1. Introduction

During the past two decades, considerable efforts have been made by many researchers in various fields of physics and chemistry to reach relativistic and nonrelativistic solutions using many potentials by adopting different methods such as the Nikiforov-Uvarov method [¹], the Wentzel-KramersBrillouin method [²], the proper quantization rule [³], and the exact quantization rule [⁴], in addition to many other methods. The exact solutions of the fundamental equations are only possible in some exceptional cases like the harmonic oscillator and the Hydrogen atom as two typical models. As for most of the cases treated by researchers, it is done by using approximations and numerical methods such as the Pekeris approximation [⁵], the Greene and Aldrich approximation [⁶], the good approximation proposed by B.H. Yazarloo et al. in the study of the oscillator strengths based on the Mobius square potential under Schrödinger equation [⁷] and the new approximation of the centrifugal term proposed C. S. Jia et al. [⁸]. The Manning-Rosen potential has been of relevant interest in recent years as it can be applied to various fields such as atomic, condensed matter, particle, and nuclear physics in both relativistic and non-relativistic regimes [^9-12]. Furthermore, it is used to describe the vibrations of diatomic molecules such as HCl, CH, LiH, CO, NO, O2, I2, N2, H2, and Ar2 [^13,14]. Many authors have studied Manning-Rosen potential in the nonrelativistic case, in both the s-wave and lwave cases (see for example [^15-18]. Furthermore, this potential was also studied in the relativistic Klein-Gordon, and Dirac equations [^19-23].

As a result of several considerations and many physical problems apparat at the level of the non-renormalizable of the electroweak interaction, quantum gravity, string theory, where the idea of non-commutativity resulting from properties of deformation of space-space (W. Heisenberg in 1930 is the first to suggest the idea and then it was developed by H. Snyder in 1947) was one of the major solutions to these problems. In the past two decades, in particular, it has attracted a great attention [^24-35].

The main objective of this work is to develop the study of B.J. Falaye et al., A.I. Ahmadov et al., and Z. H. Chen et al. [^13,18,19] within the framework of the Klein Gordon and Schrödinger equations. But in the context of symme- tries of noncommutative quantum mechanics for the purpose to get more investigation in the microscopic scales and from achieving more scientific knowledge of elementary particles in the field of nano-scales. The relativisti energy levels under the modified Manning-Rosen potential have not been obtained yet in the context of the RNCQM and NRNCQM. Furthermore, we hope to find new applications and profound physical interpretations using a new, updated model of the modified Manning-Rosen potential, which takes the form:

And

where the parameter β relates to the potential range while A and á are two dimensionless parameters, and 1/α is its range and r is the distance between the two particles. The coupling LΘ→→ equals L _x Θ₁₂ + L _y Θ₂₃ + L _z Θ₁₃, where L _x , L _y , and L _z are the usual components of angular momentum operator L→ in RQM while the new noncommutativity parameter Θ _µν equals θ _µν /2. The new structure of (RNCQM) based on new covariant noncommutative canonical commutations relations (CNCCRs) in Schrödinger , Heisenberg and interactions pictures (SP, HP, and IP), respectively, as follows [^36-46]:¨

We generalize the CNCCRs to include HP and IP. It should be noted that, in our calculation, we have used the natural units c= ℏ = 1. Here ℏeff=ℏ1+Trθθ-/4 ℏ is the effective Planck constant and here θ- observed in the NC commutator

and θ ^µν = ε ^µν θ (θ is the non-commutative parameter), which is an infinitesimals parameter if compared to the energy values and elements of antisymmetric 3 × 3 real matrix and δ _µν is the identity matrix. The symbol (∗) denotes the Weyl Moyal star product, which is generalized between two ordinary functions f(x)g(x) to the new modified form f̂(x̂)ĝ(x̂) ≡f(x) *g(x) in the symmetries of RNCQM as follows [^47-55]:

The indices µ,ν ≡ 1,3¯ 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 space-space noncommutativity properties. Furthermore, the new unified two operators ξμH^t=xμ^orpμ^t and ξμI^t=xμI^orpμI^t in HP and IP are depending on the corresponding new operators ξμH^t≡xμ^~\txor~pμ^ in SP from the following projection relations, respectively:

and

where T = t − t ₀ and the three unified coordinates ξμS≡xμorpμ, ξμHt≡xμorpμt and ξμIt≡xμI  or  pμIt are represented in three relativistic quantum mechanics pictures, while the dynamics of new systems dξH^ (t)/dt are described from the following motion equations in the modified Heisenberg picture as follows:

The operators (Ĥ _or ^mp and Ĥ _r ^mp ) are the free and global Hamiltonian for equal vector scalar Manning-Rosen potential while (Ĥ _nc−or ^mp and Ĥ _{nc− r} ^mp ) the corresponding Hamiltonians for the modified Manning-Rosen potentials. The present investigation aims at constructing a relativistic noncommutative effective scheme for the modified Manning-Rosen potential.

The paper is sketched in six sections. The rest of the five sections is organized as follows: We briefly review the usual relativistic Klein-Gordon equation with equal vector scalar Manning-Rosen potentials in the next section. Section 3 is devoted to the solutions of the deformed Klein-Gordon equation with the modified equal vector and scalar equal vector scalar ManningRosen potentials using Bopp’s shift method and improved approximation of the centrifugal term to obtain the corresponding effective potential in addition to the standard perturbation theory in the first-order in the noncommutativity parameters (Θ,σ,χ) we find the expectation values of some radial terms. Section 4 is reserved to present the new main global energy shift and the global energy spectra of the molecular physics such as (HCl, CH, LiH, CO, NO, O2, I2, N2, H2, and Ar2, and HCl) produced studied potential in the RNCQM symmetries. In Sec. 5, we apply our study for determining the energy spectra under this potential in the nonrelativistic noncommutative quantum mechanics (NRNCQM). Finally, we conclude this paper in Sec. 6.

2. Revised of the eigenfunctions and the energy eigenvalues for equal vector scalar Manning-Rosen in RQM

To achieve the main objective of the current study of finding solutions of deformed Klein-Gordon equation (KGE) in the RNCQM symmetries under the modified equal vector and scalar modified Manning-Rosen potential, it is helpful for the reader to see solutions in RQM. The equal vector and scalar modified Manning-Rosen potential [¹⁹] is given as:

and

To achieve this goal of our current research it is useful to make a summary for the Klein?Gordon equation KGE threedimensional relativistic quantum mechanics:

The vector potential V _mp (r) due to the four-vector linear momentum operator A ^µ (V _mp (r),A→ = 0) and the space-time scalar potential S _my (r) whereas the interaction of scalar and vector bosons are considering by usual substitutions (M → M + S _mp and p ^µ → p ^µ − A ^µ ), E _nl is the relativistic energy eigenvalues, ∇→ is the ordinary 3-dimensional Nabla operator while (n = 0,1,2,... and l) are represents the principal and orbital quantum numbers, respectively. Since equal vector scalar,Manning-Rosen potential has spherical symmetry, allowing the solutions of the time-independent KGE of the known form Ψ(r,θ,φ) = (U _nl (r)/r)Y _l ^m (θ,φ) to separate the radial U _nl (r) and angular Y _l ^m (θ,φ) parts of the wave function, thus Eq. (8) becomes:

Using the shorthand notation

And Eeffmp≡M2-Enl2, we obtain the following second-order Schrödinger -like equation:¨

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:

The Ref. [¹⁹] gives the total wave function and the corresponding energy eigenvalues ^E _nl of the KGE with equal scalar and vector scalar and vector modified Manning-Rosen potential as follows:

and

where s = e ^−r/β , λ _nl ≡ βM2-Enl2,ϑnl=4ηnlαα-1+2l+12 , η _nl = (E _nl + M/M), N _nl is a normalization constant and ₂ F ₁(−n,2λ _nl + ϑ _l + n + 1;1 + 2ε _nl ,s) are the hypergeometric polynomials. From the definition of Jacobi polynomials [⁵⁶]:

In terms of the definition of Jacobi polynomials, Eq. (12) can be written as:

3.The solution of DKGE under modified Manning-Rosen potential in RNCQM

The beginning of this section is devoted to reformulate the Manning-Rosen potential in the relativistic noncommutative quantum mechanics symmetries (RNCQM). We achieve this goal by rewriting the KGE by applying the notion of the Weyl-Moyal star product introduced previously (see Eq. (3)) on the differential equation that satisfied by the radial wave function U _nl (r) in the second section (see Eq. Eq. (9)); thus, the radial wave function U _nl (r) in the RNCQM symmetries becomes as follows [^56-67]:

It is known to the specialized physicists that F. Bopp was the first to propose pseudo-differential operators obtained from a symbol by the quantization rules (x → x _nc = x − (i/2)∂/∂p) and (p → p _nc = p − (i/2)∂/∂x) instead of the ordinary correspondence x → x and (p → (i/2)∂/∂x); the latter are known as Bopp’s shifts and the quantization procedure is the so-called Bopp quantization [^55,68-70]. This method has attracted the attention of many researchers and is used as an alternative to the complicated star product calculations. As a consequence, we can rewrite the deformed Schrödinger equation, deformed Klien-Gordon equation, and deformed Dirac equation with the notion of star product to the Schrödinger equation, Klien-Gordon equation, and Dirac equation with the notion of ordinary product, respectively. This useful simplification can be achieved through reformulating the new algebraic relations which are known as noncommutative canonical commutation relations in the symmetries of relativistic noncommutative quantum mechanics with star product in Eqs. (2) and (3.1) and (3.2) without the notion of star product as follows (see, e.g., [⁵⁶,⁵⁹,⁶¹,⁶²]:

The generalized positions and momentum coordinates: x^μS,p^μS, x^μH,p^μHt in the symmetries of RNCQM are defined in terms of the corresponding coordinates xμS,pμS, xμH,pμHt and xμI,pμIt in the symmetries of RQM via, respectively [^39-49]:

This allows us to find the operator r _nc ² = r ² − LL→ΘΘ in the symmetries of RNCQM [^59-61]. It is convenient to introduce a shorthand notation which will save us a lot of writing r _nc → rˆthe previously relation reduced to the rˆ² = r ² −LL→ΘΘ. According to the Bopp shift method, Eq. (17) becomes similar to the following like the Schrödinger equation (without the notions of star product):

The new operators Vmpr_and1/r2 are expressed as in RNCQM symmetries as follows:

and

Consequently, we can rewrite:

Moreover, to illustrate the above equation in a simple mathematical way and attractive form, it is useful to enter the following symbol Vnc-effmpr, thus the radial Eq. (20) becomes:

with

where Vpertmpr is given by the following relation:

It becomes obvious that the radial modified differential equation obtained in Eq. (22) cannot be solved analytically for any state l 6= 0 because of the centrifugal term. The effective perturbative potential, given in Eq. (24), has a strong singularity r → 0; we need to use the new approximation of the centrifugal term proposed by C. S. Jia et al. [⁸] for a short-range potential, an excellent approximation to the centrifugal term. Unlike the following new approximation used in the previous work in ordinary quantum mechanics [⁸,²⁰,²¹,²³]:

where ω is an adjustable dimensionless parameter. This allows us to obtain:

which after straightforward calculations we obtain ∂V _my (r)/∂r as follows:

with (1/β) = δ. The above equation can be simplified to the following form :

with

and

Obviously, Eq. (25.1) cannot be determined from 1/r. Therefore, we must use the improved approximation of the centrifugal term proposed by Badawi et al. [⁷¹]; this method proved its power and efficiency when compared with the Greene and Aldrich approximation for a short-range potential [⁶]. Unlike the following approximation used in the previous work in QM and NCQM [¹⁷,¹⁸,⁵⁹,⁶⁰,⁶²,⁷²] :

This allows us to obtain:

The approximation (25.1) reduces to Eq. (28.1) when the adjustable dimensionless parameter ω = 1. Inserting Eqs. (25.2), (27), (28.2) into Eq. (24) allows us to obtain the perturbed effective potential in the symmetries of RNCQM as follows:

The Manning-Rosen potential is extended by including new terms proportional to the radial terms

and

to become the modified Manning-Rosen potential in RNCQM symmetries. Obviously, the newly generated effective potential Veffmpr for the modified Manning-Rosen potential is also proportional to the infinitesimal vector ΘΘ, allowing us to consider it as a perturbation potential compared with the parent potential operator Vpertmpr in the symmetries RNCQM, that is, the inequality Vpertmpr≪Veffmpr has become achieved. In other words, all the physical justifications for applying the timeindependent perturbation theory become satisfied. Now, we apply the perturbative theory, in the case of RNCQM, we find the expectation values of the radial terms

and

taking into account the wave function of Manning-Rosen potential which we have seen previously in the second section. After straightforward calculations, we obtain the following expectations values:

and

We have used useful abbreviations ⟨n,l,m|B̂|n,l,m⟩ ≡ ⟨B̂⟩ _(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

We have s = exp(−δr) (with δ = 1/β), implying dr = −(1/δ)(ds/s). After introducing a new variable z = 1 − 2s, we have s = (1−z)/2, dr = (1/δ)(dz/1−z) and 1−s = (z+1)/2. From the asymptotic behavior of s = exp(−δr) and z = 1−2s,

when r → 0 (z → −1) and r → +∞ (z → 1) this allows the reformulation of Eqs. (30, i=1,4¯) as follows:

and

For the ground state n = 0, we have Pn=02λ0l,ϑ0lz=1, thus the above expectation values in Eqs. (31, i=1,4¯) are reduced to the following simple form:

and

where λ0l≡βM2-E0l2=Aη0l-ωll+1/1+ϑ0l, ϑ0l=4η0lαα-1+2l+12 and η _0l = (E _0l + M)/M

Comparing Eqs. (31, i=1,3¯) with the integral of the form [⁷³]:

A direct calculation gives the expectation values in Eqs. (34. i=1,3¯). Namely,

where ρ0l equal 2λ0l+ϑ0l For the first excited state n= 1, the Jacobi polynomial reduced to Pn=12λ1l,ϑ1lz=Ω1l+Ξ1l1-z, here Ω _1l =ϑ _1l + 1, Ξ1l=-2λ1l+ϑ1l+2 , with λ1l≡βM2-E1l2=-3/2ϑ1l-2+Aη1l-ωll+1/3+ϑ1l, ϑ1l=4η1lαα-1+2l+12, η1l=E1l+M/M. Thus, the expectation values in (33 i=1,6¯). Are reduces to the following simple form:

and

where the 12-factors Tiji=1,4¯,j=1,3 are given by:

and

By using the integral formula in Eq. (33) we obtain the analytical expressions of the 12-factors Tiji=1,4¯,j=1,3 as follows:

and

with ^ω _1l ≡ 2 ^λ _1l + ^ϑ ₁ l. The substitution of Eqs. (37.1) (37.2), (37.3) and (37.4) into Eqs. (35.1), (35.2) (35.3), and (35.4) gives the expectation values in the first excited state (1,l,m):

and

The relativistic study of the modified equal vector scalar Manning-Rosen potential is divided into three principal parts. The first one is devoted to studying the spin-orbit effect generated by the noncommutativity space-space. This is achieved by replacing the coupling of the angular momentum operator with noncommutativity coupling L→  Θ→ by the new equivalent coupling ΘΘ→  S → (with Θ = (Θ² ₁₂ + Θ² ₂₃ + Θ² ₁₃) ^1/2 ). We have oriented the spin vector of the diatomic molecules such as HCl, CH, LiH, CO, NO, O2, I2, N2, H2, and Ar2 to the direction of the vector Θ→ under modified equal vector scalar Manning-Rosen potential. Then we replace it with the corresponding value: (Θ/2) Θ/2  J2→-  L2→-  S2→ Furthermore, in quantum mechanics, the operators H^nc-rmp, J2, L2, S2, and J _z ) forms a complete set of conserved physics quantities, the eigenvalues of the operator  J2→-  L2→-  S2→ are equal the values 2k(l) ≡j(j+1) −l(l+1) −s(s+1), with |l−s| ≤j≤ |l+s|. Consequently, the energy shift Δ E^mypso n= 0, Θ, j , l, s   ≡ Δ E^mypso 0, Θ, j , l, s  and Δ E^mypso n= 1, Θ, j , l, s   ≡ Δ E^mypso 1, Θ, j , l, s  due to the perturbed effective potential produced ^V _pert ^mp (r) for the ground state and the first excited state, respectively, in RNCQM symmetries as follows:

where the global expectation value X0,l,mR is determined from the following expression:

While X1,l,mrmp=X1→0,l,mrmp. We can now generalize the above results ΔEmpSOn,Θ,j,l,s to the case of n ^th excited states in RNCQM symmetries as follows:

Thus, we can express the general expectation value as follows: Xn,l,mrmp

The second main part in the relativistic study of the modified equal vector scalar Manning-Rosen potential is corresponding to replace both (L→Θ→ and Θ₁₂) by (σℵL _z and σℵ), respectively, here ℵ and σ are, respectively symbolize the intensity of the induced magnetic field by the effect of deformation of space-space geometry and a new infinitesimal noncommutativity parameter, so that the physical unit of the original noncommutativity parameterD E Θ₁₂ (length)² is the same unit of σℵ, we have also need to apply n,l,mLz^n',l',m'=m'δnn'δll'δmm' (with − (l,l′) ≤ (m,m′) ≤ + (l,l′)). All of this data allows for the discovery of the new energy shift ΔEmpmn=0,σ,l,m≡Emym0,σ,l,m and ΔEmpmn=1,σ,l,m≡Emym1,σ,l,m due to the perturbed Zeeman effect created by the influence of the modified equal vector scalar Manning-Rosen for the ground state and the first excited state in (RNC: 3D-RS) symmetries as follows:

Thus, we can generalize the above particular cases to the general case ∆E _mp ^m (n,σ,l,m) of the modified equal vector scalar Manning-Rosen potential which corresponds to the n ^th excited states in (RNC: 3D-RS) symmetries as follows:

Now, for our purposes, we are interested in finding a new third automatically important symmetry for modified equal vector scalar Manning-Rosen potential at zero temperature in RNCQM symmetries. This physical phenomenon is induced automatically from the influence of a perturbed effective potential Vpertmpr, which we have seen in Eq. (31). We discover these important physical phenomena when our studied system consists of N non-interacting is consider as Fermi gas, it is formed from all the particles in their gaseous state under rotation with angular velocity ^Ω~ if we make the following two simultaneous transformations to ensure that previous calculations are not repeated:

Here χ is just infinitesimal real proportional constants. We can express the effective potential Vpert-rotmpr, which induced the rotational movements under the effect of modified equal vector scalar Manning-Rosen potential at zero temperature for the diatomic molecules as follows:

To simplify the calculations without compromising physical content, we choose the rotational velocity Ω→ = Ωe _z . The next step is to transform the spin-orbit coupling to the new physical phenomena as follows:

with

All of this data allows for the discovery of the new energy shift ΔEmpfn=0,χ,l,m≡ΔEmpf0,χ,l,m and ΔEmpfn=1,χ,l,m≡ΔEmpf1,χ,l,m due to the perturbed Fermi gas effect which generated automatically by the influence of the modified equal vector scalar Manning-Rosen potential for the ground state and the first excited state in RNCQM symmetries as follows:

and

Thus, we can generalize the above particular cases to the general case which correspond to the n ^th excited states in RNCQM symmetries as follows:

It is worth mentioning that K. Bencheikh et al. [^74,75] studied 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 χg(s)Ω→L→ was added to the Hamiltonian operator. In contrast to our case, where this rotation term automatically appears due to the large symmetries resulting from the deformation of the space-phase.