This study rigorously investigates the improved exponential Kratzer–Feus potential (IEKFP) model combined with exponential Kratzer–Feus potential (EKFP) and other terms produced by the effect of phase–space deformation [Formula: see text]. The new energy equation in three-dimensional non-relativistic non-commutative phase–space (3D(NR-NCPS)) symmetries is obtained using the parametric generalized Bopp’s shifts method and standard independent time perturbation theory of hydrogen-related molecules (CH, H 2 , NO, HCL, and LiH) diatomic molecules. This is achieved by applying a Green–Aldrich approximation scheme to the centrifugal terms. The new non-relativistic energy equation for the IEKFP in the presence of deformation phase–space is dependent on the discrete atomic quantum numbers ([Formula: see text] and [Formula: see text], the dissociation energy [Formula: see text], the equilibrium bond length [Formula: see text], the screening parameter [Formula: see text], and the deformation phase–space parameters ([Formula: see text]. The new resulting energy equation is utilized to calculate the partition function, from which thermodynamic properties such as mean energy, specific heat capacity, entropy, and free energy are derived in both three-dimensional non-relativistic quantum mechanics 3D(NR-QM) and (3D(NR-NCPS)) symmetries. This study has multiple applications in various domains, including atomic and molecular physics. We have indicated that our work deals with the study of the possibility of the effect of various phase–space deformations on various physical values such as energy and various thermodynamic properties under the influence of exponential Kratzer–Feus potential. So we completed the study of Amadi et al.

In this paper, we examine the bound state solutions of the three-dimensional deformed Klein-Gordon equation (3D-DKGE) for an improved deformed Deng-Fan potential in generalized form along with an improved class Yukawa potential (I(DGDF-YCP)). The standard perturbation theory and Bopp's shifts method are employed, utilizing a proper correspondence to the centrifugal potential term. We obtain new energy eigenvalues for all atomic quantum numbers (j,l,s, and m) quantum states. In addition, we found that the new energy is related to the potential depths (V0,V01,V02), the adjustable constants (q,alpha,b,c), the deformation parameter q and the noncommutativity parameters (Phi, chi, and zeta). We noticed that the energy value reflects its known value in the literature (absence of deformation), in addition to an additional correction resulting from the effect of space properties, which clearly revealed the effect of spin-orbit and intrinsic magnetic effects resulting from the movement of electrons, as these effects appeared automatically. The analysis is performed on the nonrelativistic limit of new energy spectra under I(DGDF-YCP). By appropriately adjusting the combined potential parameters (V0,V01,V02), we analyze the obtained new bound state eigenvalues of the 3D-DKGE and three-dimensional deformed Schr & ouml;dinger equation (3D-DSE) with the I(DGDF-YCP) in 3D-(R/NR)NCQs symmetries and obtain the modified Yukawa potential, the modified class of Yukawa potential, the modified Deng-Fan Yukawa potential model and the modified Deng-Fan potential model. The new analytical energy expression was used to predict the spectral masses (M-qQ(nc) and M-qq(nc)) of the heavy-light mesons (cs, cq, bs, and bq) and heavy quarkonia such as charmonium cc and bottomonium bb in 3D-(NR)NCQs regimes. We found that the spectral masses consist of their fundamental terms (MqQ and Mqq) in the framework of 3D-(NR)Qm regimes plus the impact of spatial deformation. Within the framework of the 3D-(NR)NCQs regime, the homogenous diatomic molecules (H2, I2); the heterogeneous diatomic molecules (CO, HCl, LiH); the neutral transition metal hydrides (ScH, TiH, VH, CrH); the transition-metal lithide CuLi; the transition-metal carbides (TiC, NiC); the transition metal nitrite ScN and the transition metal fluoride ScF under I(DGDF-YCP) under I(DGDF-YCP) models are examined.

The impact of deformation space on the physical characteristics of diverse physics systems has been thoroughly investigated in research papers. In this work, we study the deformed Klein-Gordon equation (DKGE) in the three-dimensional relativistic non-commutative quantum space (3D-RNCQS) regime by using the improved Hua plus modified Eckart potential (IHPMEP) model. For this consideration, the DKGE in the 3D-RNCQS regime is solved using the standard perturbation theory and the well-known Bopp's shifts method with the Greene-Aldrich approximation to the centrifugal barrier. The new relativistic energy equation and eigenfunction for the IHPMEP in the presence of deformation space-space for the heterogeneous (CO, HF, and NO) and homogeneous (N-2, H-2, and Li-2) diatomic molecules are obtained to be sensitive to the atomic quantum numbers (j,l,s, and m), the mixed potential depths (V-0,V-1,V-2, and V-3), the inverse of the screening parameter alpha, and non-commutativity parameters (Phi, chi, and zeta). Analysis is performed on the non-relativistic limit of new energy spectra. By appropriately adjusting the combined potential parameters, we analyze the obtained new bound state eigenvalues of the DKGE and deformed Schr & ouml;dinger equation with the IHPMEP in 3D-NCQS symmetries and obtain the new modified Eckart potential, the modified Hua potential, the modified Morse potential, and the modified P & ouml;schl-Teller potential. Within the framework of the 3D-NRNCQS regime, the homogeneous and heterogeneous composite systems under IHPMEP models are examined. A thorough investigation is carried out into the impact of space-space deformation on the thermal parameters of the IHPMEP, including the partition function, mean energy, free energy, specific heat, and entropy. This work is of a fundamental absorbability nature and pedagogical interest in atomic and molecular physics.

This work proposes an improved linear plus modified Yukawa potential (I(L-MYP)) model in the framework of three-dimensional relativistic and non-relativistic noncommutative quantum space (3D-(R/NR)NCQs) regimes and determine its complete eigenvalue solutions based on the corresponding solutions within the framework of the symmetries of relativistic and relativistic commutative quantum mechanics (3D-(R/NR)Qm) found in the literature. The new energy spectrum in relativistic and non-relativistic regimes is obtained using the parametric Bopp shift method and standard perturbation theory. The new (R/NR) energy equation I(L-MYP) model in the presence of deformation space-space depends on atomic quantum numbers (j,l,s and m), the mixed potential depths (A1, A2, A3 and A4), the inverse of the screening parameter alpha, and non-commutativity space-space parameters (theta, chi and zeta). We recovered various potentials such as the modified scalar-vector Yukawa potential, the new modified Yukawa potential and the deformed class of Yukawa potentials from I(L-MYP) model. We also obtained a new energy spectrum in 3D-(R/NR)NCQs for all the deduced potentials from I(L-MYP) model. The new non-relativistic eigenvalue solution of the I(L-MYP) model is then applied to obtain the modified masses MqQ of the heavy-light mesons (cs, cq, bc and bq) in the quarkonium system in 3D-(NR)NCQs regimes. If we ignore all the deformations of space-space (theta,chi,zeta)->(0,0,0), it is natural that all the physical values obtained will be identical to their corresponding values known in 3D-(NR)NCQs regimes.

Using two physical models, we investigate the three-dimensional deformed Schrödinger equation (3D-DSE) in the context of three-dimensional non-relativistic noncommutative quantum phase-space (3D(NR)-NCPS) symmetries. The Mobius square potential is what they have in common. The first is the improved modified Mobius square plus Eckart potential (IMMSEPM), and the second is the improved Mobius square plus screened-Kratzer potential (IMSSKPM). This study solves the 3D-DSE in the 3D(NR)-NCPS regime using the well-known generalized Bopp’s shifts method and conventional perturbation theory. For the homogeneous (K2 and H2) and heterogeneous (LiH, HCl, HF, DF, and CsO) diatomic molecules, the new non-relativistic energies equations and eigenfunction for the IMMSEPM and the IMSSKPM models in the presence of deformation phase-space are obtained to be sensitive to the atomic quantum numbers (j, l, s, m), the mixed potential depths (V0, V1, V2, A, B) and (A, B, De, re), the screening parameter α, and non-commutativity parameters (Θ/Φ, χ/χ, ζ/ζ) for the IMMSEPM and the IMSSKPM, respectively. By appropriately adjusting the improved modified Mobius square potential, the improved Eckart potential, the improved Hulthén potential, and the improved Poschl-Teller potential, we investigate the obtained new bound state eigenvalues of the DSE with the IMMSEPM and the IMSSKPM in 3D(NR)-NCPS symmetries. In addition, the partition function, mean energy, free energy, specific heat, and entropy of the IMMSEPM and IMSSKPM are investigated in 3D(NR)-NCPS symmetries. The current research has many potential applications in atomic and molecular physics.

In this paper, within the three-dimensional non-relativistic non-commutative phase-space (3D-NR(NCPS)) symmetries, we examined the 3D deformed Schr & ouml;dinger equation (3D-DSE) using the improved Coshine Yukawa potential (ICYP) model composed from Coshine Yukawa potential (CYP) (-4dearexp(-alpha r)cosh(alpha r)) and other terms produced from the effect of phase-space deformation (ader3L.Theta+2dea alpha exp(-2 alpha r)r2L.Theta and deaexp(-2 alpha r)r3L.Theta). For this study, the 3D-DSE in the 3D-NR(NCPS) symmetry is solved and discussed with standard independent time perturbation theory and the generalized Bopp's shifts method. For the homogeneous (H2 and N2) and heterogeneous (LiH, ScH, and HCL) diatomic molecules, it is obtained that the improved non-relativistic energy equation and eigenfunction for the ICYP in the presence of deformation phase-space are dependent on the discrete atomic quantum numbers (j,l,s,m), the dissociation energy de, the equilibrium bond length the screening parameter re, the deformation phase parameters (eta,chi,zeta) and the deformation space parameters (Phi,chi,zeta). Additionally, the thermal properties of the CYP and ICYP with 3D Schr & ouml;dinger equation (3D-SE) and 3D-DSE are thoroughly examined in the three-dimensional non-relativistic quantum mechanics (3D-NR(QM)) known in the literature symmetry and 3D-NR(NCPS) symmetries, including the partition function, mean energy, free energy, specific heat, and entropy. Furthermore, we discussed particular cases of thermodynamic characteristics for the ICYP model. In our study, we have shown that all the physical values related to energy and thermodynamic properties within the framework of the 3D(NR)NCPS symmetry are equal to the corresponding values within the framework of 3D-NR(QM) symmetry known in the literature, in addition to minor effects resulting from their interaction with the topological properties of the deformed phase-space. This study has multiple applications in various domains, including atomic and molecular physics.

Within the framework of three-dimensional non-relativistic noncommutative quantum phase-space (3D-NRNCPS) symmetries, we study the three-dimensional deformed Schrödinger equation (3D-DSE) using the improved modified Kratzer plus generalized inverse quadratic Yukawa potential (IMK-GIQYP) and the improved modified screened Kratzer plus inversely quadratic Yukawa potential (IMSK-IQYP) models. For this consideration, the well-known generalized Bopp’s shifts method and standard perturbation theory are used to solve the DSE in the 3D-NRNCPS regime. For the homogeneous (H2, N2 and I2) and heterogeneous (CO, CH and NO) diatomic molecules, the new non-relativistic energy equation and eigenfunction for the IMK-GIQYP and the IMSK-IQYP models in the presence of deformation phase-space are obtained to be sensitive to the atomic quantum numbers ([Formula: see text] and m), the mixed potential depths ([Formula: see text] and V) and ([Formula: see text] and [Formula: see text]), the screening parameters ([Formula: see text] and [Formula: see text]), and non-commutativity parameters ([Formula: see text] and [Formula: see text]) for the IMK-GIQYP and the IMSK-IQYP, respectively. We investigate the newly obtained bound state eigenvalues of the DSE in 3D-NRNCPS symmetries using the IMK-GIQYP and the IMSK-IQYP, with appropriate adjustments made to the improved modified Kratzer potential, improved modified screened Kratzer potential, improved generalized inverse quadratic Yukawa potential model and improved inversely quadratic Yukawa potential model. Additionally, in 3D-NRNCPS symmetries, the thermal properties of the IMK-GIQYP and the IMSK-IQYP, including their partition function, mean energy, free energy, specific heat and entropy, are thoroughly examined. Significant areas, including atomic and molecular physics, find many uses for this study.

Relativistic and non-relativistic quantum mechanics formulated in a non-commutative space-space have recently become the object of renewed interest. In the context of three-dimensional relativistic non-commutative quantum space ( 3D-RNCQS ) symmetries with arbitrary spin–orbit coupling quantum number k , we approximate the solution of the deformed Dirac equation for a new modified Eckart plus Hulthen potentials ( NMEHPs ) for interaction between the core and single nucleon of ( 17 O, 41 Ca, 49 Ca, and 57 Ni) isotopes with one additional nucleon (valence) in the (1 d 5/2 , 2 s 1/2 ), (1 f 7/2 , 2 p 3/2 ), (2 p 3/2 , 1 f 5/2 ) and (2 p 3/2 , 1 f 5/2 ) levels that correspond the ground-state and the first excited energy, respectively. In the framework of the spin and pseudospin ( p -spin) symmetry, we obtain the global new energy eigenvalue, which equals the energy eigenvalue in the usual relativistic QM as the main part plus three corrected parts produced from the effect of the spin–orbit interaction, the new modified Zeeman, and the rotational Fermi term. The new values that we get appear to be sensitive to the quantum numbers (j,k,l,l^p,s,s^p,m,m^p), the mixed potential depths (v_0,v_1,α), and non-commutativity parameters (NP) (Θ ,τ ,χ) . We recovered several potentials, including the NMEHPs . We were able to define intervals for the NP. In addition, we were able to reduce the difference between our new theoretical values and their experimental values by comparing this difference with what is found in the literature.

In this paper, we perform a complete non-relativistic study of the improved class of inversely quadratic Yukawa plus Hulthén potential (ICIQYHP) model in the context of three-dimensional non-relativistic non-commutative quantum phase-space (3D-NRNCPS) symmetries impacted by perturbed spin–orbit interaction and the external magnetic fields for the homogeneous (N 2 and O 2 ) and heterogeneous (CO and NO) diatomic molecules and the heavy meson systems, such as charmonium (cc) and bottomonium (bb) using generalized Bopp’s shifts method and standard perturbation theory with the Greene–Aldrich approximation to the centrifugal barrier. The new non-relativistic energy equation and eigenfunction for the ICIQYHP in the presence of deformation phase–space are obtained to be sensitive to the atomic quantum numbers ([Formula: see text] and [Formula: see text]), the mixed potential depths ([Formula: see text] and [Formula: see text]), the screening parameter [Formula: see text], and non-commutativity parameters ([Formula: see text] and [Formula: see text]). The critical particular cases in 3D-NRNCPS symmetries have been obtained by adjusting the parameters of the ICIQYHP, such as the improved Hulthén potential, the improved Coulomb potential, and the improved inversely quadratic Yukawa potential. We have also studied the spin-averaged mass spectra of the heavy mesons and the thermo-magnetic properties under the class of inversely quadratic Yukawa plus Hulthén potential model in 3D-NRQM and 3D-NRNCPS symmetries. This research can potentially be applied to atomic, condensed matter, nuclear, molecular physics, and chemical physics.

This study examines the 3D deformed Klien-Gordon and Schrodinger equations (DKGE and DSE), taking into account the effect of non-commutativity space-space in the regime. The investigation is done using the improved Eckart-Hellmann potential (IEHP) model. The DKGE and DSE in the 3D-(R/NR) NCQS regime for this consideration are solved using the well-known Bopp's shifts method and standard perturbation theory. For the homogeneous (I-2, N-2, H-2) and heterogeneous (CO, NO, VH, TiH, NiC, TiC, and CuLi) diatomic molecules, the new relativistic and non-relativistic energy equations under the IEHP in the presence of deformation space-space are obtained to be sensitive to the discrete quantum numbers (j, l, s, m), the mixed potential depths (U-0, U-1, U-2, U-3), the screening parameter a, and the non-commutativity parameters (Phi, x, zeta). The non-relativistic limit of new energy spectra is analyzed. We examine the obtained new bound state eigenvalues of the DKGE and deformed Schrodinger equation with the IEHP in 3D-(R/NR) NCQS symmetries by suitable adjustment of the combined potential parameters and get the new modified Hellmann potential, the new modified Eckart potential, the new modified Coulomb potential, and the new modified Yukawa potential. The homogeneous and heterogeneous composite systems under the IEHP model are investigated in the context of the 3D-NRNCQS regime. Under the IEHP model in 3D-NRNCQS symmetries, the influence of space-space deformation on the spin-averaged mass spectra of the heavy mesons, such as charmonium and bottomonium, is examined. Furthermore, the thermal properties such as partition function, mean energy, free energy, specific heat, and entropy physics.

Within the framework of non-relativistic non-commutative phase–space, the new bound state approximate solution of the deformed Schrödinger equation is solved for the improved screened modified Kratzer and a class of Yukawa potential (ISMK-CYP) models using the GBSM and standard perturbation theory. By employing the Greene–Aldrich-type approximation scheme, we have obtained the new explicit energy eigenvalues for the newly proposed ISMK-CYP for the homogeneous (I _2 , N _2 ) and heterogeneous (Hcl, ScH, LiH, CO, NO, and CH) diatomic molecules and the heavy mesons system such as c c and bb . Our new results show that the bound state energy is highly sensitive to the atomic quantum numbers ( j,l,s, and m), the mixed potential depths ( a,b,c,f, and g), the screening parameter ϕ _a , and non-commutativity parameters ( Φ ,χ ,ζ) and ( η ,χ,ζ) . We examine the obtained new bound state eigenvalues of the DSE with the IIQH-KP in 3D-NRNCQPS symmetries by suitable adjustment of the combined potential parameters and get the improved screened modified Kratzer model and the improved class of Yukawa potential model. The influence of the deformation of phase–space on the spin-averaged mass spectra of the heavy–light mesons such as cc and bb under the ISMK-CYP model in 3D-NRNCQPS symmetries was investigated. The vibrational partition function, vibrational mean energy, vibrational mean free energy, vibrational entropy, and vibrational specific heat capacity are among the novel thermodynamic parameters that are evaluated as well. The current study has several applications in various areas, including atomic and molecular physics.

We report the new bound state solutions of novel non-central potentials, the improved Eckart plus an improved class of Yukawa potential (IECYP), in the deformation Klein-Gordon equation by using Bopp's shifts method and standard perturbation theory in the symmetries of three-dimensional relativistic noncommutative quantum space-space (3D-RNCQS). We apply a developed scheme to overcome centrifugal terms such as (r12 and r14). The analytical expression of the energy spectra under the IECYP of (NI2 , K2 , VH, TiH, NiC, TiC, and CuLi) diatomic molecules in 3D-RNCQS and three-dimensional non-relativistic noncommutative quantum space-space (3D-NRNCQS) was obtained to be sensitive to the atomic quantum numbers (j, l, s, m) , the mixed potential depths (V1 , V2 , V3 , V4), the screening parameter's inverse 8 , and non-commutativity parameters (O , x , ? ).We also discuss vari-ous special cases related to our considered combined potentials, which have utility for other physical and chemical systems, and the results are in excellent agreement with the existing literature. We have also studied the effect of deformation in space-space on thermodynamic quantities such as partition function, mean energy, free energy, specific heat, and entropy.

In this study, the solution of the deformed Klein-Gordon and deformed Schrodinger equations (DKGE and DSE for short) for the modified Eckart potential plus a new modified deformed Hylleraas potential (MEPNMDHP, for short) with the improved approximation to deal with the centrifugal term is investigated using Bopp's shift and standard perturbation theory methods in the symmetries of relativistic noncommutative quantum mechanics RNCQM and nonrelativistic noncommutative quantum mechanics NRNCQM. To the best of our knowledge, this problem is examined in literature in usual RQM and NRQM. The new potential suggested describes some selected diatomic molecules such as the homogeneous diatomic molecules (N2 and H2) and heterogeneous diatomic molecules (HCl, HBr, SO, NO, and HI). The new values that we get appeared sensitive to the quantum numbers (݆, ݏ, ݉) in addition to the usual states' numbers (݊, ݈), the potential depths of the potential (ܸ଴,ܸଵ,ܸଶ), the range of the potential ߙ and noncommutativity parameters (߆, ߪ, ߯). We have highlighted three physical phenomena that automatically generate a result of the topological properties of non-commutativity. The first physical phenomenon is the perturbative spin-orbit coupling, the second is the magnetic induction, and the third is the rotational proper phenomenon. 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 ݈ = 0 and ݈ & NOTEQUAL; 0, so the approximate solutions are available. We have observed that the DKGE under the MEPNMDHP has a physical behavior similar to the Duffin-Kemmer equation that can describe the dynamic state of a particle with spin-1 in the symmetries of RNCQM. Four special cases; i.e., l wave is investigated in the context of DKGE and Schrodinger theories. The new relativistic and nonrelativistic energy for some potentials, such as only modified Eckart potential and only new modified Hylleraas potential, have also been obtained by varying some potential parameters.

The deformed Klein-Gordonequation has been solved in three-dimensional extended relativistic quantum mechanics (3D-ERQM) symmetries for the improved modified Yukawa-Kratzer potential (IMYKP) model under the influence of the deformation space-space symmetries. The new relativistic energy eigenvalues were calculated using the parametric Bopp’s shift method and standard perturbation theory in addition to the approximation scheme suggested by Greene and Aldrich for the inverse square terms. The new relativistic energy eigenvalues of (LiH, HCl, CO and H2) molecules under the IMYKP model it was shown to be sensitive to the atomic quantum numbers (j, l, s, m), mixed potential depths (V0, De, re), the screening parameter’s inverse α and noncommutativity parameters (Θ,τ ,χ). In addition, we analyzed the nonrelativistic energy values by applying the well-known transmission rules known in the literature. In addition, we studied many special cases useful to researchers in the framework of the new extended symmetries, such as the improved modified Kratzer potential, the improved generalized Kratzer potential, the improved Kratzer potential, the improved modified Kratzer plus screened Coulomb potential, the improved Hellmann potential, the improved Yukawa potential, and improved inversely square Yukawa potential. We noticed that these particular results are identical to our previous work and other known works in the literature. The study is further extended to calculate the mass spectra of mesons of charmonium (cc) and bottomonium (bb) within the framework of the IMYKP model in three-dimensional extended non-relativistic quantum mechanics (3D-ENRQM) symmetries.

In this work, the modified approximation to the centrifugal barrier term is applied to find approximate bound-state solutions of the deformed Dirac equation for the spin and pseudospin symmetries in a model with the improved hyperbolic Hulth´en and hyperbolic exponential inversely quadratic potentials (IHHEIQPs) using the parametric method of Bopp’s shift and the standard perturbation theory in the extended relativistic quantum mechanics (ERQM). Our results indicate that the new energy eigenvalues are highly sensitive to the potential parameters (ν1, A) and to the values of quantum atomicnumbers (j, k, l, m,̃︀ l, ̃︀ m, s, ̃︀s), range of the potential ν, and noncommutativity parameters (θ, σ, β). We found that the effect of a space-space deformation gives a correction in the energy spectrum, where the main energy term remains due to the effect of the hyperbolic Hulth´en and hyperbolic exponential inversely quadratic potentials known in the literature. The new nonrelativistic energies are obtained by applying the nonrelativistic limit to the relativistic spin-energy equation in the extended nonrelativistic quantum mechanics (ENRQM). The proposed potential model reduces to the improved Hulth´en and exponential inversely quadratic potentials as special cases in ERQM. The present results are applied for calculating the new mass spectra Mhiqpnc−nl of heavy mesons such as cc, bb, bc, bs, cs, and bq,q = (u, d) in ENRQM. It turns out that the values of masses come from the contribution of the mass spectra Mhiqpnl in NRQM, while the effect of a space-space deformation δMhiqpnc−nl is an infinitesimal correction as compared with Mhiqpnl. Our results seem to be significant and agree perfectly with the ones in the literature.

The analytical solutions of the mass-dependent Klien-Gordon equation (KGE) with the improved harmonic oscillator potential and the improved vector quark-antiquark interaction are derived from the symmetries of threedimensional relativistic noncommutative quantum mechanics (3D-RNCQM symmetries) using Bopp's shift method and perturbation theory.The energy state equations are sensitive to the global parameters characterizing the noncommutativity space-space (Θ, σ) and the potential parameter (E nl , a,b, c, m0, k) in addition to the discrete atomic quantum numbers (j, l, s, m).The energy spectra for the mass-dependent Shrödinger equation with improved vector quarkantiquark interaction and harmonic oscillator potential were found using nonrelativistic limit principles.We also used the current findings to determine the heavy-meson masses of charmonium cc and bottomonium bb in both ordinary nonrelativistic quantum mechanics symmetries and three-dimensional nonrelativistic noncommutative 3D-NRNCQM symmetries.

Under the influence of the deformation space-space symmetries, the improved Mobius square plus generalized Yukawa potentials (IMSGYPs) have been employed to solve the deformed Klien–Gordon equation in three-dimensional noncommutative relativistic quantum space (3D-RNCQS) symmetries. Combined with the approximation approach suggested by Greene and Aldrich, we also employ the parametric Bopp’s shift approach and standard perturbation theory to derive novel relativistic energy eigenvalues. The new relativistic energy eigenvalues of (N 2 , K 2 , NI, ScI, and RbH) diatomic molecules under the IMSGYPs were shown to be sensitive to the atomic quantum numbers ([Formula: see text]), the mixed potential depths ([Formula: see text]), the screening parameter’s inverse [Formula: see text] and non-commutativity parameters ([Formula: see text], [Formula: see text], [Formula: see text]). In addition, we analyzed the new non-relativistic energy values in three-dimensional noncommutative non-relativistic quantum space (3D-NRNCQS) symmetries, by applying the well-known mapping in the literature. Furthermore, we studied many special cases useful to researchers in the framework of the new extended symmetries, such as the newly generalized Mobius square potential, the newly generalized Yukawa potential, and the newly generalized Deng-Fan potential. The study is further extended to calculate the mass spectra of mesons of the heavy quarkonium system, such as [Formula: see text], bottomonium [Formula: see text], [Formula: see text] and light mesons [Formula: see text] and [Formula: see text], that have the quark and antiquark flavors within the framework of the IMSGYPs model in 3D-NRNCQS symmetries.

The bound state solutions of the deformed Klien-Gordon equation (DKGE) have been determined in the extended relativistic quantum mechanics ERQM symmetries using the improved spatially-dependent mass Coulomb potential with mixed scalar-vector Coulomb potentials (ISDM-SVCPs) model. The spatially-dependent mass Coulomb potential, as well as a combination of ((1/(r³)) and (1/(r⁴))), are included in the ISDM-SVCPs model, which is coupled with the coupling LΘ, which explains the interaction of the physical features of the system with the topological deformations of space-time. The new relativistic energy eigenvalues for the ISDM-CP have been derived using the parametric Bopp's shift method and standard perturbation theory. Quantum numbers (j,l,s,m), mixed potential depths (q/s_{c},m₀,m₁), and noncommutativity parameters (Θ,τ,χ) seemed to affect the new values we obtained. Within the framework of relativistic extended quantum mechanics, we have addressed certain significant particular instances that we hope will be valuable to the specialized researcher. In DKGE symmetries, we've also looked at the improved pure scalar Coulomb-like potential. The formulation of total energy was also discovered in the context of extended symmetries, which unified the energies of bosonic particles and antiparticles into a single mathematical formula. When the three simultaneous limits (Θ,τ,χ) were applied, we recovered the normal results of relativistic in the literature (0,0,0).

Relativistic and nonrelativistic quantum mechanics formulated in a noncommutative space-space have recently become the object of renewed interest. In the context of extended relativistic quantum mechanics (ERQM) symmetries with arbitrary spin-orbit coupling quantum number [Formula: see text], we approximate to solve the deformed Dirac equation (DDE) for a new suggested new generalized Schiöberg and Manning–Rosen potentials within the generalized (Coulomb and Yukawa)-like tensor interactions (NGSM-GLTs). In the framework of the spin and pseudospin (p-spin) symmetry, we obtain the global new energy eigenvalue which equals the energy eigenvalue in usual relativistic quantum mechanics (RQM) as the main part plus three corrected parts produced from the effect of the spin-orbit interaction, the new modified Zeeman, and the rotational Fermi term, by using the parametric of the well-known Bopp’s shift method and standard perturbation theory using Greene–Aldrich approximation to handle [Formula: see text], [Formula: see text] and other terms in the effective potential. The new values that we got appeared sensitive to the quantum numbers ([Formula: see text]), the mixed potential depths ([Formula: see text],[Formula: see text],[Formula: see text],[Formula: see text],[Formula: see text]), the range of the potential [Formula: see text], and noncommutativity parameters ([Formula: see text],[Formula: see text],[Formula: see text]). We recovered several potentials, including the improved Schiöberg and Manning–Rosen potentials within the improved Yukawa-like tensor interaction, the new Schiöberg and Manning–Rosen potentials within the improved Coulomb-like tensor interaction, the new Schiöberg potential within the improved Yukawa-like tensor interaction, the new Manning–Rosen potential within the improved Yukawa-like tensor interaction, and the new Schiöberg and Manning–Rosen potentials potential problems in the context of nonrelativistic extended quantum mechanics symmetries.

The deformed Klein–Gordon equation has been solved in three-dimensional extended relativistic quantum mechanics (3D-ERQM) symmetries for the newly improved screened Kratzer potential (NISKP) model under the influence of the deformation space–space symmetries. The new relativistic energy eigenvalues were calculated using the parametric Bopp’s shift method and standard perturbation theory in addition to the approximation scheme suggested by Greene and Aldrich for the inverse square terms. The new relativistic energy eigenvalues of (LiH, HCl, CO and H 2 ) molecules under the NISKP model were shown to be sensitive to the atomic quantum numbers ([Formula: see text]), mixed potential depths ([Formula: see text]), the screening parameter’s inverse [Formula: see text], control parameter [Formula: see text] and non-commutativity parameters ([Formula: see text], [Formula: see text], [Formula: see text]). In addition, we analyzed the non-relativistic energy values by applying the well-known transmission rules known in the literature. Furthermore, we studied many special cases useful to researchers in the framework of the new extended symmetries, such as the new screened cosine Kratzer potential, the new screened Kratzer potential (SKP), the new Kratzer potential, the new coshine Yukawa potential, the new coshine Yukawa potential, the new shifted improved SKP and the new shifted SKP. The study is further extended to calculate the mass spectra of mesons of the heavy quarkonium system such as charmonium [Formula: see text], bottomonium [Formula: see text], [Formula: see text] and light mesons [Formula: see text] and [Formula: see text] that have the quark and antiquark flavor within the framework of the NISKP model in 3D-ENRQM symmetries.