors have studied the finitetemperature SE for hot quark-gluon plasma, heavy quarkonia in quark-gluon plasma, (electron and proton systems), and so on [2][3][4][5].The problem of calculating the energy spectra of the SE with various types of potentials such as the internal energy potential and the Cornell potential at finite temperature has been attracting interest in recent years [2][3][4][5][6][7][8].Abu-Shady has studied heavy-quarkonium mesons (HLM) using an internal energy potential and obtained wave function and energy spectra by solving SE using AEIM when the finite temperature is included [7].The main objective is to develop the research article [7] and expand it to the large symmetry known by nonrelativistic noncommutative phase-space (NRNCPS) to achieve a more accurate physical vision so that this study becomes valid in the field of nanotechnology.Noncommutative quantum mechanics is an old idea that has been extensively discussed in the literature.It should be noted that noncommutativity (NC) was first introduced by Heisenberg in 1930 [9] and then by Snyder in 1947 [10].It has appeared since the beginning of ordinary quantum mechanics.There has been a growing interest in this field since the discovery of string theory and the modified uncertainty principle.In addition, the NC idea is suggested as a result of the production of quantum gravity.It would provide a natural background for finding a suitable solution for a possible regularization of QFT [11][12][13][14][15][16][17][18][19][20][21][22][23].During the past three decades, the NC theory has been the focus of extensive investigation and has produced a very interesting new class of quantum field theories with intriguing and occasionally unexpected properties [24].Thus, the topographical properties of the NC space-space and phase-phase have a clear effect on the various physical properties of quantum systems and this has been a very interesting field in many fields of physics.The idea of noncommutativity has been studied in many articles, such as [24][25][26][27][28][29][30][31][32][33][34][35][36].On the other hand, we explore the possibility of creating new applications and more profound interpretations in the sub-atomics and nanoscales using a new version of the improved internal energy potential, which has the following form: The second and third terms in the above equation are the present effects of (space-space) and (phase-phase) noncommutativity properties.However, the new operators  () =  ∨ ̂ () in HP are depending on the corresponding new operators  =  ∨ ̂ in SP from the following projection relations: Here  =  ∨  and() =  ∨  ().The dynamics of the new systems ( ) dt are described by the following motion equations in 3D-NRNCPS symmetries: The two operators ( and  ) are present as the quantum Hamiltonian operators for the internal energy potential and the improved internal energy potential in the 3D-NRNCPS symmetries and their extension.This paper consists of five sections and the organization scheme is given as follows: In the next section, the theory part, we briefly review the SE with internal energy potential at finite temperature based on refs.[7][8].Section 3 is devoted to studying the DSE by applying the generalized Bopp's shift method and obtaining the improved internal energy potential and the modified spinorbit operator at finite temperature.Then, we applied the standard perturbation theory to find the quantum spectrum of the ground state, the first excited state, and the (, , ) ℎ excited state produced by the effects of modified spin-orbit and newly modified Zeeman interactions.In the fourth section, a discussion of the main results is presented in addition to determining the new formula for determining the mass spectra of the quarkonium system in the 3D-NRNCPS symmetries framework.Finally, in the last section, a summary and conclusions are presented.

THEORY 2.1. Overview of the eigenfunctions and energy eigenvalues for the internal energy potential at finite temperature in the 3D-NRNCPS symmetries framework
As already mentioned, our objective was to obtain the spectrum of the improved internal energy potential at finite temperatures.To achieve this goal, it is useful to summarize the time-independent Schrödinger equation for the internal energy potential at a finite temperature [7][8]: where  (, ) is determined from: The reduced mass  for the quarkonium particle for example (and) equal .The complete wave function  (, , ) = ( )  (, ) is given by [7]: Also, the energy  of the potential in Eq. ( 7) is determined from the following equation: where  is a normalizing constant,  is a natural number accounting for the radial excitation, while  is a non-negative integer number that represents the orbital angular momentum,

The exact modified spin-orbit operator for heavy-quarkonium systems with improved internal energy potential in the pNRQCD system:
In this subsection, we apply the same strategy that we have seen exclusively in some of our published scientific works [46,[55][56][57][58][59][60][73][74][75][76].Under such a particular choice, one can easily reproduce both couplings ( ), respectively.Thus, the perturbative Hamiltonian operator  (, ,  ) for the heavy quarkonium systems will be transformed into a modified spin-orbit operator  (, ,  ), under the improved internal energy potential at a finite temperature as follows: here  and  are equals  +  +  and  +  +  , respectively, and is a new constant, which plays the role of strong coupling in quantum chromodynamics or QCD theory, we have chosen two vectors ( → and  → ) parallel to the spin-s of the heavy quarkonium system.Furthermore, the above perturbative terms  () can be rewritten to the following new form: where  ≡ We will see them in detail in the next subsection.After profound calculation, one can show that the new radial function ()satisfies the following differential equation for the improved internal energy potential: Through our observation of the expression of  (, ,  ), which appears in equation ( 25), we see it as proportionate to the two infinitesimals parameters ( and ), thus, in what follows, we proceed to solve the modified radial part of the DSE that is, equation (30) by applying standard perturbation theory to find an acceptable solution at the first order of two parameters  and .The proposed solutions for DSE under improved internal energy potential include energy corrections, which are produced automatically from two principal physical phonemes', the first one is the effect of modified spin-orbit interaction and the second is the modified Zeeman effect while the stark effect can appear in the linear part of improved internal energy potential at finite temperature model.

The exact modified spin-orbit spectrum for a heavy-quarkonium system under improved internal energy potential in pNRQCD
The purpose here is to give a complete prescription for determining the energy levels of the ground state, the first excited state, and (, , ) the excited state, of heavy quarkonium systems.We first find the corrections ( ( (), , , ),  ( (), , , ) and  ( (), , , )) for heavy quarkonium systems such as (charmonium and bottomonium) mesons that have the quark and antiquark flavor under a new improved internal energy potential at finite temperature, which have three polarities up and down  =  + 1 (spin great),  =  (spin middle) and  =  − 1 (spin little), respectively, at the first order of two parameters ( and ).Moreover, by applying the perturbative theory, in the case of perturbed non-relativistic quantum chromodynamics pNRQCD framework, we obtained the following results: For the ground state  = 0, the expressions of the 6-factors  = 1,6 will be simplified to the following form: ) where = 2 +  ().It is convenient to apply the following special integral [82]: where  √ and ()denote the parabolic cylinder functions and the Gamma function.After straightforward calculations, we can obtain the explicit results: Evaluating the integral in Eq. ( 38) and applying the special integration, which is given by Eq. ( 33), we obtain the following results: with  = ∑  .

The exact modified magnetic spectrum for heavy quarkonium systems under improved internal energy potential in pNRQCD
In addition to the important results obtained previously, now we consider another important physically meaningful phenomenon produced by the effect of the improved internal energy potential at finite temperature on the perturbative NRQCD related to the influence of an external uniform magnetic field  → .To avoid repetition in the theoretical calculations, it is sufficient to apply the following replacements: (45) Eq. ( 45) can be rewritten as follows: The 6-factors   = 1,6 are given by Eq. ( 33).Then we find the magnetic specters of energy produced by the operator  (, , ) for the ground state and first excited states repeating the same calculations in the previous subsection.
Having completed the first and second-induced perturbed both spin-orbit interaction and self-magnetic phenomena, now, for our purposes, we are interested in finding a new third automatically important symmetry for improved internal energy potential at zero temperature in RNCQM symmetries.This physical phenomenon is induced automatically by the influence of the perturbative Hamiltonian operator  (, , ), which we can obtain from the initial perturbed Hamiltonian operator in Eq. ( 25).We discover these important physical phenomena when our studied system the quarkonium particle such as ( and) undergoing rotation with angular velocity  → if we make the following two transformations to ensure that previous calculations are not repeated: ) Here (,) are just two infinitesimal real proportional constants.We can express the perturbative Hamiltonian operator  (, , ) which induced the rotational movements of the quarkonium particle as follows: To simplify the calculations without compromising physical content, we choose the rotational velocity  → =  .
It is important to note that in Ref. [83], rotating isotropic and anisotropic harmonically confined ultra-cold Fermi gases were studied in two and three dimensions at absolute zero, but in that study, the rotational term had to be manually added to the Hamiltonian operator.In contrast, in our study, the rotation operator appears automatically because of the phase-space deformation caused by the improved internal energy potential models in the 3D-NRNCPS symmetries.It is crucial to note that perturbation theory cannot be utilized to find corrections of the second order ( and  ) because we have only employed corrections of the first order of infinitesimal noncommutative parameters ( and ).

MAIN RESULTS
In the previous subsections, we obtained the solution of the modified Schrödinger equation for new improved internal energy potential, which is given in Eq. ( 25) by using the generalized Bopp's shift method and standard perturbation theory in pNRQCD by the feature of 3D-NRNCPS symmetries.with spin-1 for heavy quarkonium systems  ( = , ) with improved internal energy potential at finite temperature are obtained in this paper based on our original results presented in Eqs. ( 37), ( 40), ( 41), (44), and (50) in addition to the ordinary energy  for the improved internal energy potential at a finite temperature which is presented in Eq. ( 11):  For the ground state: (, ,  = 0,  = 0, ) =  +  ( 5 3 .2 )  , , , ,  = −, + ,  =  +   ( () +  + ) + ( () +  + ) , ( 5 3 . 3 ) where  and  are the energy of the ground state and the first excited state of heavy quarkonium systems in the symmetries of quantum mechanics under internal energy potential at finite temperature: This is one of the main objectives of our research and by noting that the obtained eigenvalues of energy are real's and then the NC diagonal Hamiltonian   ,  is Hermitian.Furthermore, it's possible to write the three elements  ,  and  as follows: ,  →   ,  ≡   ,  ,  , (55) where In the symmetries of 3D-NRNCPS, the new kinetic term can be expressed as: The three modified interaction elements  ,  ,  are given by the following expressions: interactions modified to the new form  ,  ,  in 3D-NRNCPS symmetries.On the other hand, it is evident consider the quantum number  takes (2 + 1) values and we have also three values for( =  ± 1, ), thus every state in the usually three-dimensional space of energy for a heavy quarkonium system under improved internal energy potential will be 3(2 + 1)sub-states.To obtain the complete total degeneracy of the energy level of the improved internal energy potential in 3D-NRNCPS symmetries, we need to sum all allowed values .Total degeneracy is thus, Note that the obtained new energy eigenvalues  ,  ,  , , , ,  = −, + ,  now depend on new discrete atomic quantum numbers (, , , ) and in addition to the parameters of the internal energy potential.It is pertinent to note that when the atoms have spin-0, the total operator can be obtained from the interval | − | ≤  ≤ | + |, which allows us to obtain the eigenvalues of the operator  as (, , ) ≡ 0 and then the nonrelativistic energy spectrum  ,  ,  , , ,  = −, + ,  reads [68,69]: ,  ,  , , , ,  = −, + ,  =  +    + .
It is important to apply the present results ( 53) and ( 58) to quarkonium mesons.One of the most important applications in the extended model of pNRQCD is to calculate the modified mass spectra of the heavy quarkonium systems (the mass of the quarkonium bound state), such as charmonium and bottomonium mesons, which have the quark and antiquark flavor in the symmetries of NCQM under improved internal energy potential at finite temperature.To achieve this goal, we generalize the traditional formula [84][85][86][87][88][89][90][91],  = 2 +  which defines the total mass of the different quarkonium states (resonance masses), to the new form: Here  is the bare mass of quarkonium or twice the reduced mass of the system.Moreover,   , , , ,  = −, + ,  is the non-polarized energies, which can determine from Eqs. ( 49) and (54).Thus, at finite temperature, the modified mass of the quarkonium system  obtain: Hereis the heavy quarkonium system at a finite temperature with improved internal energy potential in commutative quantum mechanics, which is defined in ref. [7].If we consider(, , ) → (0,0,0), we recover the results of the commutative space of ref. [7] for the improved internal energy potential, which means that our calculations are correct.The novelty in this work is that the generalized Bopp shift method is successfully applied to find the solution of the 3-radial DSE at finite temperature in the symmetries of the 3D-NRNCPS framework.The automatic appearance of the spin in the term of improved energy as a quantum number clearly shows that the deformed Schrödinger equation under the influence of the improved energy potential model at finite temperature rises to the descriptor of the Dirac equation, meaning that this system can be valid in the field of high energies.

CONCLUSION
In the present work, the 3-dimensional deformed Schrodinger equation is analytically solved using the generalized Bopp's shift method and standard perturbation theory by the feature of 3D-NRNCPS symmetries.The improved internal energy potential at finite temperature is extended to include the effect of the non-commutativity space phase based on ref. [7] and the (, , ) ℎ excited state with (spin-1 and spin-0) for the heavy quarkonium system under the influence of the improved internal energy potential model at finite temperature are obtained. We have obtained, at finite temperature, the modified mass of the quarkonium system  which equals the sum of the corresponding values  in the 3D-NRNCPSsymmetries, and two perturbative terms proportional with two parameters ( and ).
 Since the main quantum number, spin, appears clearly and automatically in the expression of the global Hamiltonian and its eigenvalues, this is an indication that our results are valid in the field of high energies where the Dirac equation is applied.
Through high-value results, which we have achieved in the present work, we hope to extend our recent work for further investigations of particle physics and other characteristics of quarkonium at finite temperatures.
The modified eigenenergies   ; we resume the main results: The ordinary Hamiltonian operator at finite temperature   ,  in 3D-NRNCPSsymmetries was replaced by a new modified operator   ,  which equals →) to a heavy quarkonium system under the influence of the improved internal energy potential at the finite-temperature model.Wehave obtained the perturbative corrections (   , , , ,  = −, + ,  ,   , , , , ,  = 1,  = −, + ,  and   E E E , , , ,  = −, + ,  for the ground state, the first excited state,