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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:
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
While the uncertainty relations will be changed into the following formula in the new symmetries as follows:
With
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
And
Moreover, the dynamics of new systems
Here (
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]:
and
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:
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,
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
where F
nk
(r) and G
nk
(r) represent the upper and lower components of the Dirac spinors Ψ
nk
(r,Ω) while
Using the shorthand notation
We obtain the following second-order Schrödinger-like equation in RQM and NRQM symmetries, respectively:
With
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,
and
For the spin symmetry, the equation of energy is given by:
For the pseudospin symmetry, the equation of energy is given by:
with
and
The Pochhammer symbol is (m)
n
= Γ[m + n]/Γ[m + n], and s = e
−αr
,
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]:
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 x → x − (i/2)(∂/∂p), and p → p+(i/2)(∂/∂x) instead of the ordinary correspondence x → x 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]):
The generalized positions and momentum coordinates
This allows us to find the operator
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):
The new operators
and
Here q = e αre − 1. Therefore, 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
with:
Moreover,
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]:
It is important to mention here that the above approximations are valid in short when αr ≪ 1. This allows us to obtain:
After straightforward calculations (∂V hmp (r)/∂r) we obtain as follows:
Upon invoking the Greene-Aldrich approximation scheme, the expression (E nl + M/r)(∂V hmp (r)/∂r) reduces to the form:
By making the substitution Eq. (32) into Eq. (28), we find the perturbed effective potential
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
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:
with
We can use the method proposed by Dong et al. [100] and applied by Zhang [101], we calculate the integrals in Eqs.
here 2 F 1 (c 1 ,c 2;c 3;s) is the generalized hypergeometric function:
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:
for p = 3 and 1 = 2 while Γ(σ) denoting the usual Gamma function. We obtain from Eqs.
with
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
The global expectation value
The second part is obtained from the magnetic effect of the perturbative effective potential
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
Here χ is just infinitesimal real proportional constants. We can express the effective potential
To simplify the calculations without compromising physical content, we choose the rotational velocity Ω parallel to the (Oz) axis. Then we 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
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
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
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:
The perturbed effective potential in Eq. (33) turns to perturbed effective potential in the symmetries of RNCQM as follows:
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:
Thus, the corresponding global expectation value
The new relativistic energy in Eq. (47) reduces to the new energy
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:
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:
The perturbed effective potential in Eq. (33) turns to perturbed effective potential in the symmetries of RNCQM as follows:
In this case, the additive energy shift under the influence of improved equally mixed generalized Morse potentials in REQM symmetries is given by:
Thus, the corresponding global expectation value is determined from the following expression:
The new relativistic energy in Eq. (47) reduces to the new energy under improved equal scalar and vector Hellmann potentials in RNCQM, as follows:
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:
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
It is worth motioning that Bopp’s shift method permutes to reduce the above equation to the simplest form:
The new Hamiltonian operator for fermionic massive spin 1/2 particles interacting with the relativistic improved Hellmann-generalized Morse potential model is given by:
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:
with
The new operators
and
Substituting Eqs. (65) and (66) into Eqs. (63.1) and (63.2), we obtain:
By comparing (Eqs. (63.1) and (63.2)) and (Eqs. (69.1) and (69.2)), we get an effective deformed spin symmetry
A direct calculation gives:
We apply the Greene and Aldrich approximation to the Eqs. (70.1) and (70.2) to obtain:
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:
Then the recorded results in relations (37. i=1,2,3,4) will turn out to be appropriate to Dirac’s deformed theory:
with
With the help of the special integral that we saw in Eq. (36), we obtain easily:
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
where
and
We have seen that in the previous section that the eigenvalues of the operator
and
The new relativistic energy
and
where
and
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:
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):
From Eqs. (1.2) and (23) we can write this potential in the ENRQM symmetries as follows:
where
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:
Now, we apply the Greene and Aldrich approximation for the centrifugal term in the perturbed potential
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
And by following the same method used in the relativistic study, we obtain the nonrelativistic energy corrections
According to the standard perturbation theory. The new generalized nonrelativistic energy
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
The new relativistic energy in Eq. (92) reduces to the new energy
where
while
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
In this case, the nonrelativistic energy correction under the influence of improved Morse potentials in ENRQM symmetries is given by:
Thus, the corresponding global expectation value
The new nonrelativistic energy in Eq. (92) reduces to the new energy under IHGMP in ENRQM symmetries, as follows:
while
6.4 Study the nonrelativistic fermion cases
We have seen that in the previous section that the eigenvalues of the operator
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]:
with
with
and
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
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 ,σ c ,χ 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,σ 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 ,σ c ,χ 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.
