SciELO - Scientific Electronic Library Online

 
vol.68 número2Quasi uniformly accelered motion of quasi normal magnetic biharmonic particles in Heisenberg space with cold plasmaAn inexpensive contact angle measurement system índice de autoresíndice de materiabúsqueda de artículos
Home Pagelista alfabética de revistas  

Servicios Personalizados

Revista

Articulo

Indicadores

Links relacionados

  • No hay artículos similaresSimilares en SciELO

Compartir


Revista mexicana de física

versión impresa ISSN 0035-001X

Rev. mex. fis. vol.68 no.2 México mar./abr. 2022  Epub 27-Mar-2023

https://doi.org/10.31349/revmexfis.68.020801 

RESEARCH

HIGH ENERGY PHYSICS

Diatomic molecules and fermionic particles with improved Hellmann-generalized Morse potential through the solutions of the deformed Klein-Gordon, Dirac and Schrödinger equations in extended relativistic quantum mechanics and extended nonrelativistic quantum mechanics symmetries

A. Mairechea 

aDepartment of Physics, M’sila University, Laboratory of Physics and Material Chemistry, M’sila University, BP 239, Algeria. ∗e-mail: abdelmadjid.maireche@univ-msila.dz


Abstract

In this paper, we investigate the new approximate bound state solution of deformed Klein-Gordon, Dirac and Schrödinger equations in the symmetries of extended relativistic quantum mechanics ERQM and extended nonrelativistic quantum mechanics ENRQM have been obtained with a newly proposed potential called improved Hellmann-generalized Morse potential (IHGMP, for short). To the best of our knowledge, this problem is examined in literature in the usual RQM and NRQM with Hellmann-generalized Morse potential. The potential is a superposition of Hellmann potential, generalized Morse or Deng-Fan potential, and some other exponential terms. By employing the improved approximation to deal with the centrifugal term, Bopp’s shift and standard perturbation theory method. The new approximate analytical energy shift and the corrections of bound state energy eigenvalues in ERQM and ENRQM are obtained for some selected diatomic molecules such as (HCl, LiH, H2, ScH, TiH, VH, CrH, CuLi, TiC, NiC, ScN and ScF). The new values that we get are sensitive to the quantum numbers (j,l,s,m), the potential depths of the improved Hellmann-generalized Morse potential (α,b), the range of the potential α, the dissociation energy D e , the equilibrium bond length r e , and noncommutativity parameters(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. In the new symmetries of NCQM, it is not possible to get the exact analytical solutions for l = 0 and l ≠ 0, the approximate solutions are available. Four special cases, i.e., l wave are investigated in the context of deformed Klein-Gordon and Schrödinger theories. The relativistic energy equations and the new nonrelativistic energy for some potentials such as improved Hellmann potential and improved generalized Morse potential have also been obtained by varying some potential parameters. We have clearly shown that the Schrödinger and Klein Gordon equations in the new symmetries can physically describe each of the two Dirac equations and the Duffin-Kemmer equation under the effect of IHGMP.

Keywords: Klein-Gordon equation; Schrödinger equation; Morse potential; Hellmann potential; the diatomic molecules; noncommutative geometry; star products; Bopp’s shift method

1. Introduction

Two scientific revolutions took place at the beginning of the last century; the first was embodied by the general and special theories of relativity and the second was the development of quantum mechanics. The Schrödinger equation was the first used to probe matter at the smallest scales [1], after which came the Klein-Gordon, Duffin-Kemmer, and Dirac equations. For the case of neutral or charged particles with spin zero, one can deal with the Klein-Gordon equation [2,3]. In contrast, for the case of a particle with spin-1, the Duffin-Kemmer equation [4] is necessary for mesons. As for the case in the case of spin-1/2 as electrons and their antagonists (positron), the Dirac equation [5] is the tool used to access the physical and chemical information of the system. All fundamental equations that we have referred to are normally solved using different potentials, depending on the nature of the problem being studied. Exponential potentials have been, and still are, the tool through which researchers have sought to study molecules. We will devote our current study to two types of potentials of great importance in this field, the Hellmann and generalized Morse potential. Many researchers have previously studied them in different energy levels, either separately for each of them or in combination, but it was in the framework of usual nonrelativistic quantum mechanics (NRQM) and relativistic quantum mechanics (RQM). Our study will be the focus of a case for combining them in the framework of a large quantum symmetry that is known by nonrelativistic noncommutative quantum mechanics (or extended nonrelativistic quantum mechanics, ENRQM) and relativistic noncommutative quantum mechanics (ERQM) symmetries (or extended relativistic quantum mechanics) using deformed Schrödinger, Klein-Gordon, and Dirac equations. It is well known that the Hellmann potential [6-8], is a superposition of the attractive Coulomb potential and a Yukawa potential [9]. Ikhdair et al. [10] have studied the bound state energies of the Hellmann potential by using the SUSY perturbation formalism. In 2007, Koncak et al. [11] studied the radial Schrödinger equation for the Hellmann potential within the framework of the asymptotic iteration method and obtained the bound state energy eigenvalues. This potential has many important applications; we mention one of the solidstate physics [12-14], alkali hydride molecules [15], innershell ionization problem [16], nuclear physics [17], among other applications. Furthermore, it was studied in both relativistic and non-relativistic quantum mechanics (see, e.g., [17-20]).

Deng-Fan potential originally appeared many decades ago. This potential was proposed by Deng and Fan [21] as a molecular potential, which improved the Morse potential [22], is known as the generalized Morse potential. Dong has used this potential as a suitable alternative potential to the Morse potential in the study of diatomic molecules to describe the vibrational spectrum and electromagnetic transitions [23,24]. Moreover, it can be used to study the diatomic molecular and obtain their energy spectra [25]. In 2008, Dong and Gu [26] obtained a bound state solution of the Schrödinger equation with the Deng-Fan molecular potential. Dong [27] trained the relativistic of spinless particles subject to a rotating Deng-Fan Oscillator. Oluwadare et al. [24] obtained the exact l-wave solutions of the Klein-Gordon and Dirac equations with equally mixed scalar and vector Deng-Fan molecular potentials, the normalized wave function, and the corresponding energy equations. Hassanabadi et al. analyzed the relativistic spinless particles under Deng-Fan potential [28]. By using the asymptotic iteration method, Ortakaya et al. [29] obtained the approximate analytical solutions of the Dirac equation with the Deng-Fan potential including a Coulomb tensor interaction in the presence of spin symmetry and pseudo-spin symmetry. In 2009, Zhang et al. [30] obtained the approximate analytical solutions of the Dirac equation with the generalized Morse potential model in the presence of spin symmetry and pseudo-spin symmetry by using the supersymmetric shape invariance formalism. Also, Daif [31] obtained l-state solutions of the Feynman propagator with the Deng-Fan molecular potential. Moreover, Maghsoodi et al. obtained. By employing the Pekeris-type approximation, Oyewumi et al. [32] obtained bound state solutions of the Deng-Fan molecular potential using the Nikiforov- Uvarov method for diatomic molecules (HCl, LiH, H2, ScH, TiH, VH, CrH, CuLi, TiC, NiC, ScN and ScF). Very recently, Ekwevugbe [33] obtained a nonrelativistic energy spectrum of the Deng-Fan Oscillator via the WKB approximation. Currently, the idea of combining more than two potentials has attracted interest. This combination expands the application scope to include new fields. And as a successful model for this combination, Okoia et al. and Ebomwonyi et al. studied the Hellmann generalized and Morse potentials in the case of the relativistic Klein-Gordon equation, Dirac equation, and nonrelativistic Schrödinger equation; this combination can be applied in different branches of physics, including molecular and atomic physics [34,35].

In recent work, we combine Hellmann-generalized Morse potentials and explore the corresponding deformed Klein-Gordon, Dirac, and Schrödinger equations in the symmetries of ERQM and ENRQM. The idea of non-commutative quantum mechanics is old and dates to the early years of ordinary quantum mechanics, originally from Snyder [36] and later developed. The non-commutative quantum theory contributed positively to overcoming many problems that ordinary quantum mechanics could not solve. Quantum gravity, string theory, and the divergence problem of the standard model new data made this new quantum theory a refuge for physicists to find solutions and discoveries hoped for on the other hand [38-48]. In recent years, a lot attention has been drawn to this development [49-59].

Concerning the combination of Hellmann and generalized Morse potentials, the subject of the current study, I have previously dealt with the non-specific study of each of them separately, but I have not dealt with them in a combined way, and neither has any other researcher done so far. We have treated the generalized Hellmann potential in the symmetries of NERQM [60]. Moreover, we have applied the Hellmann potential on the Mirror Nuclei 17O and 17F in the symmetries of NERQM [61]. Moreover, we have studied the deformed Schrödinger equation with the generalized Hellmann-Kratzer potential model in the symmetries of ENRQM [62]. Very recently, we have investigated the bound-state solutions of the deformed Klein-Gordon and Schrödinger equations for arbitrary l-states with the modified Morse potential in the symmetries of noncommutative quantum mechanics [54]. From what we have seen so far that most of the studies concerning improved Hellmann-generalized Morse potential were within the framework of ordinary quantum mechanics. The above works motivated us to investigate the approximate solutions of the 3-dimensional deformed Klein-Gordon equation, Dirac and Schrödinger equations for¨ improved Hellmann-generalized Morse potential offered in Refs. [34, 35] in RQM and NRQM. The potential focus of study and interest can be applied for some selected diatomic molecules such as (HCl, LiH, H2, ScH, TiH, VH, CrH, CuLi, TiC, NiC, ScN and ScF) in ERQM and ENRQM symmetries. The research reported in the present article was motivated by the fact that the study of the MHGPs in the ERQM and ENRQM symmetries has not been reported in the available literature. Here, our focus was on the MHGPs, which has the following form in the new symmetry:

V,Shmpr=De,Se[1+-a+be-αrrDe,Se-2eαre-1eαr-1+eαre-1eαr-12], (1.1)

V,Shmprnc=V,Shmpr-V,ShmprrLΘ2r+OΘ2, (1.2)

where (D e ,S e ) are the dissociations energies, r e is the equilibrium bond length, (α and b) are the potential strengths, α is the screening parameter, r nc and r is the distance between the two particles in EQM and QM symmetries, respectively. The coupling equals LΘLΘ equals L x , Θ12 + L y Θ23 + L z Θ13 with L x , L y and L z are present the usual components of the angular momentum operator L while the new noncommutativity parameter Θ ij equals θ ij /2. The new algebraic structure of noncommutative covariant canonical commutations relations NCNCCRs in the three representations of Schrödinger, Heisenberg, and interactions pictures, as follows (It should be noted that, in our calculation, we have used the natural units ħ = c = 1) [63-70]:

xμS,H,I,pνS,H,I=iδμνxncμS,H,I,*pncνS,H,I=iδμν, (2.1)

xμS,H,I,xνS,H,I=0xncμS,H,I,*xncνS,H,I=iθμν. (2.2)

While the uncertainty relations will be changed into the following formula in the new symmetries as follows:

ΔxμS,H,IΔpνS,H,Iδμν2    ΔxμS,H,IΔpνS,H,Ieffδμν2ΔxμS,H,IΔpνS,H,Iθμν2. (3)

With xncμS,H,I=xncμS,xncμH,xncμI are the generalized coordinates in NCQM symmetries and pncμS,H,I=pncμS,pncμH,pncμI are the corresponding generalizing coordinates in the usual QM symmetries. It is important to note that Eq. (2.2) is a covariant equation (the same behavior of xμS,H,I) under Lorentz transformation, which includes boosts and/or rotations of the observer’s inertial frame. We are generalizing the NCNCCRs to include Heisenberg and interaction pictures. Here eff is the effective Planck constant, θ µν = є µν θ (θ is the non-commutative parameter) which is an infinitesimal parameter if compared to the energy values and elements of antisymmetric 3 × 3 real matrices and δ µν is the identity matrix. The symbol ∗ denotes the Weyl Moyal star product, which is generalized between two ordinary functions f(x)h(x) to the new deformed form f^(xnc)h^(xnc) which expressed with the Weyl Moyal star product f(x) ∗ h(x) in the symmetries of NCQM as follows θ µν = є µν θ (θ is the non-commutative parameter) which is an infinitesimal parameter if compared to the energy values and elements of antisymmetric 3 × 3 real matrices and δ µν is the identity matrix. The symbol ∗ denotes the Weyl Moyal star product, which is generalized between two ordinary functions A(x)B(x) to the new deformed form A^(xnc)B^(xnc) which expressed with the Weyl Moyal star product A(x) ∗ B(x) in the symmetries of NCQM as follows [37-43]:

A*Bx=expiϵμνθμxμxABxfhx

-iϵμνθ2μxAμxBxμ=xν+Oθ2. (4)

The indices (µ,ν = 1,2,3) and O θ 2 stand for the second and higher-order terms of the NC parameter. Physically, the second term in Eq. (4) presents the effects of spacespace noncommutativity. Furthermore, it is possible to unify the operators X^μHt=xncμHpncμHt and X^μIt=xncμIpncμIt in the Heisenberg and the interaction pictures using the following projection relations, respectively:

XμHt=exp(iH^rnchmpT)XμSexp(-iH^rnchmpT)QM-symmetry (5.1)

X^μHt=exp(iH^rnchmpT)X^μSexp(-iH^rnchmpT)NCQM-symmetry, (5.2)

And

XμIt=exp(iH^orhmpT)XμSexp(-iH^orhmpT)QM-symmetry (6.1)

X^μIt=exp(iH^nc-orhmpT)X^μSexp(-iH^nc-orhmpT)NCQM-symmetry. (6.2)

Moreover, the dynamics of new systems dX^μIt/dt can be described by the following motion equations in the deformed Heisenberg picture as follows:

dXμHtdt=XμHt,H^hmp+XμHttQM-symmetry (7.1)

dX^μItdt=X^μIt,*H^nchmp+X^μIttExtendedQM-symmetry. (7.2)

Here (H^orhmp and H^nc-orhmp) are the free and total Hamiltonian operators for equal vector scalar of the Hellmann-generalized Morse potential in QM and EQM symmetries, while (H^hmp and H^nchmp) the Hamiltonians in QM and EQM symmetries. The purpose of this paper is to investigate the l-state solution of the deformed Klein-Gordon and Schrödinger equations within¨ Bopp’s shift and standard perturbation theory methods to generate an accurate new energy spectrum in ERQM and ENRQM symmetries. Our current work is structured in eight sections. The first one includes the scope and purpose of our investigation, while the remaining parts of the paper are structured as follows. A review of the Klein-Gordon, Dirac and Schrödinger equations with Hellmann-generalized Morse potential is presented in Sec. 2. Section 3 is devoted to studying the deformed Klein-Gordon equation by applying the ordinary Bopp’s shift method and the Greene and Aldrich approximation for the centrifugal term to obtain the effective potential of the improved Hellmann-generalized Morse potential in RNCQM symmetries. Besides, via perturbation theory, we find the expectation values of some radial terms to calculate the energy shift produced by the effect of the perturbed effective potential of the improved Hellmann-generalized Morse potential. Section 4 is devoted to present the global energy shift and the global energy spectra produced by improved Hellmann-generalized Morse potential in the deformed Klein-Gordon symmetries. In Sec. 5, we examine some particular relativistic important cases in the context of the deformed Klein-Gordon theory. In the next section, we derive the global energy shift and the global energy spectra produced with improved Hellmann-generalized Morse potential in the deformed Dirac symmetries. In Sec. 7, we apply our study for determining the energy spectra of some selected diatomic molecules such as (HCl, LiH, H2, ScH, TiH, VH, CrH, CuLi, TiC, NiC, ScN and ScF) in the ENRQM under improved Hellmann-generalized Morse potential, also, to study the composite systems. In Sec. 8, our conclusive remarks and future directions are given.

2. Revised of Klein-Gordon, Dirac and Schrödinger equations under Hellmann-generalized Morse potential

Before we start constructing the new solutions of the deformed Klein-Gordon, Dirac, and Schrödinger equations under the¨ improved Hellmann-generalized Morse potential MGHPs, we give a summary of the corresponding usual solutions in ordinary relativistic quantum mechanics and nonrelativistic quantum mechanics. The Hellmann-generalized Morse potential in the symmetries of RQM and NRQM is given by [34,35]:

Vhmpr=De1+-a+be-αrrDe-2eαre-1eαr-1+eαre-1eαr-12, (8)

and

Shmpr=Se1+-a+be-αrrSe-2eαre-1eαr-1+eαre-1eαr-12. (9)

The 3-dimensional Klein-Gordon equation, Dirac equation with a scalar potential S hmp (r) and a vector potential V hmp (r), and the Schrödinger equation with the vector potential V hmp (r) for the diatomic molecule (or fermionic particles) with reduced mass M and wave function Ψ nk (r,Ω) are given as:

-Δ+M+Shmpr2-Enl-Vhmpr2Ψr,Ω=0, (10.1)

αp+βM+ShmprΨnkr,Ω=Enk-VhmprΨnkr,Ω, (10.2)

-Δ2M+VhmprΨr,Ω=EnlnrΨr,Ω. (10.3)

Here ∆ is the ordinary 3-dimensional Laplacian operator, the vector potential V hmp (r) due to the four-vector linear momentum operator A µ (V hmp (r), A = 0) and space-time scalar potential S hmp (r) due to the mass, Enl,Enk and Enlnr are represents the relativistic and nonrelativistic energy eigenvalues, (n,l) are the principal and orbital quantum numbers, respectively.

αi=0σiσi0,β=I2×200I2×2,

and σ i are the usual Pauli matrices. Since the Hellmann-generalized Morse potential has spherical symmetry, allowing the solutions of the time-independent Klein-Gordon equation and Schrödinger equation of the known form Ψr,Ω=(ψnlr/r)YlmΩ to separate the radial ψ nl (r) and angular parts YlmΩ of the wave function Ψ(r,Ω). For the Dirac equation,

Ψnkr,Ω=1rFnkrYjmlΩiGnk~rYjml~Ω,

where F nk (r) and G nk (r) represent the upper and lower components of the Dirac spinors Ψ nk (r,Ω) while YjmlΩ and Yjml~Ω are the spin and pseudospin spherical harmonics and m is the projection on the z-axis. Thus, Eqs. (10.1), (10.2) and (10.3) can be expressed as:

d2dr2-M2-Enl2-2EnlVhmp{r}+MShmp{r}+Vhmp2r-Shmp2r-l[l+1]r2ψnlr=0, (11.1)

d2dr2-kk+1r2-M+Enk-Δ{r}M-Enk+Σ{r}+dΔrdrddr+kr-ψrM+Enk-ΔrFnk(r)=0, (11.2)

d2dr2-kk-1r2-M+Enk-Δ{r}M-Enk+Σ{r}+dΣrdrddr-kr+ψ{r}M+Enk-ΣrGnk~r=0, (11.3)

d2dr2+2MEnlnr-Vhmp{r}-l{l+1}2Mr2ψnlr=0. (11.4)

Using the shorthand notation Eeffhmp=M2-Enl2 and:

Veffhmpr=2EnlVhmpr+MShmpr-Vhmp2r+Shmp2r+l(l+1)r2, (12.1)

Σr=De1+-a+be-αrrDe-2eαre-1eαr-1+eαre-1eαr-12and dΔrdr=0Δr=CsFor Spin Symmetry LimitΔr=De1+-a+be-αrrDe-2eαre-1eαr-1+eαre-1eαr-12anddΣrdr=0Σr=CpsFor Pseudospin Symmetry Limit (12.2)

Veff-nrhmpr=Vhmpr+l(l+1)r2. (12.3)

We obtain the following second-order Schrödinger-like equation in RQM and NRQM symmetries, respectively:

d2dr2-Eeffhmp+Veffhmp(r)ψnl(r)=0, (13.1)

d2dr2-kk+1r2-M+Enks-CsM-Enks-ar+be-αrr+De{1+qe-αr-1}2Fnkr=0, (13.2)

d2dr2-kk-1r2-M-Enkps+ΣcM+Enkps+ar-be-αrr-De{1-qe-αr-1}2Gnk~r=0, (13.3)

d2dr2+2MEnlnr-Veff-nrhmp{r}ψnlr=0. (13.4)

With kk-1=l~(l~-1) and k (k + 1) = l(l + 1). When the vector potential is equal to the scalar potential V mp (r) = S mp (r) the effective potential leads to the following simple form

Veffhmpr=2Enl+MVhmpr+l(l+1)r2. (14)

The authors of Refs. [34, 35] using both Nikiforov-Uvarov method and the Greene and Aldrich approximation for the centrifugal term to obtain the expressions for the wave function as hypergeometric polynomials and the corresponding energy values for improved Hellmann-generalized Morse potential, in RQM and NRQM symmetries as,

Ψr,Ω=n!Γn+Anl+1NnlΓ2Anl+1sAnl/21-sVnl+122F1-n,n+Vnl+Anl+1;1+Anl;sYlmΩ, (15.1)

Fnkr=Nnk2ωnk+1nn!sωnk1-sλnk+122F1-n,n+2ωnk+2λnk+1;1+ωnk;sYlmΩ, (15.2)

Gnk~r=Nnk2Ωnk+1nn!sΩnk1-sβnl+122F1-n,n+2Ωnk+2βnl+1;1+Ωnk;sYlmΩ, (15.3)

and

Enl2-M2=De-αaEnl+M+α2ll+1-14αn+12+δnl-Enl+Ma-b+2Deαeαre-1n+12+ll+1

-Enl+MDeαeαre-12+αll+1n+12+ll+12, (16.1)

Enlnr=De-αa+α2ll+12M-α28MΛn,l-2MηlΛn,l2. (16.2)

For the spin symmetry, the equation of energy is given by:

M+Enksp-CsM-Enksp+De=M+Enksp-Csαa-α2kk+1+α24Λs2n,k+M-Enksp+DeηskΛsn,k2. (16.3)

For the pseudospin symmetry, the equation of energy is given by:

De-M-EnkpsM-Enkps-Cps=M-Enkps-Cpsαa-α2kk-1

+α24Λs2n,k+M-Enkps+DeηskΛsn,k2, (16.4)

with

δnl=14+Deα2Enl+Meαre-12+αll+1, (17.1)

Anl=ϵnl-aα2Enl+M+4ll+1, (17.2)

Vnl=ϵnl+Deq2α2Enl+M+4ll+1, (17.3)

ωnk=M+Enksp-CsM-Enksp+Deα2-Enksp+Maα+kk+1, (17.4)

Ωnk=M+Enkps-DeM-Enkps+Cpsα2-Enkps+M+Cpsaα+kk-1, (17.5)

βnk=14+Deα2M-Enkps+CpsDeq2+ll+1, (17.6)

ηl=bα-2Deq2α2-αα+2Deq2α2+ll+1, (17.7)

Λnl=n+12+14+Deq2α2+αll+1, (17.8)

Λsn,k=n+12+14+Deα2M+Enksp-Cs+kk+1, (17.9)

Λpsn,k=n+12+14+Deα2M+Enkps-Cps+kk-1, (17.10)

and

ηsk=bα-2Deq2α2-αα2+2Deq2α2+kk+1, (18.1)

ηpk=bα-2Deq2α2-αα2+2Deq2α2+kk-1. (18.2)

The Pochhammer symbol is (m) n = Γ[m + n]/Γ[m + n], and s = e −αr , Nnlm2 and N nkl are the normalization constants, (see Refs. [34,35]).

3. The new solutions of DKGE under IHGMP in the EQM

3.1. Review of Bopp’s shift method

Let us begin in this subsection by finding the deformed Klein Gordon equation (DKGE) in the symmetries of relativistic noncommutative quantum mechanics or the extended quantum mechanics under the improved Hellmann-generalized Morse potential (IHGMP). Our goal is achieved by applying the new principles which we have seen in the introduction, Eqs. (2.1), (2.2) and (3), summarized in new relations and the notion of the Weyl-Moyal star product. These data allow us to rewrite the usual radial Klein-Gordon equation in Eq. (13.1) in the ERQM symmetries as follows [54-57,76-85]:

d2dr2-Eeffhmp+Veffhmpr*ψnlr=0. (19)

It is established extensively in the literature that star products can be simplified by Bopp’s shift method. The physicist Fritz Bopp was the first to consider pseudo-differential operators obtained from a symbol by the quantization rules xx − (i/2)(∂/∂p), and pp+(i/2)(∂/∂x) instead of the ordinary correspondence xx and p → (i/2)(∂/∂x). This is known by Bopp’s shifts and this quantization procedure is called Bopp quantization [85-87]. It is known to the specialists that Bopp’s shift method has been applied effectively and has succeeded in simplifying the three basic equations: the deformed Klein-Gordon equation [54-57,76-85], deformed Dirac equation [88-91], deformed Schrödinger equation [92-95] and Duffin-Kemmer-Petiau equation [81,82] with the notion of star product to the Klein-Gordon equation, the Dirac equation and the Schrödinger equation with the notion of ordinary product. Thus, Bopp’s shift method is based on reducing second order linear differential equations of the deformed Klein-Gordon equation, the deformed Dirac equation, and the deformed Schrödinger equation with star product¨ to second-order linear differential equations of Klein-Gordon equation, Dirac equation, and Schrödinger equation without star¨ product with simultaneous translation in the space-space. The CNCCRs with star product in Eqs. (2.1) and (2.2) become new CNCCRs without the notion of star product as follows (see, e.g., [54-57,76-85]):

xncμS,H,I,pncνS,H,I=xncμS,H,IpncνS,H,I-pncνS,H,IxncμS,H,I=ieffδμν, (20.1)

xncμS,H,I,xncνS,H,I=xncμS,H,IxncνS,H,I-xncνS,H,IxncμS,H,I=iθμν. (20.2)

The generalized positions and momentum coordinates xncμS,H,I=xncμS,xncμH,xncμI and pncμS,H,I=pncμS,pncμH,pncμI, in the symmetries of ERQM are defined in terms of the corresponding coordinates in the symmetries of RQM xμS,H,I=xμS,xμH,xμI and pμS,H,I=pμS,pμH,pμI via, respectively [54-57,76-85]:

xncμS,H,I=xμS,H,I-3ν=1iθμν2pνS,H,Iand  pncμS,H,I=pμS,H,I. (21)

This allows us to find the operator rnc2 equal r 2LΘ in NCQM symmetries [54-57,76-85]. New effective potential for MHGPs in ERQM symmetries

3.2. The new effective potential of the improved Hellmann-generalized Morse potential model in DRKGT symmetries

According to the Bopp shift method, Eq. (19) with star product becomes similar to the following like the Schrödinger equation (without the notions of star product):

d2dr2--M2-Enl2-ll+1rnc2-VhmprncEnl+Mψnlr=0. (22)

The new operators Vhmprnc and (ll+1/rnc2) are expressed as in ERQM symmetries as follows:

Vhmprnc=De-ar+be-αrr-2Deqe-αr1-e-αr+Deq2e-2αr1-e-αr2-VhmprrLΘ2r+OΘ2, (23)

and

ll+1rnc2=ll+1r2+ll+1r4LΘ+OΘ2. (24)

Here q = e αre − 1. Therefore, we can rewrite:

2VhmprncEnl+M=2VhmprEnl+M-Enl+MrVhmprrLΘ+OΘ2. (25)

Moreover, to illustrate the above equation in a simple mathematical way and attractive form, it is useful to enter the following symbol Vnc-effhmpr, thus the radial Eq. (22) becomes:

d2dr2-Eeffhmp+Vnc-effhmprψnlr=0, (26)

with:

Vnc-effhmpr=Veffhmpr+Vperthmpr. (27)

Moreover, Vperthmpr is given by the following relation:

Vperthmpr=ll+1r4-Enl+MrVhmprrLΘ+OΘ2. (28)

It should be noted that when l = 0 the Eq. (13.1) can be exactly solved analytically for the case l ≠ 0, the authors of Refs. [34,34] approximatively solved the equation using the Greene and Aldrich approximation in relativistic and nonrelativistic quantum mechanics symmetries. In the new form of radial like-Schrödinger equation written in Eq. (26), we have observed new terms including ( (1/r), (1/r 4) and other Columbia-like terms) which make this equation impossible to solve analytically for and, it can only be solved approximately. From this point of view, we can consider the Greene and Aldrich approximation [96]. It is also used in many other works [97-99]:

1r2α21-e-2αr2=α21-s2. (29)

It is important to mention here that the above approximations are valid in short when αr ≪ 1. This allows us to obtain:

1rα1-e-2αr=α1-s. (30)

After straightforward calculations (∂V hmp (r)/∂r) we obtain as follows:

Vhmprr=ar2-bαe-αrr-bαe-αrr2-2Deαqe-αr1-e-αr+Deq1-qe-2αr1-e-αr2-2Deαq2e-3αr1-e-αr3. (31)

Upon invoking the Greene-Aldrich approximation scheme, the expression (E nl + M/r)(∂V hmp (r)/∂r) reduces to the form:

Enl+MrVhmprr=αEnl+Ma1-s3-bαs1-s3+2Deq-bαs1-s2+2Deαq1-qs21-s3-2Deαq2s31-s4. (32)

By making the substitution Eq. (32) into Eq. (28), we find the perturbed effective potential Vperthmps generated from noncommutativity properties of space-space that produced by the effect of IHGMP in the symmetries of ERQM as follows:

Vperthmps=ll+1α41-s4-αEnl+Ma1-s3-bαs1-s3

+2Deq-bαs1-s2+2Deαq1-qs21-s3-2Deαq2s31-s4LΘ+OΘ2. (33)

We have replaced the term (l[l + 1]/r 4) with the Greene and Aldrich approximation in Eq. (28). The equal scalar and vector improved Hellmann-generalized Morse potentials are extended by including new terms proportional to the radial terms (1/[1 − s]4), (1/[1 − s]3), (s/[1 − s]3, (s/[1 − s]2) , (s 2 /[1 − s]3) and (s 3 /[1 − s]4) become the improved Hellmann-generalized Morse potential in ERQM symmetries. The generated new effective potential Vperthmps is also proportional to the infinitesimal vector Θ. This allows us to consider the new additive part of the effective potential Vperthmps as a perturbation potential compared with the main potential Veffhmps (the parent potential operator in the symmetries of ERQM, that is, the inequality has become achieved VperthmpsVeffhmps. That is all physical justifications for applying the time-independent perturbation theory become satisfied. This allows us to give a complete prescription for determining the energy level of the generalized n th excited states.

3.3. The expectation values under IHGMP in the deformed Klein-Gordon symmetries

In this subsection, we want to apply the perturbative theory, in the case of ERQM symmetries, we find the expectation values of the radial terms (1/[1 − s]4), (1/[1 − s]3), (s/[1 − s]3, (s/[1 − s]2) , (s 2 /[1 − s]3) and (s 3 /[1 − s]4) taking into account the wave function which we have seen previously in Eq. (15.1). Thus after straightforward calculations, we obtain the following results:

11-s4nlm=Nnlm20+s21-s4sAnl1-s2Vnl+1 2F1-n,n+Vnl+Anl+1;1+Anl;s2dr, (34.1)

11-s3nlm=Nnlm20+11-s3sAnl1-s2Vnl+1 2F1-n,n+Vnl+Anl+1;1+Anl;s2dr, (34.2)

s1-s3nlm=Nnlm20+s1-s3sAnl1-s2Vnl+1 2F1-n,n+Vnl+Anl+1;1+Anl;s2dr, (34.3)

s1-s2nlm=Nnlm20+s1-s2sAnl1-s2Vnl+1 2F1-n,n+Vnl+Anl+1;1+Anl;s2dr, (34.4)

s21-s3=Nnlm20+s21-s3sAnl1-s2Vnl+1 2F1-n,n+Vnl+Anl+1;1+Anl;s2dr, (34.5)

s31-s4=Nnlm20+s31-s4sAnl1-s2Vnl+1 2F1-n,n+Vnl+Anl+1;1+Anl;s2dr, (34.6)

with Nnlm2=n!Γn+Anl+1Nnl/Γ2Anl+12 and we have used useful abbreviations 〈X (nlm) = 〈n,l,m X n,l,m〉 to avoid the extra burden of writing equations. Furthermore, we have applied the property of the spherical harmonics, which has the form YlmΩ'Yl'm'Ωd2Ω=δll'δmm'. We have s = e −αr , this allows us to obtain dr = −(1)(ds/s). From the asymptotic behavior of s = e −αr when (r → 0) (s → +1)and when (r → +∞) (s → 0), this allows to reformulate Eqs. (34,i=1,6¯) as follows:

11-s4nlm=Nnlm2α0+1sAnl-11-s2Vnl-3 2F1-n,n+Anl+Vnl+1;1+Anl,s2ds, (35.1)

11-s3nlm=Nnlm2α0+1sAnl-11-s2Vnl-2 2F1-n,n+Anl+Vnl+1;1+Anl,s2ds, (35.2)

s1-s3nlm=Nnlm2α0+1sAnl1-s2Vnl-2 2F1-n,n+Anl+Vnl+1;1+Anl,s2ds, (35.3)

s1-s2nlm=Nnlm2α0+1sAnl1-s2Vnl-1 2F1-n,n+Anl+Vnl+1;1+Anl,s2ds, (35.4)

s21-s3=Nnlm2α0+1sAnl+11-s2Vnl-2 2F1-n,n+Anl+Vnl+1;1+Anl,s2ds, (35.5)

s31-s4=Nnlm2α0+1sAnl+21-s2Vnl-3 2F1-n,n+Anl+Vnl+1;1+Anl,s2ds. (35.6)

We can use the method proposed by Dong et al. [100] and applied by Zhang [101], we calculate the integrals in Eqs. (35,i=1,6¯). With the help of the special integral formula

0+1sξ-11-sσ-1 2F1c1,c2;c3;s2ds=ΓξΓσΓξ+σ 3F2c1,c2,σ;c3,σ+ξ;1, (36.1)

here 2 F 1 (c 1 ,c 2;c 3;s) is the generalized hypergeometric function:

 3F2c1,c2,σ;c3,σ+ξ;1=+n=0c1nc2nσnc3nσ+ξn!. (36.2)

Is obtained from the generalized hypergeometric function which has parameters p of type 1 and q parameters of type 2 of the form the following formula:

 pFqα1,α2,...,αp;β1,β2,...,β1;1=+n=0α1n...αpnβ1n...βpnn!, (36.3)

for p = 3 and 1 = 2 while Γ(σ) denoting the usual Gamma function. We obtain from Eqs. (35,i=1,6¯) the following results:

11-s4nlm=NnlvirΓAnlΓ2Vnl-2ΓKnl-2 3F2-n,n+Anl+Vnl+1,2Vnl-2;1+Anl,Knl-2;1, (37.1)

11-s3nlm=NnlvirΓAnlΓ2Vnl-1ΓKnl-1 3F2-n,n+Anl+Vnl+1,2Vnl-1;1+Anl,Knl-1;1, (37.2)

s1-s3nlm=NnlvirΓAnl+1Γ2Vnl-1ΓKnl 3F2-n,n+Anl+Vnl+1,2Vnl-1;1+Anl,Knl;1, (37.3)

s1-s2nlm=NnlvirΓAnl+1Γ2VnlΓKnl+1 3F2-n,n+Anl+Vnl+1,2Vnl-1;1+Anl,Knl+1;1, (37.4)

s21-s-3nlm=NnlvirΓAnl+2Γ2Vnl-1ΓKnl+1 3F2-n,n+Anl+Vnl+1,2Vnl-1;1+Anl,Knl+1;1, (37.5)

s31-s4nlm=NnlvirΓAnl+3Γ2Vnl-2ΓKnl+1 3F2-n,n+Anl+Vnl+1,2Vnl-1;1+Anl,Knl+1;1, (37.6)

with Nnlvir=n!Γn+Anl+1Nnl/Γ2Anl+12(1/α) and Knl=Anl+2Vnl.

3.4. The energy shift for the MHGPs in ERQM symmetries

The global relativistic energy shift for the improved Hellmann-generalized Morse potential model in ERQM symmetries is composed of three principal parts. The first one is produced from the effect of the generated spin-orbit effective potential. This effective potential is obtained by replacing the coupling of the angular momentum operator and the noncommutative vector LΘ with the new equivalent coupling ΘLS (with Θ2=Θ122+Θ232+Θ132). This degree of freedom comes considering that the infinitesimal noncommutative vector Θ is arbitrary. We have chosen it to a parallel of the spin S of the diatomic molecules under Hellmann-generalized Morse potential. Furthermore, we replace the new spin-orbit coupling ΘLS with the corresponding physical form (Θ/2)G 2, with G 2 = J 2L 2S 2. Moreover, in quantum mechanics, the operators (H^rnchmp,J2, L2,S2 and Jz) forms a complete set of conserved physics quantities, the eigenvalues of the operator G 2 are equal to the values τ (j,l,s) = [j(j + 1) − l(l + 1) − s(s + 1)]/2, with |ls| ≤ j ≤ |l + s|. As a direct consequence, the partial energy shift ΔEhmpson,α,a,b,De,re,Θ,j,l,s due to the perturbed effective potential Vperthmps produced for the n th excited state, in DRKGT symmetries as follows:

ΔEhmpson,α,a,b,De,re,Θ,j,l,s=Θj(j+1)-l(l+1)-s(s+1)ZnlmRHMPn,n,α,a,b,De,re. (38)

The global expectation value ZnlmRHMPn,n,α,a,b,De,re is determined from the following expression:

ZnlmRHMPn,n,α,a,b,De,re=ll+1α411-s4nlm-Enl+Mαa11-s3nlm

-bs1-s3nlm+2Deq-bs321-s4nlm+2αDeq1-qs21-s3nlm

-2αDeq2s31-s4nlm. (39)

The second part is obtained from the magnetic effect of the perturbative effective potential Vperthmps under the improved Hellmann-generalized Morse potential model. This effective potential is achieved when we replace both (LΘ and Θ12) by (σL z and σℵ), respectively, here (ℵ and σ) are symbolize the intensity of the magnetic field induced by the effect of the deformation of space-space geometry and a new infinitesimal noncommutativity parameter, so that the physical unit of the original noncommutativity parameter Θ12 (length)2 is the same unit of σℵ, we have also need to apply 〈n,l’,m’ L z n,l,m〉 = m’m δ l’l δ n’n ( −l’m’l and −lml). All of this data allows for the discovery of the new energy shift ΔEhmpmagn,α,a,b,De,re,σ,m due to the perturbed Zeeman effect which created by the influence of the improved Hellmanngeneralized Morse potential model for the n th excited state in ERQM symmetries as follows:

ΔEhmpmagn,n,α,a,b,De,re,σ,j,l,s=σZnlmRHMPn,n,α,a,b,De,rem. (40)

Now, for our purposes, we are interested in finding a new third automatically important symmetry for the improved Hellmann-generalized Morse potential model at zero temperature in ERQM symmetries. This physical phenomenon is induced automatically from the influence of a perturbed effective potential Vperthmps which we have seen in Eq. (33). We discover these important physical phenomena when our studied system consists of non-interacting is considered as Fermi gas, it is formed from all the particles in their gaseous state (HCl, LiH, H2, ScH, TiH, VH, CrH, CuLi, TiC, NiC, ScN and ScF) undergoing rotation with angular velocity if we make the following two simultaneous transformations to ensure that the previous calculations are not repeated:

ΘχΩandLΘχLΩ. (41)

Here χ is just infinitesimal real proportional constants. We can express the effective potential Vperthmp-rots which induced the rotational movements of the diatomic molecules as follows:

Vperthmp-rots=ll+1α41-s4-αEnl+Ma1-s3-bαs1-s3+2Deq-bαs1-s2+2Deαq1-qs21-s3-2Deαq2s31-s4χLΩ. (42)

To simplify the calculations without compromising physical content, we choose the rotational velocity parallel to the (Oz) axis. Then we transform the spin-orbit coupling to the new physical phenomena as follows:

Λ(s)LΩ=χΛ(s)ΩLz. (43)

With

Λ(s)=ll+1α41-s4-αEnl+Ma1-s3-bαs1-s3+2Deq-bαs1-s2+2Deαq1-qs21-s3-2Deαq2s31-s4. (44)

All of this data allows for the discovery of the new energy shift ΔEhmpf-rotn,n,α,a,b,De,re,χ,m due to the perturbed Fermi gas effect Vperthmp-rotr which generated automatically by the influence of the Hellmann-generalized Morse potential for the n th excited state in REQM symmetries as follows:

ΔEhmpf-rotn,α,a,b,De,re,χ,m=χZnlmRHMPn,α,a,b,De,re,V2Ωm. (45)

It is worth mentioning that the authors in Refs. [102,103] studied a rotating isotropic and anisotropic harmonically confined ultra-cold Fermi gas in a two and three-dimensional space at zero temperature, but in this study, the rotational term was added to the Hamiltonian operator, in contrast to our case, where this rotation term χΛ(s)LΩ automatically appears due to the large symmetries resulting from the deformation of space-phase.

4. Relativistic results of IHGMP in the deformed Klien-Gordon theory symmetries

In this section of the paper, we summarize our obtained results ΔEhmpson,α,a,b,De,re,j,l,s , ΔEhmpmagn,α,a,b,De,re,m and ΔEhmpf-rotn,α,a,b,De,re,m) for the nth excited state due to the spin-orbital coupling, modified Zeeman effect, and perturbed Fermi gas potential induced by Vperthmps on based to the superposition principle in the deformed relativistic Klein-Gordon theory under the improved Hellmann-generalized Morse potential model. This allows us to deduce the additive energy shift ΔEhmptotn,α,a,b,De,re,j,l,s,m under the influence of the improved Hellmann-generalized Morse potential model in ERQM symmetries as follows

ΔEhmptotn,α,a,b,De,re,Θ,σ,χ,j,l,s,m=ZnlmRHMPn,α,a,b,De,reΘτj,l,s+σm+χΩm. (46)

The above results present the global energy shift, which generated with the effect of noncommutativity properties of space-space; it depended explicitly on the noncommutativity parameters Θ,σ,χ, the parameters of the Hellmann-generalized Morse potential n,α,a,b,De,re in addition to the atomic quantum numbers (j,l,s,m). We observed that the obtained global effective energy under the Hellmann-generalized Morse potential has a carry unit of energy because it is combined with the carrier of energy (M2-Enl2). As a direct consequence, the energy Er-nchmpn,n,α,a,b,De,re,j,l,s,m produced with the improved Hellmann-generalized Morse potential model, in the symmetries of ERQM is the sum of the root quart of the shift energy Δ[Ehmpf-rotn,n,α,a,b,De,re,χ,m]1/2 and the relativist energy E nl produced by the effect due to the effect of the Hellmann-generalized Morse potential in RQM, as follows:

Er-nchmpn,n,α,a,b,De,re,Θ,σ,χ,j,l,s,m=Enl+ZnlmRHMPn,n,α,a,b,De,reΘτj,l,s+σm+χΩm1/2. (47)

The relativistic energy E nl is determined from the energy Eq. (16.1).

Equation (47) describes the relativistic energy of some diatomic molecules such as HCl, LiH, H2, ScH, TiH, VH, CrH, CuLi, TiC, NiC, ScN and ScF under the improved Hellmann-generalized Morse potential model in the DRKGT symmetries.

4.1. Relativistic particular cases under IHGMP in ERQM symmetries

After examining the bound state solutions of any l-state DKGE with IHGMP, our task is now to discuss some particular cases below. By adjusting potential parameters for each case, some familiar potentials, which are useful for other physical systems, can be obtained:

1. Setting D e to zero the potential in Eq. (8) turns to the Hellmann potential [6-8], in RQM symmetries, as follows:

VhmprVhpr=-ar+be-αrr. (48)

The perturbed effective potential in Eq. (33) turns to perturbed effective potential in the symmetries of RNCQM as follows:

Vperthps=ll+1α41-s4-αEnl+Ma1-s3-bαs1-s3-bαs1-s2LΘ+OΘ2. (49)

In this case, the additive energy shift under the influence of the improved equally mixed Hellmann potential in ERQM symmetries is determined from the following formula:

Ehmpf-rotn,n,α,a,b,De,re,χ,m=ZnlmRHPn,n,α,a,bΘτj,l,s+σm+χΩm. (50)

Thus, the corresponding global expectation value ZnlmRHPn,n,α,a,b is determined from the following expression:

ZnlmRHMPn,n,α,a,b=ll+1α411-s4nlm

-Enl+Mαa11-s3nlm-bs1-s3nlm-bs3/21-s4nlm. (51)

The new relativistic energy in Eq. (47) reduces to the new energy Er-nchpn,n,α,a,b,Θ,σ,χ,j,l,s,m under modified equal scalar and vector improved Hellmann potential in ERQM symmetries, as follows:

Er-nchpn,n,α,a,b,Θ,σ,χ,j,l,s,m=Enlhp+ZnlmRHPn,n,α,a,bΘτj,l,s+σm+χΩm1/2. (52)

Making the corresponding parameter replacements in Eq. (16.1), we obtain the energy equation for the improved Hellmann potential in the Klein-Gordon theory with equally mixed potentials in RQM symmetries as:

Enlhp2-M2=De-αaEnlhp+M+α2ll+1-14αn+12+δnl-Enlhp+Ma-bn+12+ll+1-αll+1Enlhp+Mn+12+ll+12. (53)

2. Setting the parameters α = b = 0, the potential in Eq. (8) turns to the equal scalar and vector generalized Morse potential or Deng-Fan potential [21], in RQM symmetries, as follows:

Vmpr=De-2Deqe-αr1-e-αr+Deq2e-2αr1-e-αr2. (54)

The perturbed effective potential in Eq. (33) turns to perturbed effective potential in the symmetries of RNCQM as follows:

Vpertmps=ll+1α41-s4-αEnl+M2Deqαs1-s2+2Deαq1-qs21-s3-2Deαq2s31-s4LΘ+OΘ2. (55)

In this case, the additive energy shift under the influence of improved equally mixed generalized Morse potentials in REQM symmetries is given by:

ΔEmptotn,α,De,re,Θ,σ,χ,j,l,s,m=ZnlmRMPn,α,De,reΘτj,l,s+σm+χΩm. (56)

Thus, the corresponding global expectation value is determined from the following expression:

ZnlmRMPn,α,a,b,De,re=ll+1α411-s4nlm-2DeqEnl+Mα

×s3/21-s4nlm+α1-qs21-s3nlm-αqs31-s4nlm. (57)

The new relativistic energy in Eq. (47) reduces to the new energy under improved equal scalar and vector Hellmann potentials in RNCQM, as follows:

Er-nchmpn,α,De,re,Θ,σ,χ,j,l,s,m=Enlmp+ZnlmRMPn,n,α,De,reΘτj,l,s+σm+χΩm1/2. (58)

Making the corresponding parameter replacements in Eq. (16.1), we obtain the energy equation for the generalized Morse potential in the Klein-Gordon theory in RQM symmetries as:

Enlmp2-M2=DeEnl+M+α2ll+1-14αn+12+δnl-2Deαeαre-1Enlmp+Mn+12+ll+1

-Enlmp+MDeαeαre-12+αll+1n+12+ll+12. (59)

5. Fermionic massive spin 1/2 particles interacting with relativistic IHGMP model in the deformed Dirac theory

To obtain the improved Dirac equation (IDE) for the improved Hellmann-generalized Morse potential model in the symmetries of deformed Dirac theory (DDT), we replace both the ordinary Hamiltonian operators H^pi,xi, ordinary spinors Ψnkr,θ,φ, and ordinary energy Enk by NC Hamiltonian operator H^pnci,xnci, the new spinor Ψnkrnc, and new energy Enc-nkmp and the ordinary product will be replaced by star product *, respectively. This allows us to write the IED for improved Hellmann-generalized Morse potential as follows [88-91]:

H^pnci,xnci*Ψnkrnc=Enc-nkmpΨnkrnc. (60)

It is worth motioning that Bopp’s shift method permutes to reduce the above equation to the simplest form:

Hmppnci,xnciΨnkrnc=Enc-nkmpΨnkr. (61)

The new Hamiltonian operator for fermionic massive spin 1/2 particles interacting with the relativistic improved Hellmann-generalized Morse potential model is given by:

Hmppnci,xnci=αpnc+βM+Shmprnc+Vhmprnc. (62)

By applying the notion of the Weyl-Moyal star product which has been seen previously in Eqs. (2)-(4), the differential equations that are satisfied by the radial wave function in Eqs. (13.2) and (13.3), for the spin symmetry solutions and pseudospin symmetry of MDE with IHGMP, respectively, in relativistic noncommutative quantum mechanics symmetries in the deformed Dirac theory (RNCQM-DT, in short), becomes as follows:

d2dr2-kk+1rnc2-M+Enks-CsM-Enks+ΣhmprncFnkr=0, (63)

d2dr2-kk-1rnc2-M+Enkps-ΔhmprncM-Enkps+CpsGnkr=0, (64)

with

Σhmprnc=Δhmprnc=De-arnc+be-αrncrnc+De1+qe-αrnc-12. (65)

The new operators Σhmprnc, Δhmprnc, (kk+1/rnc2) and (kk-1/rnc2) in the deformed Dirac theory symmetries, are expressed as:

Σhmprnc=De-ar+be-αrr+De1+qe-αr-12-ΣhmprrLΘ2r+OΘ2, (66)

kk-1rnc2=kk-1r2+kk-1r4LΘ+OΘ2, (67)

and

kk+1rnc2=kk+1r2+kk+1r4LΘ+OΘ2. (68)

Substituting Eqs. (65) and (66) into Eqs. (63.1) and (63.2), we obtain:

d2dr2-kk+1r2-M+Enksp-CsM-Enksp+Σhmpr

    -kk+1r4-M+Enksp-Cs2rΣhmprrLΘFnkr=0, (69.1)

d2dr2-kk-1r2-M+Enkps-ΔhmprM-Enkps+Cps

    -kk-1r4-M-Enkps+Cps2rΔhmprrLΘGnk~r=0. (69.2)

By comparing (Eqs. (63.1) and (63.2)) and (Eqs. (69.1) and (69.2)), we get an effective deformed spin symmetry Σhmp(r) and pseudospin symmetry Δhmp(r), similar to the perturbative effective potential of Eq. (28),

Σhmppertr=kk+1r4-M+Enksp-Cs2rΣhmprrLΘ, (70.1)

Δhmppertr=kk+1r4-M-Enksp+Cps2rΔhmprrLΘ. (70.2)

A direct calculation gives:

Σhmprr=ar2+bαe-αrr-be-αrr2+2Deαqe-αr1-e-αr2-2Deαq2e-αr1-e-αr3. (71)

We apply the Greene and Aldrich approximation to the Eqs. (70.1) and (70.2) to obtain:

Σhmppertr=α2kk+1α21-s4-M+Enksp-Cs

×aα21-s3-bαs21-s2-Deq-bα/2s1-s3-Dq2s1-s4LΘ, (72.1)

Δhmppertr=α2kk+1α21-s4-M-Enkps-Cps

×aα21-s3-bαs21-s2-Deq-bα/2s1-s3-Dq2s1-s4LΘ. (72.2)

Thus, we need to find the expectation values of the radial terms (1/[1 − s]4), (1/[1 − s]3), (s/[1 − s]2), (s/[1 − s]3) and (s/[1 − s]4) taking into account the wave function which we have seen previously in Eqs. (15.2) and (15.3). We have calculated the expectation values of the radial terms of the first four terms, and but for the last term, we have not done so yet because it did not appear in the Klein Gordon deformed theory using the wave function in Eq. (15.1). To avoid repeating the previous work, it is sufficient to make the following changes to find the four first expectation values as follows:

Fnkr=Nnk2ωnk+1nn!sωnk1-sλnk+122F1-n,n+2ωnk+2λnk+1;1+ωnk;sYlmΩ, (15.2)

Gnk~r=Nnk2Ωnk+1nn!sΩnk1-sβnl+122F1-n,n+2Ωnk+2βnl+1;1+Ωnk;sYlmΩ, (15.3)

{Anl2ωnkΩnk and

Vnlλnkβnkn!Γn+Anl+1NnlΓ2Anl+1Nnk2ωnk+1nn!Nnk2Ωnk+1nn!. (73)

Then the recorded results in relations (37. i=1,2,3,4) will turn out to be appropriate to Dirac’s deformed theory:

11-s4nlmsp=NnkvirΓ2ωnkΓ2λnk-2ΓKnknew-2

× 3F2-n,n+2ωnk+λnk+1,2λnk-2;1+2ωnk,Knknew-2;1, (74.1)

11-s3nlmsp=NnkvirΓ2ωnkΓ2λnk-1ΓKnknew-1

× 3F2-n,n+2ωnk+λnk+1,2λnk-1;1+2ωnk,Knknew-1;1, (74.2)

s1-s3nlmsp=NnkvirΓ2ωnk+1Γ2λnk-1ΓKnknew

× 3F2-n,n+2ωnk+λnk+1,2λnk-1;1+2ωnk,Knknew;1, (74.3)

s1-s2nlmsp=NnkvirΓ2ωnk+1Γ2λnkΓKnknew+1

× 3F2-n,n+2ωnk+λnk+1,2λnk-1;1+2ωnk,Knknew+1;1, (74.4)

11-s4nlmpsp=Nnk'virΓ2ΩnkΓ2βnk-2ΓKnk'new-2

× 3F2-n,n+2Ωnk+βnk+1,2βnk-2;1+2Ωnk,Knk'new-2;1, (74.5)

11-s3nlmpsp=Nnk'virΓ2ΩnkΓ2βnk-1ΓKnk'new-1

× 3F2-n,n+2Ωnk+βnk+1,2βnk-1;1+2Ωnk,Knk'new-1;1, (74.6)

s1-s3nlmpsp=Nnk'virΓ2Ωnk+1Γ2βnk-1ΓKnk'new

× 3F2-n,n+2Ωnk+βnk+1,2Ωnk-1;1+2Ωnk,Knk'new;1, (74.7)

s1-s2nlmpsp=Nnk'virΓ2Ωnk+1Γ2βnkΓKnk'new+1

× 3F2-n,n+2Ωnk+βnk+1,2βnk-1;1+2Ωnk,Knk'new+1;1, (74.8)

with

Nnkvir=n!Γn+2ωnk+1NnkΓ4ωnk+121α,Nnkvir=n!Γn+2Ωnk+1NnkΓ4Ωnk+121α,

Knknew=2ωnk+2λnk and Knk'new=2Ωnk+2βnk The four first results (74.1), (74.2), (74.3) and (74.4) are present the expectation values for deformed spin symmetry while the last four terms (74.5), (74.6), (74.7) and (74.8) related to the pseudospin symmetry. Now, the expectation value for (s/1-s4) deformed spin symmetry and pseudospin symmetry is determined from the equation taking into account the wave (15.2) and (15.3) as follows:

s1-s4nlmsp=Nnkvir0+1s2ωnk1-s2λnk-3 2F1-n,n+2ωnk+2λnk+1;1+ωnk;s2ds, (75.1)

s1-s4nlmpsp=Nnkvir0+1s2Ωnk1-s2βnk-3 2F1-n,n+2Ωnk+2βnl+1;1+Ωnk;s2ds. (75.2)

With the help of the special integral that we saw in Eq. (36), we obtain easily:

s1-s4nlmsp=NnkvirΓ2ωnk+1Γ2λnk-2ΓKnknew-13F2-n,n+Knknew+1,2λnk-2;1+ωnk,Knknew-1;1, (76.1)

s1-s4nlmpsp=Nnk'virΓ2Ωnk+1Γ2βnk-2ΓKnk'new-13F2-n,n+Knk'new+1,2βnk-2;1+Ωnk,Knk'new-1;1. (76.2)

Moreover, and by applying the same method that we saw in the previous section related to the deformed Klein-Gordon theory, taking into account statistical differences between this theory and deformed Dirac theory. The global additive energy ΔEhmpD-totn,α,a,b,De,re,Θ,σ,χ,j,l,s,m , which produced with effective two perturbative terms Σhmppertr and Δhmppertr for deformed spin symmetry and pseudospin symmetry, in RNCQM-DT symmetries, is as follows

ΔEhmptotn,α,a,b,De,re,Θ,σ,χ,j,l,s,m=Θτj,l,s+σm+χΩmZnlmspn,n,α,a,b,De,reFor deformed spin symmetryZnlmpsn,n,α,a,b,De,reFor deformed p-spin symmetry, (77)

where Znlmspn,n,α,a,b,De,re and Znlmpsn,n,α,a,b,De,re are determined by the following relations

Znlmspn,n,α,a,b,De,re=α2kk+1α211-s4nlmsp-M+Enksp-Csaα211-s3nlmsp

-bαs2s1-s2nlmsp-Deq-bα/2s1-s3nlmsp-Dq2s1-s4nlmsp. (78)

and

Znlmpsn,n,α,a,b,De,re=α2kk-1α211-s4nlmsp-M-Enkps-Cpsaα211-s3nlmps

-bαs2s1-s2nlmps-Deq-bα/2s1-s3nlmps-Dq2s1-s4nlmps. (79)

We have seen that in the previous section that the eigenvalues of the operator G2=J2-L2-S2 are equal to the values τj,l,s=j(j+1)-l(l+1)-3/4/2 and τj,l~,s~=[jj+1-l~l~+1-34]/2, thus, for the case of spin-1/2, the possible values of j are l±1/2 and l~±1/2 for spin symmetry τj,l,s and pseudospin symmetry τ(j,l~,s~), which allows us to get and as follows:

τj=l±1/2,l,s=1/2=l2            For up polarity  j=l+1/2-l+12              For dawn polarity  j=  l-1/2, (80.1)

and

τj=l~±1/2,l~,s~=1/2=l~2            For up polarity        j=l~+1/2-l~+12      For dawn polarity    j=l~-1/2. (80.2)

The new relativistic energy Er-ncspn,α,a,b,De,re,Θ,σ,χ,j,l,s,m and Er-ncps n,α,a,b,De,re,Θ,σ,χ,j,l~,s~,m for the case of spin-1/2 with improved Hellmann-generalized Morse potential, in the symmetries of the deformed Dirac symmetries, corresponding to the generalized n th excited states:

Er-ncspn,α,a,b,De,re,Θ,σ,χ,j,l,s,m=Enks+Znlmspn,n,α,a,b,De,re)l2            For up polarity  j=l+1/2-l+12     For dawn polarity  j=  l-1/2, (81.1)

and

Er-ncps(n,α,a,b,De,re,Θ,σ,χ,j,l~,s~,m)=Enkps+Znlmpsn,n,α,a,b,De,rel~2            For up polarity        j=l~+1/2-l~+12      For dawn polarity    j=l~-1/2, (81.2)

where Enks and Enkps are usual relativistic energies within the Dirac theory obtained from Eqs. (16.3) and (16.4) while k and k~ are determined from the following relations:

k=k1=-l+1=l+1/2            For    s1/2,p3/2...etc.j=l+1/2                Aligned spin                  k<0k1=-l+1=l+1/2            For    s1/2,p3/2...etc.j=l-1/2                Aligned spin        k>0, (82)

and

k~=k1=-l~=-j+1/2            For    s1/2,p3/2...etc.j=l~-1/2            Aligned spin  k~<0k1=-l+1=l+1/2            For    s1/2,p3/2...etc.j=l-1/2            Un aligned spin  k~>0. (83)

6. Nonrelativistic study of improved Hellmann-generalized Morse potential

6.1 Nonrelativistic improved Hellmann-generalized Morse potential

In this subsection section, we want to derive the nonrelativistic spectrum, which is produced by the effect of the IHGMP for some diatomic molecules such as HCl, LiH, H2, ScH, TiH, VH, CrH, CuLi, TiC, NiC, ScN, and ScF by applying the notion of the Weyl Moyal star product which have seen previously in Eqs. (2.1), (2.2), and (4) to the differential equation that satisfied by the radial wave function ψ nl (r) in Eq. (13.4), the radial wave function in extended nonrelativistic quantum mechanics (ENRQM, for short) symmetries becomes as follows:

d2dr2+2MEnlnr-Vhmpr-l{l+1}2Mr2*ψnlr=0. (84)

According to Bopp’s shift method, Eq. (84) becomes similar to the following like the Schrödinger equation (without the notion of the Weyl-Moyal star product):

d2dr2+2MEnlnr-Vhmprnc-l{l+1}2Mrnc2ψnlr=0. (85)

From Eqs. (1.2) and (23) we can write this potential in the ENRQM symmetries as follows:

Vhmprnc=De-ar+be-αrr-be-αrr2+2Deqe-αr1-e-αr2+Deq2e-αr1-e-αr2+Vnr-perthmpr, (86)

where Vnr-perthmpr is infinitesimal value if compared with the main part Vhmpr. Furthermore, it presents the global perturbative potential of IHGMP in the ENRQM symmetries:

Vnr-perthmpr=ll+1r4LΘ-VhmprrLΘ2r+OΘ2. (87)

The first term in Eq. (87) due to the centrifugal term (l[l + 1]/r nc 2 ) in ENRQM symmetries (see Eq. (24)) which equals the usual centrifugal term (l[l + 1]/r 2) plus the perturbative centrifugal term (l[l + 1]/r 4)LΘ while the second term in Eq. (87) is produced with the effect of IHGMP. We have seen in Eq. (31) the expression ∂V hmp (r)/∂r, allows us to get as follows:

Vnr-perthmpr=ll+1r4LΘ-ar2-bαe-αrr-bαe-αrr2-2Deαqe-αr1-e-αr

+Deq1-qe-2αr1-e-αr2-2Deαq2e-3αr1-e-αr3LΘ2r+OΘ2. (88)

Now, we apply the Greene and Aldrich approximation for the centrifugal term in the perturbed potential Vnr-perthmps, we obtain:

Vnr-perthmps=ll+1α41-s4-α2a1-s3-bαs1-s3+2Deq-bαs1-s2

+2Deαq1-qs21-s3-2Deαq2s31-s4LΘ+OΘ2. (89)

6.2. Nonrelativistic bound state correction under the improved Hellmann-generalized Morse potential

In this subsection, we want to generate nonrelativistic bound state corrections under improved Hellmann-generalized Morse potential. The expression of perturbative potential in Eq. (89) needs to calculate the expectation values of (1/[1 − s]4), (1/[1 − s]3), (s/[1 − s]3, (s/[1 − s]2), (s 2 /[1 − s]3) and (s 3 /[1 − s]4) to find the nonrelativistic energy corrections produced by the perturbative potential Vnr-perthmps. We have seen the expectation values of these terms in Eqs. (35). i=1,2,3,4,5,6), allow us to get the global nonrelativistic expectation ZnlmNRHMPn,n,α,a,b,De,re value is determined from the following expression:

ZnlmNRHMPn,α,a,b,De,re=ll+1α411-s4nlm-α2a11-s3nlm

-bs1-s3nlm+2Deq-bs321-s4nlm

+2αDeq1-qs21-s3nlm-2αDeq2s31-s4nlm. (90)

And by following the same method used in the relativistic study, we obtain the nonrelativistic energy corrections ΔEhmptot-nr n,α,a,b,De,re,Θ,σ,χ,j,l,s,m for the generalized excited states due to the spin-orbit coupling, improved Zeeman effect and nonrelativistic perturbed Fermi gas potential under the influence of the improved Hellmann-generalized Morse potential in ENRQM symmetries are as follows:

ΔEhmptot-nrn,α,a,b,De,re,Θ,σ,χ,j,l,s,m=ZnlmNRHMPn,α,a,b,De,reΘτj,l,s+σm+χΩm. (91)

According to the standard perturbation theory. The new generalized nonrelativistic energy Enr-nchmp(n,n,α,a,b,De,re,Θ, σ,χ,j,l,s,m) for the excited n th states, which, produced by the effect of the improved Hellmann-generalized Morse potential, is the sum of the nonrelativistic energies Enlnr (see Eq. (16.2)) due to the effect of Hellmann-generalized Morse potential in NRQM and the above corrections in Eq. (91):

Enr-nchmpn,α,a,b,De,re,Θ,σ,χ,j,l,s,m=Enlnr+ZnlmNRHMPn,α,a,b,De,reΘτj,l,s+σm+χΩm. (92)

6.3 Nonrelativistic particular cases under the IHGMP

After examining the bound state solutions of any l-state deformed Schrödinger equation with the improved Hellmann-generalized Morse potential, our task is now to discuss some particular cases below. By adjusting the potential parameters for each case, some familiar potentials, which are useful for other physical systems, can be obtained.

1. Setting D e to zero the potential in Eq. (8) turns to the Hellmann potential (Eq. (48)) in NRQM symmetries. The perturbed effective potential in Eq. (89) turns to perturbed effective potential Vnr-perthps in the symmetries of RNCQM as follows:

Vnr-perthps=ll+1α41-s4-α2a1-s3-bαs1-s3-bαs1-s2LΘ+OΘ2. (93)

The new relativistic energy in Eq. (92) reduces to the new energy Enr-nchpn,α,a,b,Θ,σ,χ,j,l,s,m under improved equal scalar and vector improved Hellmann potential in ENRQM symmetries, as follows:

Enr-nchpn,α,a,b,Θ,σ,χ,j,l,s,m=Ehp-nlnr+ZnlmNRHPn,n,α,a,bΘτj,l,s+σm+χΩm, (94)

where ZnlmNRHPn,n,α,a,b is determined as a particular case from the global nonrelativistic expectation value ZnlmNRHMP n,n,α,a,b,De,re:

ZnlmNRHMPn,α,a,b=ll+1α411-s4nlm-α2a11-s3nlm

-bs1-s3nlm-bs3/21-s4nlm, (95)

while Ehp-nlnr can be obtained directly from Eq. (16.1) by replace D e with zero.

2. Setting α = b =0, the potential in Eq. (8) turns to the generalized Morse (Eq. (54)). The perturbed effective potential in Eq. (33) turns to perturbed effective potential Vnr-pertmps in the symmetries of RNCQM as follows:

Vnr-pertmps=ll+1α41-s4-Deqα2αs1-s2+1-qs21-s3-qs31-s4LΘ+OΘ2. (96)

In this case, the nonrelativistic energy correction under the influence of improved Morse potentials in ENRQM symmetries is given by:

ΔEmptot-nrn,α,De,re,Θ,σ,χ,j,l,s,m=ZnlmNRMPn,α,De,reΘτj,l,s+σm+χΩm. (97)

Thus, the corresponding global expectation value ZnlmNRMPn,n,α,De,re is determined from the following expression:

ZnlmNRMPn,α,De,re=ll+1α411-s4nlm-α2Deqbs321-s4nlm

+1-qs21-s3nlm-qs31-s4nlm. (98)

The new nonrelativistic energy in Eq. (92) reduces to the new energy under IHGMP in ENRQM symmetries, as follows:

Enr-ncmpn,α,De,re,Θ,σ,χ,j,l,s,m=Emp-nlnr+ZnlmNRMPn,α,De,reΘτj,l,s+σm+χΩm, (99)

while Emp-nlnr can be obtained directly from Eq. (16.1) by replace α = b =0 with zero.

6.4 Study the nonrelativistic fermion cases

We have seen that in the previous section that the eigenvalues of the operator G2=J2-L2-S2, for the case of spin-1/2, are determined by Eq. (80.1) thus, the nonrelativistic energy in Eq. (92) can be generalized to the case of spin-1/2 with an improved Hellmann-generalized Morse potential, in the symmetries of ENRQM, corresponding to the generalized excited states:

Enr-nchmpn,α,a,b,De,re,Θ,σ,χ,j,l,s,m=De-αa+α2ll+12M-α28MΛn,l-2MηlΛn,l2

+ZnlmNRHMPn,α,a,b,De,rel2+σm+χΩmFor up polarity  j=l+1/2-l+12  +σm+χΩmFor dawn polarity  j=l-1/2. (100)

We conclude with all merit that the combination of potentials in the new symmetries of ENRQM brought about an upgrade to the ordinary Schrödinger equation to become similar in the description ability of the Dirac equation. The spin condition was clearly shown in the above result in Eq. (98).

6.5 Composite systems in ENRQM symmetries

Now, considering composite systems such as molecules made of N = 2 particles of masses m n (n=1,2) in the frame of noncommutative algebra, it is worth taking into account features of descriptions of the systems in the space. In NRQM symmetries, it was obtained that composite systems with different masses are described with different noncommutative parameters [49, 52, 54, 104]:

xncμS,H,I,*xncνS,H,I=iθμνc. (101)

with

  θμνc=2n=1μn2θμνn, (102)

with μn=(mn/mnn), the indices (n=1,2) label the particle, and θμνn is the parameter of noncommutativity, corresponding to the particle of mass mn. Note that in the case of a system of two particles with the same mass m1=m2 such as the homogeneous (N2, H2 and I2) diatomic molecules the parameter θμνn=θμν. Thus, the two parameters Θ and σ which appears in Eq. (92) are changed to the new form:

Θc2=2n=1μn2Θ12n2+2n=1μn2Θ23n2+2n=1μn2Θ13n2, (103.1)

σc2=2n=1μn2σ12n2+2n=1μn2σ23n2+2n=1μn2σ13n2. (103.2)

and

χc2=2n=1μχ12n2+2n=1μχ23n2+2n=1μχ13n2. (103.3)

As it is mentioned above, in the case of a system of two particles with the same mass m 1=m 2 such as the homogeneous (N2, H2 and I2) diatomic molecules Θμνn=Θμν and σμνn=σμν. Finally, we can generalize the nonrelativistic global energy Enr-nchmp ( n,α,A,B,V0,Θ,σ,χ,j, l,s,m) under the improved Hellmann-generalized Morse potential model considering that composite systems with different masses are described with different noncommutative parameters for the diatomic (HCl, LiH, ScH, TiH, VH, CrH, CuLi, TiC, NiC, ScN and ScF) as:

Enr-nchmpn,α,a,b,De,re,Θ,σ,χ,j,l,s,m=Enlnr+ZnlmNRHMPn,n,α,a,b,De,re

×Θcτj,l,s+σcm+χcΩm. (104)

Finally, we arrived at the important results achieved in this new work. The KGE, as the most well-known relativistic wave equation, describes spin-zero particles, but its extension to the RNCQM symmetric deformed Klein-Gordon equation under improved Hellmann-generalized Morse potential has a physical behavior similar to the Duffin-Kemmer equation for a meson with spin-s that can describe a dynamic state of a particle with spin one in the symmetries of relativistic noncommutative quantum mechanics. This is one of the most important new results of this research. It is worthwhile to mention that for the two simultaneous limits (Θ,σ,χ) and (Θ c c c )→(0,0,0) we recover the results of Refs. [34, 35].

7. Summary and Conclusions

This main part of our paper gives a summary of the basic points in our work. In this work, we have found the approximate bound state solutions of the deformed Klein-Gordon, deformed Dirac and Schrödinger equations of the improved Hellmann-¨ generalized Morse potential, which correspond to high and low energy physics for the diatomic molecules (HCl, LiH, H2, ScH, TiH, VH, CrH, CuLi, TiC, NiC, ScN and ScF), in the symmetries of extended KG theory, Dirac theory and Schrödinger¨ theory. We have used Bopp’s shift method, stationary, perturbation theory, and the improved approximation scheme to deal with the centrifugal term. In addition to the usual state numbers (n,l) , the new energy equations have appeared sensitive to discrete atomic numbers (j,l,s,m), the parameters for the quantum states (α, α,b,D e ,r e ) in addition to noncommutativity parameters (Θ and χ). This new behavior is in the symmetries of extended relativistic and relativistic quantum mechanics equivalent to a conventional physical system under the influence of at least three perturbative systems with the perturbed spinorbit, improved Zeeman effect and the perturbed Fermi gas effect. The main difference lies in the fact that these perturbations appear automatically in the case of the new system of ERQM and ENRQM symmetries. Furthermore, we can conclude that the deformed Klein-Gordon equation under the improved Hellmann-generalized Morse potential becomes similar to Duffin- Kemmer equation for a meson with spin-s, it can describe the dynamic state of a particle with spin-s in this symmetry. We also noted how the MHGPs model can be reduced to the improved Hellmann potential and improved Morse potential by applying appropriate potential constant values. Moreover, we have applied our results to composite systems such as molecules made of N = 2 particles of masses m n (n = 1,2) such as HCl, LiH, ScH, TiH, VH, CrH, CuLi, TiC, NiC, ScN and ScF. It is worth mentioning that, for all cases, to make the two simultaneous limits (Θ,σ,χ) and (Θ c c c ) → (0,0) , the ordinary physical quantities are recovered in refs. [34,35]. Finally, given the effectiveness of the methods that we followed in achieving our goal in this research, we advise researchers to apply the same methods in other studies, whether in the relativistic and nonrelativistic regimes for others potentials.

Acknowledgments

This work was supported by the Research Fund of Laboratory of Physics and Material Chemistry, University of M’sila, and DGRSDT in Algeria with the project number B00L02UN280120180001. The author wishes to give their sincere gratitude to the referees for their kind comments which improved the manuscript.

References

1 E. Schrödinger, An Undulatory Theory of the Mechanics of Atoms and Molecules, Phys. Rev. 28 (1926) 1049, https://doi.org/10.1103/PhysRev.28.1049. [ Links ]

2 O. Klein, Quantentheorie und funfdimensionale Relativitätstheorie, Z. Physik 37 (1926) 895, https://doi.org/10.1007/BF01397481 [ Links ]

3 W. Gordon, Der Comptoneffekt nach der Schrödingerschen Theorie, Z. Physik 40 (1926) 117, https://doi.org/10.1007/BF01390840 [ Links ]

4 N. Kemmer, The Particle Aspect of Meson Theory, Proc. R. Soc. Lond. A 173 (1939) 91, https://doi.org/10.1098/rspa.1939.0131. [ Links ]

5 P.A.M. Dirac, The Quantum Theory of the Electron, Proc. R. Soc. Lond. A 117 (1928) 610, https://doi.org/10.1098/rspa.1928.0023. [ Links ]

6 H. Hellmann, A combined approximation procedure for calculation of energies in the problem of many electrons, Acta Physicochim, U.R.S.S. 1 (1935) 913; ibid. 4 (1936) 225; 4 (1936) 324. [ Links ]

7 H. Hellmann, A New Approximation Method in the Problem of Many Electrons, J. Chem. Phys. 3 (1935) 61, https://doi.org/10.1063/1.1749559. [ Links ]

8 H. Hellmann and W. Kassatotchkin, Metallic Binding According to the Combined Approximation Procedure, J. Chem. Phys. 4 (1936) 324, https://doi.org/10.1063/1.1749851. [ Links ]

9 H. Yukawa, On the interaction of elementary particles I, Proc. Phys. Math. Soc. Japan 17 (1935) 48, https://doi.org/10.11429/ppmsj1919.17.048. [ Links ]

10 S.M. Ikhdair and R. Sever, A perturbative treatment for the bound states of the Hellmann potential, J. Mol. Struct. 809 (2007) 103, https://doi.org/10.1016/j.theochem.2007.01.019. [ Links ]

11 G. Kocak, O. Bayrad and I. Boztosum, Arbitrary l-state solutions of the Hellmann potential, J. Theor. Comput. Chem. 06 (2007) 893, https://doi.org/10.1142/S0219633607003313. [ Links ]

12 S.M. Ikhdair, and B.J. Falaye. Two Approximate Analytic Eigensolutions of the Hellmann Potential with any Arbitrary Angular Momentum, Z. Naturforsch. A 68 (2013) 701, https://doi.org/10.5560/zna.2013-0054 [ Links ]

13 J.C. Philips, and L. Kleinmann, New Method for Calculating Wave Functions in Crystals and Molecules, Phys. Rev. A 116 (1959) 287; https://doi.org/10.1103/PhysRev.116.287; L. Kleinman and J. C. Phillips, Crystal Potential and Energy Bands of Semiconductors. III. Self-Consistent Calculations for Silicon, Phys. Rev. 118 (1960) 1153, https://doi.org/10.1103/PhysRev.118.1153. [ Links ]

14 A.J. Hughes, and J. Callaway, Energy Bands in Body-Centered and Hexagonal Sodium, Phys. Rev. A 136 (1964) A1390, https://doi.org/10.1103/PhysRev.136.A1390 [ Links ]

15 Y.P. Varshni, and R.C. Shukla, Alkali Hydride Molecules: Potential Energy Curves and the Nature of their Binding, Rev. Mod. Phys. 35 (1963) 130, https://doi.org/10.1103/RevModPhys.35.130. [ Links ]

16 J.N. Das, and S. Chakraborty, Atomic inner-shell ionization, Phys. Rev. A 32 (1985) 176, https://doi.org/10.1103/PhysRevA.32.176. [ Links ]

17 M. Mousavi and M.R. Shojaei, Mirror Nuclei of 17O and 17F in Relativistic and Nonrelativistic Shell Model, Adv. High Energy Phys. 2017 (2017) 5841701, https://doi.org/10.1155/2017/5841701. [ Links ]

18 M. Hamzavi, K.E. Thylwe, and A.A. Rajabi, Approximate Bound States Solution of the Hellmann Potential. Commun. Theor. Phys. 60 (2013) 1, https://doi.org/10.1088/0253-6102/60/1/01. [ Links ]

19 C.A. Onate, M.C. Onyeaju, A.N. Ikot and O. Ebomwonyi, Eigen solutions and entropic system for Hellmann potential in the presence of the Shrodinger equation, Eur. Phys. J. Plus. 132 (2017) 462, https://doi.org/10.1140/epjp/i2017-11729-8 [ Links ]

20 C.O. Edet, K.O. Okorie, H. Louis and N.A. Nzeata-Ibe, Any l-state solutions of the Schrödinger equation inter-¨ acting with Hellmann-Kratzer potential model. Indian J. Phys. 94 (2020) 243, https://doi.org/10.1007/s12648-019-01467-x. [ Links ]

21 Z.H. Deng, and Y.P. Fan, A Potential Function of Diatomic Molecules, Shandong Univ. J. 7 (1957) 162. [ Links ]

22 P.M. Morse, Diatomic Molecules According to the Wave Mechanics. II. Vibrational Levels, Phys. Rev. 34 (1929) 57, https://doi.org/10.1103/PhysRev.34.57. [ Links ]

23 S.H. Dong, Factorization method in quantum mechanics Fundamental Theories in Physics. 150 (The Netherlands: Springer 2007) pp 187-213, [ Links ]

24 O.J. Oluwadare, K.J., Oyewumi, C.O. Akoshile and O.A. Babalola, Approximate analytical solutions of the relativistic equations with the Deng-Fan molecular potential including a Pekeris-type approximation to the (pseudo) centrifugal term, Physica Scripta, 86 (2012) 035002, https://doi.org/10.1088/0031-8949/86/03/035002. [ Links ]

25 A.D.S. Mesa, C. Quesne and Y.F. Smirnov, Generalized Morse potential: Symmetry and satellite potentials, J. Phys. A 31 (1998) 321, https://doi.org/10.1088/0305-4470/31/1/028. [ Links ]

26 S.H. Dong and X.Y. Gu, Arbitrary l state solutions of the Schrödinger equation with the Deng-Fan molecular potential, J. Phys. Conf. Ser. 96 (2008) 012109, https://doi.org/10.1088/1742-6596/96/1/012109. [ Links ]

27 S.H. Dong, Relativistic Treatment of Spinless Particles Subject to a Rotating Deng-Fan Oscillator, Commun. Theor. Phys. 55 (2011) 969, https://doi.org/10.1088/0253-6102/55/6/05. [ Links ]

28 H. Hassanabadi, B.H. Yazarloo, S. Zarrinkamar and H. Rahimov, Deng-Fan Potential for Relativistic Spinless Particlesan Ansatz Solution, Commun. Theor. Phys. 57 (2012) 339, https://doi.org/10.1088/0253-6102/57/3/02. [ Links ]

29 S. Ortakaya, H. Hassanabadi and B.H. Yazarloo, Bound state solutions of the Dirac equation with the Deng-Fan potential including a Coulomb tensor interaction, Chin. Phys. B 23 (2014) 030306. https://doi.org/10.1088/1674-1056/23/3/030306. [ Links ]

30 L.H. Zhang, X.P. Li and C.S. Jia, Approximate analytical solutions of the Dirac equation with the generalized Morse potential model in the presence of the spin symmetry and pseudospin symmetry, Phys. Scr. 80 (2009) 035003, https://doi.org/10.1088/0031-8949/80/03/035003. [ Links ]

31 E. Maghsoodi, H. Hassanabadi and S. Zarrinkamar, Spectrum of Dirac Equation Under Deng-Fan Scalar and Vector Potentials and a Coulomb Tensor Interaction by SUSYQM, FewBody Syst. 53 (2012) 525, https://doi.org/10.1007/s00601-012-0314-5. [ Links ]

32 K.J. Oyewumi, O.J. Oluwadare, K. D. Sen and O.A. Babalola, Bound state solutions of the Deng-Fan molecular potential with the Pekeris-type approximation using the NikiforovUvarov (N-U) method, J. Math. Chem. 51 (2012) 976, https://doi.org/10.1007/s10910-012-0123-6. [ Links ]

33 E. Omugbe, Non-relativistic Energy Spectrum of the DengFan Oscillator via the WKB Approximation Method, Asian J. Phys. Chem. Sci. 8 (2020) 26, https://doi.org/10.9734/ajopacs/2020/v8i130107 [ Links ]

34 P.O. Okoia, C.O. Edetb and T.O. Magu, Relativistic treatment of the Hellmann-generalized Morse potential, Rev. Mex. Fis. 66 (2020) 1, https://doi.org/10.31349/RevMexFis.66.1 [ Links ]

35 O. Ebomwonyi, C.A. Onate, M.C. Onyeaju and A.N. Ikot, Any l- states solutions of the Schrödinger equation interacting with Hellmann-generalized Morse potential model, Karbala Int. J. Mod. Sci. 3 (2017) 59, https://doi.org/10.1016/j.kijoms.2017.03.001. [ Links ]

36 H.S. Snyder, The Electromagnetic Field in Quantized SpaceTime, Phys. Rev. 72 (1947) 68, https://doi.org/10.1103/PhysRev.72.68. [ Links ]

37 A. Connes, M. R. Douglas and A. Schwarz, Noncommutative geometry and Matrix theory, J. High Energy Phys. 1998 (1998) 003. https://doi.org/10.1088/1126-6708/1998/02/003. [ Links ]

38 N. Seiberg and E. Witten, String theory and noncommutative geometry, J. High Energy Physics 1999 (1999) 032, https://doi.org/10.1088/1126-6708/1999/09/032. [ Links ]

39 S. Capozziello, G. Lambiase and G. Scarpetta, Generalized uncertainty principle from quantum geometry. Int. J. Theor. Phys. 39 (2000) 15, https://doi.org/10.1023/A:1003634814685. [ Links ]

40 S. Doplicher, K. Fredenhagen and J.E. Roberts, Spacetime quantization induced by classical gravity, Phys. Lett. B 331 (1994) 39, https://doi.org/10.1016/0370-2693(94)90940-7. [ Links ]

41 E. Witten, Refection on the fate spacetime, Phys. Today 49 (1996) 24, https://doi.org/10.1063/1.881493. [ Links ]

42 A. Kempf, G. Mangano and R.B. Mann, Hilbert space representation of the minimal length uncertainty relation, Phys. Rev. D 52 (1995) 1108, https://doi.org/10.1103/PhysRevD.52.1108. [ Links ]

43 F. Scardigli, Some heuristic semi-classical derivations of the Planck length, the Hawking effect and the unruh effect, Nuovo Cimento B 110 (1995) 1029, https://doi.org/10.1007/BF02726152. [ Links ]

44 R.J. Adler and D.I. Santigo, On gravity and the uncertainty principal, Mod. Phys. Lett. A 14 (1999) 1371, https://doi.org/10.1142/S0217732399001462. [ Links ]

45 T. Kanazawa, G. Lambiase, G. Vilasi and A. Yoshioka, Noncommutative Schwarzschild geometry and generalized uncertainty principle, Eur. Phys. J. C. 79 (2019) 95, https://doi.org/10.1140/epjc/s10052-019-6610-1. [ Links ]

46 F. Scardigli, Generalized uncertainty principle in quantum gravity from micro-black hole Gedanken experiment, Phys. Lett. B 452 (1999) 39, https://doi.org/10.1016/S0370-2693(99)00167-7. [ Links ]

47 P.M. Ho and H.C. Kao, Noncommutative quantum mechanics from noncommutative quantum field Theory, Phys. Rev. Lett. 88 (2002) 151602. https://doi.org/10.1103/PhysRevLett.88.151602. [ Links ]

48 O. Bertolami, G.J. Rosa, C.M.L. Dearagao, P. Castorina and D. Zappala, Scaling of variables and the relation between noncommutative parameters in noncommutative quantum mechanics, Mod. Phys. Lett. A 21 (2006) 795, https://doi.org/10.1142/S0217732306019840. [ Links ]

49 P. Gnatenko, Parameters of noncommutativity in Liealgebraic noncommutative space, Phys. Rev. D 99 (2019) 026009, https://doi.org/10.1103/PhysRevD.99.026009. [ Links ]

50 O. Bertolami and P. Leal, Aspects of phase-space noncommutative quantum mechanics, Phys. Lett. B 750 (2015) 6, https://doi.org/10.1016/j.physletb.2015.08.024. [ Links ]

51 M.A. De Andrade and C. Neves, Noncommutative mapping from the symplectic formalism, J. Math. Phys. 59 (2018) 012105, https://doi.org/10.1063/1.4986964. [ Links ]

52 Kh.P. Gnatenko and V.M. Tkachuk, Upper bound on the momentum scale in noncommutative phase space of canonical type, (Europhys. Lett.) 127 (2019) 20008, https://doi.org/10.1209/0295-5075/127/20008. [ Links ]

53 Kh.P. Gnatenko and TV.M. kachuk, Composite system in rotationally invariant noncommutative phase space, Int. J. Mod. Phys. A 33 (2018) 1850037, https://doi.org/10.1142/S0217751X18500379. [ Links ]

54 A. Maireche, Bound-state solutions of the modified KleinGordon and Schrödinger equations for arbitrary l-state with the modified Morse potential in the symmetries of noncommutative quantum mechanics, J. Phys. Stud. 25 (2021) 1002, https://doi.org/10.30970/jps.25.1002. [ Links ]

55 A. Maireche, A Theoretical Model of Deformed Klein-Gordon Equation with Generalized Modified Screened Coulomb Plus Inversely Quadratic Yukawa Potential in RNCQM Symmetries, Few-Body Syst 62 (2021) 12. https://doi.org/10.1007/s00601-021-01596-2 [ Links ]

56 A. Maireche, Solutions of Klein-Gordon equation for the modified central complex potential in the symmetries of noncommutative quantum mechanics, Sri Lankan J. Phys. 22 (2021) 1, Doi:http://doi.org/10.4038/sljp.v22i1.8079 [ Links ]

57 A. Maireche, Theoretical Investigation of the Modified Screened cosine Kratzer potential via Relativistic and Nonrelativistic treatment in the NCQM symmetries, Lat. Am. J. Phys. Educ. 14 (2020) 3310. [ Links ]

58 A. Maireche, Modified unequal mixture scalar vector Hulthén-Yukawa potentials model as a quark-antiquark interaction and neutral atoms via relativistic treatment using the improved approximation of the centrifugal term and Bopp’s shift method, Few-Body Syst. 61 (2020) 30. https://doi.org/10.1007/s00601-020-01559-z. [ Links ]

59 J. Gamboa, M. Loewe and J.C. Rojas, Noncommutative quantum mechanics, Phys. Rev. D 64 (2001) 067901. https://doi.org/10.1103/PhysRevD.64.067901. [ Links ]

60 A. Maireche, A New Approach to the approximate analytic solution of the three-dimensional Schrödinger equation for Hydrogenic and neutral atoms in the generalized Hellmann potential model Ukr. J. Phys. 65 (2020) 987. https://doi.org/10.15407/ujpe65.11.987 [ Links ]

61 A. Maireche, A New Look at a Nonrelativistic Shell Model: Study of the Mirror Nuclei 17O and 17F in the Symmetries of NCQM, To Phys. J. 5 (2020) 51. [ Links ]

62 A. Maireche Any l-States solutions of the modified Schrödinger equation with generalized Hellmann-Kratzer potential model in the symmetries of NRNCQM, To Phys. J. 4 (2019) 16, https://purkh.com/index.php/tophy/article/view/521. [ Links ]

63 E.F. Djemaï and H. Smail, On quantum mechanics on noncommutative quantum phase space, Commun. Theor. Phys. 41 (2004) 837, https://doi.org/10.1088/0253-6102/41/6/837. [ Links ]

64 Y. Yi, K. Kang, W. Jian-Hua and C. Chi-Yi, Spin-1/2 relativistic particle in a magnetic field in NC phase space, Chin. Phys. C. 34 (2010) 543, https://doi.org/10.1088/1674-1137/34/5/005. [ Links ]

65 O.G. Valencia and H.L.A. Arias, Thermodynamic properties of diatomic molecule systems under SO (2,1)-anharmonic Eckart potential, Int. J. Quantum Chem. 118 (2018) e25589. https://doi.org/10.1002/qua.25589. [ Links ]

66 O. Bertolami, J.G. Rosa, C.M.L. de Aragão, P. Castorina and D. Zappalà, Noncommutative gravitational quantum well, Phys. Rev. D 72 (2005) 025010, https://doi.org/10.1103/PhysRevD.72.025010. [ Links ]

67 J. Zhang, Fractional angular momentum in non-commutative spaces, Phys. Lett. B 584 (2004) 204, https://doi.org/10.1016/j.physletb.2004.01.049. [ Links ]

68 M. Chaichian, Sheikh-Jabbari and A. Tureanu, Hydrogen atom spectrum and the Lamb Shift in noncommutative QED, Phys. Rev. Lett. 86 (2001) 2716, https://doi.org/10.1103/PhysRevLett.86.2716. [ Links ]

69 E.M.C. Abreu, C. Neves and W. Oliveira, Noncommutativity from the symplectic point of view, Int. J. Mod. Phys. A 21 (2006) 5359, https://doi.org/10.1142/S0217751X06034094. [ Links ]

70 E.M.C. Abreu, J.A. Neto, A.C.R. Mendes C. Neves, W. Oliveira and M.V. Marcial, Lagrangian formulation for noncommutative nonlinear systems, Int. J. Mod. Phys. A 27 (2012) 1250053, https://doi.org/10.1142/S0217751X12500534. [ Links ]

71 J. Wang and K. Li, The HMW effect in noncommutative quantum mechanics, J. Phys. A Math. Theor. 40 (2007) 2197, https://doi.org/10.1088/1751-8113/40/9/021. [ Links ]

72 K. Li, and J. Wang, The topological AC effect on non-commutative phase space, Eur. Phys. J. C. 50 (2007) 1007, https://doi.org/10.1140/epjc/s10052-007-0256-0. [ Links ]

73 A. Maireche, A theoretical investigation of nonrelativistic bound state solution at finite temperature using the sum of modified Cornell plus inverse quadratic potential, Sri Lankan J. Phys. 21 (2020) 11, https://doi.org/10.4038/sljp.v21i1.8069. [ Links ]

74 A. Maireche, Extended of the Schrödinger Equation with New Coulomb Potentials plus Linear and Harmonic Radial Terms in the Symmetries of Noncommutative Quantum Mechanics, J. Nano- Electron. Phys. 10 (2018) 06015, https://doi.org/10.21272/jnep.10(6).06015 [ Links ]

75 A. Maireche, Heavy light mesons in the symmetries of extended nonrelativistic quark model, Yanbu J. Eng. Sci. 17 (2019) 51, https://doi.org/10.53370/001c.23732. [ Links ]

76 A. Maireche, The Klein-Gordon equation with modified Coulomb plus inverse-square potential in the noncommutative three-dimensional space, Mod. Phys. Lett. A . 35 (2020) 052050015. https://doi.org/10.1142/S0217732320500157. [ Links ]

77 A. Maireche, The Klein-Gordon equation with modified Coulomb potential plus inverse-square-root potential in three-dimensional noncommutative space, To Phys. J. 3 (2019) 186, https://purkh.com/index.php/tophy/article/view/489Links ]

78 A. Maireche, Bound state solutions of Klein-Gordon and Schrödinger equations with linear combination of Hulthén and Kratzer potentials, Afr. Rev. Phys. 15 (2020) 19, http://lamp.ictp.it/index.php/aphysrev/article/view/1779/620. [ Links ]

79 H. Motavalli and A.R. Akbarieh, Klein-Gordon equation for the Coulomb potential in noncommutative space, Mod. Phys. Lett. A 25 (2010) 2523, https://doi.org/10.1142/S0217732310033529. [ Links ]

80 M. Darroodi, H. Mehraban and H. Hassanabadi, The Klein-Gordon equation with the Kratzer potential in the noncommutative space, Mod. Phys. Lett. A 33 (2018) 1850203, https://doi.org/10.1142/S0217732318502036. [ Links ]

81 A. Saidi and M.B. Sedra, Spin-one (1+3)-dimensional DKP equation with modified Kratzer potential in the noncommutative space, Mod. Phys. Lett. A 35 (2020) 2050014, https://doi.org/10.1142/S0217732320500145. [ Links ]

82 H. Aounallah and A. Boumali,, Solutions of the Duffin- Kemmer Equation in Non-Commutative Space of Cosmic String and Magnetic Monopole with Allowance for the Aharonov-Bohm and Coulomb Potentials, Phys. Part. Nucl. Lett. 16 (2019) 195, https://doi.org/10.1134/S1547477119030038. [ Links ]

83 A. Maireche, A model of modified Klein-Gordon equation with modified scalar-vector Yukawa potential, Afr. Rev Phys. 15 (2020) 1, http://lamp.ictp.it/index.php/aphysrev/article/view/1777/618. [ Links ]

84 A. Maireche, The Relativistic and Nonrelativistic Solutions for the Modified Unequal Mixture of Scalar and Time-Like Vector Cornell Potentials in the Symmetries of Noncommutative Quantum Mechanics, Jordan J. Phys. 14 (2021) 59, https://doi.org/10.47011/14.1.6 [ Links ]

85 L. Mezincescu, Star Operation in Quantum Mechanics, https://arxiv.org/abs/hep-th/0007046. [ Links ]

86 L. Gouba, A comparative review of four formulations of noncommutative quantum mechanics, Int. J. Mod. Phys. A 31 (2016) 1630025, https://doi.org/10.1142/S0217751X16300258. [ Links ]

87 F. Bopp, La mécanique quantique est-elle une mécanique statistique classique particulière?, Ann. Inst. Henri Poincare´ 15, (1956) 81. [ Links ]

88 A. Maireche, New Relativistic Bound States for Modified Pseudoharmonic Potential of Dirac Equation with Spin and Pseudo-Spin Symmetry in One-electron Atoms, Afr. Rev Phys. 12 (2017) 130, http://lamp.ictp.it/index.php/aphysrev/article/view/1533/564. [ Links ]

89 A. Maireche, New Relativistic Atomic Mass Spectra of Quark (u, d and s) for Extended Modified Cornell Potential in Nano and Plank’s Scales, J. Nano-Electron. Phys. 8 (2016) 01020, https://doi.org/10.21272/jnep.8(1).01020. [ Links ]

90 A. Maireche, A New Relativistic Study for Interactions in One-electron atoms (Spin / Particles) with Modified Mie-type Potential J. Nano- Electron. Phys. 8 (2016) 04027. https://doi.org/10.21272/jnep.8(4(1)).04027. [ Links ]

91 A. Maireche, Investigations on the Relativistic Interactions in One-Electron Atoms with Modified Yukawa Potential for Spin 1/2 Particles Int. Front. Sci. Lett. 11 (201) 29, Doi:https://doi.org/10.18052/www.scipress.com/IFSL.11.29 [ Links ]

92 A. Maireche, Nonrelativistic treatment of Hydrogen-like and neutral atoms subjected to the generalized perturbed Yukawa potential with centrifugal barrier in the symmetries of noncommutative Quantum mechanics, Int. J. Geom. Methods Mod. Phys. 17 (2020) 2050067, https://doi.org/10.1142/S021988782050067X. [ Links ]

93 A. Maireche, A Recent Study of Excited Energy Levels Diatomics for Modified more General Exponential Screened Coulomb Potential: Extended Quantum Mechanics, J. NanoElectron. Phys. 9 (2017) 03021, https://doi.org/10.21272/jnep.9(3).03021. [ Links ]

94 A. Maireche, A new study of energy levels of hydrogenic atoms and some molecules for new more general exponential screened Coulomb potential, Open Acc J Math Theor Phy. 1 (2018) 232, Doi:10.15406/oajmtp.2018.01.00040 [ Links ]

95 A. Maireche, Effects of two-dimensional noncommutative theories on bound states Schrödinger diatomic molecules under¨ New modified Kratzer-type interactions, Int. Lett. Chem. Phys. Astron. 76 (2017) 1, https://doi.org/10.18052/www.scipress.com/ILCPA.76.1 [ Links ]

96 R.L. Greene and C. Aldrich, Variational wave functions for a screened Coulomb potential, Phys. Rev. A 14 (1976) 2363, https://doi.org/10.1103/PhysRevA.14.2363. [ Links ]

97 M. Badawi, N. Bessis and G. Bessis, On the introduction of the rotation-vibration coupling in diatomic molecules and the factorization method, J. Phys. B 5 (1972) L157, https://doi.org/10.1088/0022-3700/5/8/004. [ Links ]

98 W.C. Qiang, and S.H. Dong, Analytical approximations to the solutions of the Manning-Rosen potential with centrifugal term, Phys. Lett. A 368 (2007) 13, https://doi.org/10.1016/j.physleta.2007.03.057. [ Links ]

99 F.A. Serrano, X.Y. Gu and S.H. Dong, Qiang-Dong proper quantization rule and its applications to exactly solvable quantum systems, J. Math. Phys. 51 (2010) 082103, https://doi.org/10.1063/1.3466802. [ Links ]

100 S.-H. Dong, W.-C. Qiang, G.-H. Sun and V.B. Bezerra, Analytical approximations to the l-wave solutions of the Schrödinger equation with the Eckart potential, J. Phys. A: Math. Theor. 40 (2007) 10535, https://doi.org/10.1088/1751-8113/40/34/010. [ Links ]

101 Y. Zhang, Approximate analytical solutions of the KleinGordon equation with scalar and vector Eckart potentials, Phys. Scr. 78 (2008) 015006, https://doi.org/10.1088/0031-8949/78/01/015006. [ Links ]

102 S. Medjedel and K. Bencheikh, Exact analytical results for density profile in Fourier space and elastic scattering function of a rotating harmonically confined ultra-cold Fermi gas, Phys. Lett. A 383 (2019) 1915, https://doi.org/10.1016/j.physleta.2019.03.021 [ Links ]

103 K. Bencheikh, S. Medjedel and G. Vignale, Current reversals in rapidly rotating ultracold Fermi gases, Phys. Rev. A 89 (2014) 063620, https://doi.org/10.1103/PhysRevA.89.063620. [ Links ]

104 K.P. Gnatenko, Composite system in noncommutative space and the equivalence principle, Phys. Lett. A 377 (2013) 3061, https://doi.org/10.1016/j.physleta.2013.09.036. [ Links ]

Received: May 06, 2021; Accepted: July 07, 2021

Creative Commons License This is an open-access article distributed under the terms of the Creative Commons Attribution License