Arbitrary (k, l) States-Solutions of the Dirac and Schrödinger Equations Interacting with Improved Spatially-Dependent Mass Coulomb Potential with an Improved Coulomb-Like Tensor Interaction Model for H-atoms from 3D-RNCS and 3D-NRNCS Symmetries

The deformed Dirac equation has been investigated, in the context of 3D-relativistic noncommutative space (3D-RNCS) symmetries, using the improved spatially dependent mass Coulomb potential with an improved Coulomb-like tensor interaction (ISDM(CP-CLTI)) model under the conditions of spin symmetry and pseudospin symmetry. The ISDM(CP-CLTI) model is the combining the spatially dependent mass Coulomb potential with the Coulomb-like tensor interaction (CP-CLTI) and the two central terms that are generated to the topological defects of spacespace. Within the confines of the parametric Bopp shift method and conventional perturbation theory, the new relativistic and non-relativistic energy eigenvalues for the hydrogen atoms ( -atoms), such as , and under the ISDM(CP-CLTI) model have been derived. The novel values and that we discovered examined to be dependent on the noncommutativity parameters (NP) , mixed potential depths , and quantum atomic discrete quantum numbers . We have obtained several interesting special examples within the framework of relativistic extended quantum mechanics, which we believe will be of interest to the expert researcher. We were able to retrieve the typical results of relativistic and non-relativistic examples in the literature when we applied the three simultaneous constraints


TION
Among the most important physical facts known to researchers in various fields of physics and chemistry is that systems interacting with strong fields are described by applying the Klein-Gordon (KG) and Duffin-Kemmer-Petiau (DKP) wave equations for particles with spin zero and integer values, respectively.For particles with half spin such as the quark, antiquark, electrons, and positrons, the researchers agreed to apply the well-known Dirac equation in the literature.Furthermore, the Dirac equation (DE) can be generalized to describe nuclear, atomic, and plasma systems.In arbitrary dimensions, the bound states of the KG and Dirac equations with Coulomb-like scalar plus vector potentials have been investigated in many works.Gu et al.  obtained exact solutions for the DE with a Coulomb potential and presented the energy levels and the corresponding fine structure in the generalized (D+1) space-time. 1 In 2003, Dong studied the (D+1)-dimensional DE with the Coulomb potential following the Tricomi equation approach and he showed that the energy levels E(n, l, D) are dependent on the continuous dimension D 2 .In the next year, Ma et al. studied the D-dimensional KG equation with a Coulomb plus scalar potential in higher-dimensional field theory and obtained analytically the eigenfunctions, which are expressed by the confluent hypergeometric functions. 2ong et al. studied the D-dimensional KG equation with a Coulomb potential and analytically obtained and expressed the eigenfunctions by the confluent hypergeometric function. 3 Hamzavi et al. (2010) used an asymptotic iteration method with an arbitrary spin-orbit coupling number to solve the DE for spatially-dependent mass (SDM) Coulomb potentials (see Equation (3)), including a Coulomb-like tensor potential , in the pseudospin symmetry limit, and obtained the energy eigenvalues and corresponding eigenfunctions. 4For any arbitrary spin-orbit state.Ikhdair and Ramazan investigated the ef-fect of a spatially dependent mass function on the solution of the DE with the Coulomb potential in the (3+1), the analytic bound state energy eigenvalues and the corresponding upper and lower two-component spinors of the two Dirac particles were found in closed form using the Nikiforov-Uvarov (NU) approach in the context of spin and pseudospin symmetry. 5Ikhdair calculated the exact bound-state energy eigenvalues by studying the effect of spatially dependent mass functions on the solution of the Klein-Gordon (KG) equation in the (3+1)-dimensions for spinless bosonic particles with mixed scalar-vector Coulomb-like field potentials and masses that are directly proportional and inversely proportional to the distance from the force center.The NU approach is also used to obtain the related KG wave functions for mixed scalar vector and pure scalar Coulomb-like field potentials. 6Adorno et al. studied the relativistic energy levels of a hydrogen-like atom under DE with Coulomb field in the framework of -modified, due to space non-commutativity, they showed the degeneracy of levels 2S 1/2 , 2P 1/2 , and 2P 3/2 is lifted fully in the non-commutative space (NC), allowing new transition channels to emerge. 7Kupriyanov calculated the hydrogen atom spectrum on curved noncommutative space defined by commutation relations (see Eq. ( 5)) and demonstrated that the noncommutativity contribution appears as a correction to the fine structure. 8Chair and Dalabeeh used both the second-order correction of perturbation theory and the exact computation due to Dalgarno-Lewis to compute the second-order noncommutative Stark effect in the ground-state energy of the H-atoms in noncommutative space in an external electric field, and derived the sum rule of the mean oscillator strength. 9Very recently, Nagiyev et al. (2022), in the context of position-dependent mass, extended an exactly solvable model of a nonrelativistic quantum linear harmonic oscillator to the case where an external homogeneous gravitational field is applied. 10Within the noncommutative quantum electrodynamics theory, Chaichian et al. calculated the energy levels of the hydrogen atom as well as the Lamb shift, 11,12 in addition to other works which treated the hydrogen atom-influenced Coulomb potential with constant mass in noncommutative quantum mechanics (NCQM) symmetries. 13,146][17] The physical consequences of noncommutativity NC of space coordinates ( and ) have attractive to the attention of specialized researchers to justify the rigorous study of noncommutative versions of quantum field theory and ordinary quantum mechanics.The geometrical characteristics of the space-time under consideration are influenced by the presence of phase-space deformation.As a result, the topological properties influence a quantum system's physical characteristics, including the relativistic and non-relativistic energy eigenvalues that are the subject of this research.Because the eigenvalue solutions are changed by the topological defects in comparison to the results obtained in flat space, it is vital and significant to investigate quantum mechanical systems in the non-relativistic and relativistic limits under these defects. Seiberg and Witten 26 obtained a new version of gauge fields in noncommutative gauge theory by extending earlier ideas on the advent of noncommutative geometry in string theory with a nonzero B-field.One of the potential goals of NC deformation of space-space and phasephase 27 is to eliminate the observed unwanted divergences or infinities that appear to cause short-range in field theories such as gravitational theory by generating new quantum fluctuations in the domain of nano-scales.I think that this study will advance our understanding of elementary particles and subatomic-scale research.The lack of documentation of the improved SDM Coulomb potential with a Coulomb-like tensor interaction (ISDM(CP-CLTI)) model with a deformed Dirac equation (DDE) for H-atoms (He + , Li +2 , Be 3+ ) in the literature provided the impetus for the present study.The vector , and scalar potentials of the ISDM(CP-CLTI) models which we are in the process of studying and scrutinizing are presented as follows In addition to the SDM which noted with , in 3D-RNCS symmetry, expressed as Where are the vector and scalar potentials according to the view of 3D-relativistic quantum mechanics (RQM) known in the literature 4,5 are of the form where , is a scalar constant, is the integration constant that represents the static mass of the fermion particle, is the perturbed mass, , , is the Compton-like wavelength in FM units, is a vector dimensionless real parameter coupling constant.The constant mass is a dimensionless real constant that should be set to zero, while ( and ) representing the distances between the two particles in the 3D-RNCS and 3D-RQM symmetries, respectively. .The noncentral generators in the case of the noncommutative quantum group can be correctly represented as self-adjoint differential operators ( , ) existing in 3 varieties.[30][31][32][33][34][35][36][37][38][39][40] and (we have applied the natural units ), the generalized coordinates and the corresponding generalizing coordinates are and in the 3D-RNCS symmetries, while and =( , , ) are corresponding values in the 3D-RQM symmetries and denoting the complex number field.The usual uncertainty relation corresponding to the LHS of Eq.
(5) will be extended to become four uncertainties in the new form symmetries as follows and [37][38][39][40][41] with and are, respectively, equal to the average values and .It should be noted that the only uncertainty relation that is acknowledged to exist within the context of three-dimensional relativistic quantum mechanics symmetries that are known in the literature is given by It should be noted that generated uncertainty relation in Eq. ( 6) from the generalization of the left-hand side of Eq. ( 4) to the right-hand side form.The second new three-uncertainties relations in Eq. ( 7) are produced by the effect of deformation of space-space arising from the right-hand side of Eq. ( 5) that is divided into three varieties.The novel uncertainty relations in Eq. ( 7) within the context of 3D-RQM symmetries have no equivalents in the literature.To take into account both Heisenberg and interaction pictures, the modified equal-time noncommutative canonical commutation relations (METNCCCRs) were expanded in 3D-RNCS symmetry.Here are elements of antisymmetric ( ) real matrices, the effective Planck constant approximatively equal to reduced Planck constant , ( is the infinitesimal parameter NP, and for and ) is just an antisymmetric number while is the Kronecker symbol.The Weyl-Moyal star product of and is , which is generalized to the new deformed form by as follows, [38][39][40][41][42][43][44][45][46][47][48] which can be reduced to in the first order of NP as follows [46][47][48][49][50][51][52][53][54][55] The sum indices equal 1,2 and 3. Physically, the second term in Eq. ( 9) is the influence of space-space deformation.8][59][60] It is important to point out that we have treated this potential in the framework of extended quantum mechanics symmetries of the modified Klien-Gordon equation 61 (MKGE) framework for bosonic particles and antiparticles.The following is a summary of the current paper's structure.In this context, it is useful for the researcher and the reader to mention that the SDM problem has received wide attention for several decades.][64] As far as the spin or pseudospin symmetry, Wei and his coauthors have worked out some results, e.g. 65,66The first Section of the paper presents the purpose and scope of our research, and the remaining sections are organized as follows: Section 2 gives an overview of the DE with an SDM Coulomb potential and a Coulomb-like tensor interaction.Section 3 is devoted to studying the deformed Dirac equation (DDE) using the well-known Bopp shift method to obtain the ISDM (CP-CLTI) model's effective two potentials of the ISDM (CP-CLTI) model.Furthermore, using standard perturbation theory, we find the expectation values of the radial terms ( and ) to calculate the corrected relativistic energy generated by the effect of the perturbed effective potentials ( and ) of the ISDM(CP-CLTI) model, and we derive the global corrected energies and for H-atoms (He + , Li +2 , Be 3+ ) under the ISDM (CP-CLTI) model.In the next Sect., we will address some special cases of physical importance to researchers and readers.In Section 5, we study a nonrelativistic limit of spin symmetry and apply our findings to hydrogenic atoms.A brief conclusion is given in Sect. 5.

REVIEW OF DE UNDER SDM(CP-CLT) IN 3D-RELATIVIC QM REGIMES
To solve approximatively the DDE for ISDM(CP-CLT) model, it is necessary to make a suitable revision of the corresponding potential which is known in 3D-RQM regimes as SDM Coulomb potential including a Coulomb-like tensor interaction SDM(CP-CLT) model within the framework of 3D-RQM regimes which described by the following DE as follows with here the vector potential and space-time scalar potential are produced from the four-vector linear momentum operator ( , ) and the mass , respectively.The usual Dirac Hamiltonian operator for an interacted particle with the SDMC-CLT model, is the momentum, are the relativistic eigenvalues, represent the principal and spin-orbit coupling terms.The tensor interaction equals , , FM is the Coulomb radius, and denote the charges of the projectile and the target nuclei , 67 , and are three-vector spin matrices of three vectors., respectively.Researchers authors for the fifth reference used the asymptotic iteration method to get the expressions for the lower component as a function of the generalized Laguerre polynomial in 3D-RQM symmetries as, here That corresponds to the p-spin symmetry.The relativistic energy equation is determined by 4 To avoid repeating the solution of Eq. ( 12), a quick look at the relationship between the current set of parameters and the previous set reveals that the negative energy solution for pseudospin symmetry, where , can be obtained directly from the positive energy solution for spin symmetry by applying the parameter map [68][69][70] and Apply the above equations, we found the second-order Schrödinger-like equation for spin symmetry as follows

Allow us to obtain the following energy equation
The corresponding upper-spinor component wave function will be written as follows here For the absence of the tensor interaction , all results which we reviewed concern the p-spin symmetry and we deduced for the spin symmetry will be reduced in the work of Samer and Ramazan of the following form 5  4), (5), and (9), which are contained in new relationships METNCCCRs and the concept of the Weyl-Moyal star product, allows us to achieve our goal.With the help of this data, we can rewrite the typical radial DE in Eq. (10) in the 3D-RNCS symmetries as follows: In the 3D-RNCS symmetries, allow us to rewrite the upper and lower components and in the following second-order differential equations and According to employing the Connes method 24,25 or the Seiberg and Witten map, 26 one route to discovering solutions to Eqs. ( 27) and (28) are the application of these methods.Specialists recognize that the Bopp-Shift linear transformation can be used to convert the star product into the typical product described in the literature.Bopp was the first to take into account the quantization rules' ability to produce pseudo-differential operators from a symbol instead of ordinary correspondence respectively.2][73][74] Recent years have seen a lot of success with this approach.6][77][78][79] The MKGE, [80][81][82][83][84][85][86][87] the DDE, [88][89][90][91] and the modified Duffin-Kemmer-Petiau equation MDKPE 92,93 are just a few examples of the relativistic physics problems for which this method has been successful.The Bopp-Shift linear transformation is based on the reduction of second-order linear differential equations (MSE, MKGE, DDE, and DD-KPE) with the Weyl-Moyal star product to second-order linear differential equations of Schrodinger equation, Klein-Gordon equation, Dirac equation, and Duffin-Kemmer-Petiau equation without Weyl-Moyal star product with simultaneous translation in 3D-RNCQS symmetry.Interestingly, we may reduce the mentioned equations using the Bopp-Shift linear transformation.and Now, it is possible to obtain new MASCCCRs with usual known products in 3D-RQM symmetries, from the deformed algebraic structure of covariant canonical commutation relations with the notion of the Weyl-Moyal star product in Eqs. ( 5) and ( 6), as follows [66][67][68][69] In the symmetries of 3D-RNCS, the generalized positions and momentum coordinates ( , ) are defined as [71][72][73][74] This permuted us to get the new operator into two varieties in 3D-RNCQS regimes as follows  16) and ( 21) and the new sentence of Eqs.(35) and (36) lies in the appearance of two additive potentials ( and ).Additionally, these additive potentials are proportional to the infinitesimal non-commutativity couplings ( and ).From a physical point of view, this means that these additive potentials are spontaneously generated as an effect of the influence of space-space deformation.Thus, we can be considered very small compared to the original potentials ( and ), respectively.Moreover, if we use the Heaviside step function or simply the theta function, we can rewrite the globally generated two potentials ( and ) for spin and pseudospin symmetries for to upper and lower components ( , ) and ( , ), respectively as follows and Here, the step function is given by, In 3D-RNCS symmetries, the SDM Coulomb potential containing a Coulomb-like tensor interaction is prolonged by including new additive potentials ( and ) that expressed to the radial terms ( and ) which are combined with new couplings ( and ) to give the improved SDM Coulomb potential including a Coulomblike tensor interaction.The two globally generated potentials ( and ) describes the physical interaction between the SDM Coulomb potential and the physical properties that correspond to spin and pseudospin symmetries ( , ) with the influence of space-space deformation which is characterized by non-commutativity vector .The physical behavior of both two perturbed effective potentials ( , ) quite similar to their original ( ) in terms of their dependence on the two infinitesimal couplings ( and ).This permuted us to consider the new additive-generated potentials ( and ) as perturbation potentials compared with the main potentials ( and ) which are also known with the parent potential operator in the symmetries of DDE, that is, the two inequalities ( and ) have become achieved.All this physical information confirms that the application of the timeindependent perturbation theory is the physical tool that guarantees the treatment of the studied issue with high efficiency and accuracy to find various corrections for determining the energy level of the generalized excited states.

THE RELATIVISTIC EXPECTATION VALUES UNDER THE ISDM (CP-CLTI) MODEL IN THE 3D-RNCS SYMMETRIES FOR SPIN SYMMETRY
In this subsection, we want to use the time-independent perturbation theory (SPT), in the context of 3D-RNCQS symmetries, and that's to find both That corresponds to the spin symmetry taking into consideration the unperturbed form of the upper component that we have seen previously in Eq. ( 23 In this subsection, we want to use the SPT; in the case of 3D-RNCQS symmetries, we obtain the relativistic expectation values of the form That corresponds to the case of the pseudo-spin regime with tensor interaction taking into account the unperturbed wave function which we have seen previously in Eq. (17).By observing the equations expressing each of the upper and lower components ( and ) shown in Eqs. ( 23) and ( 17), we note that there is a possibility of moving from the unperturbed upper component to the other lower component by making appropriate transfers for this purpose This permuted us to get the new physical relativistic expectation values for the pseudospin symmetry from Eqs. (45.1) and (45.2) without re-calculation, and

THE NEW CORRECTED ENERGY FOR ISDM(CP-CLTI) MODEL IN 3D-RNCS SYMMETRIES
We aim through this subsection to achieve the contribution produced from deformed space-space properties based on our proper strategy, which we have successfully used in our previous reaches and which we try to develop in every new reach.We can say, that the total new relativistic energy in the perspective of deformation Dirac theory is generated under the ISDM (CP-CLTI) model.This is a direct consequence of a major contribution to relativistic energy known in the 3D-RQM symmetries under the SDMCCLT model in usual Dirac theory, which we paved for through a quick look for the spin and (pseudospin)-symmetry in Eqs. ( 18) and ( 22) (see second chapter).The new physical contribution is generated from the effect of space-space deformation, which can be evaluated through several contributions; we present the important fundamental physics contributions in three forms that are vital to the outputs of quantum mechanics.The first contribution is induced from the effect of the perturbed spin-orbit effective potentials ( and ) corresponds to spin symmetry and pseudospin symmetry.These perturbed effective potentials are obtained by replacing the coupling of the angular momentum ( and ) operators and the NC vector with the new equivalent couplings ( and ) for spin-symmetry and pseudospin-symmetry, respectively (with ).This is because the NC vector is an arbitrary value, which allows us to deal with its value according to the physical need.We have oriented the two spin-s and spin-of the fermionic particles to become parallel to the vector that interacted under the for spin and pseudospin regimes, respectively, which were produced under the ISDM (CP-CLTI) model, are given from the new expressions and Now we will discuss the second major contribution represented by the magnetic effect of the perturbative effective potentials ( and ) under the effect of ISDM(CP-CLTI) model in the 3D-RNCS regimed.These effective potentials are obtained when we replace both ( and ) with new corresponding values ( and ), respectively.We just replaced by .with ( and ) present the intensity of the magnetic field induced by the effect of the influence of space-space deformation.The new infinitesimal noncommutativity parameter so that the Arbitrary (k, l) States-Solutions of the Dirac and Schrödinger Equations Interacting with Improved Spatially… Yanbu Journal of Engineering and Science physical unit of the original noncommutativity parameter (length) 95 is the same unit of , ( = ), We'll need to apply the following relations to achieve this That corresponds to the spin and pseudospin-regimes, respectively.All of these physical considerations permuted us to get the new energy shift ( , , ) and ( , , ) due to the perturbed Zeeman effect created by the influence of the ISDM(CP-CLTI) model for the excited state in deformation Dirac theory symmetries as follows and After we have accomplished the previous two physical cases, it is useful to address another case that we think is useful for our case.This new physical phenomenon is generated automatically from the effect of perturbed effective potentials ( and ) which we have seen in Eqs. ( 37) and (38), respectively.We consider the fermionic particles undergoing rotation with angular velocity .The features of this subjective phenomenon are determined by replacing the arbitrary vector with a new vector .Allowing us to replace the two couplings ( and ) with ( and ), respectively, as follows here is a new infinitesimal real proportional parameter.The new effective generated perturbed potentials ( and ), due to the rotational movements of the fermionic particles, can be expressed as follows and We have oriented the rotational velocity to become parallel to the ( ) axis ( ) to simplify the calculations.This, of course, does not change the physical characteristics of the problem as much as it simplifies the calculations.Thus, the two rotational movements and can be transformed into a physical simplified form as follows All these physical considerations permuted us to discover the corrected energy ( , , ) and ( , , ) due to the perturbed two effective potentials ( and ) which are generated automatically with the effect of the improved spatially dependent mass Coulomb potential including a Coulomblike tensor interaction for the excited state in MDT symmetries as follows It is useful to point out that in previous studies, the researcher's authors for reference 96 investigated rotating isotropic and anisotropic harmonically confined ultra-cold Fermi gases in two-and three-dimensional space at zero temperature.In this context, the rotational term was manually to the of the Hamiltonian operator.Whereas in our current study, the two perturbed rotation operators ( , ) generated automatically as a consequence of the influence of space-space deformation under the improved spatially dependent mass Coulomb potential including a Coulomb-like tensor interaction model.For a fermionic particle and ant-particle (negative energy), the eigenvalues of the operations ( and As far as we know, the new energy spectra given in Eqs. ( 59) and (60) for the bound state solutions for the ISDM(CP-CLTI) model are new and no previous researcher has obtained them.

STUDY SOME USEFUL RELATIVISTIC APPLICATIONS IN 3D-RNCS SYMMETRIES
After studying the relativistic solutions of the ISDM (CP-CLTI) model, we examine some important cases derived from Eqs. ( 59) and (60) in the context of RQM symmetries which are treated within the context of 3D-RQM symmetries in the main Ref., 5 and we are now in the process of restudied taking into account the effect of non-commutativity influences By assuming, in Eq. ( 57), the parameters and , i.e., or one finds directly The first two parts in the right hand said of Eqs. ( 63) and ( 64) describe the usual relativistic energy of fermionic particles and fermionic anti-particles within the context of 3D-RQM symmetries.The rest terms present the effect of space-space deformation on these main energies in 3D-RQM regimes.However, it can be obtained by applying the limit If we consider the case when and , i.e., or and , then Eq. ( 57) simplified to the following expressions and as follows The first two parts in the right hand said of Eqs. ( 66) and ( 67) describe the ordinary relativistic energy of particle and anti-particle within the context of 3D-RQM regimes known in the literature.The remaining terms describe the influence of space-space deformation on these main energies.
While the new expectation value can be obtained by applying the limit For the s-wave which corresponds and , Eq. ature.The remaining terms present the influence of spacespace deformation on these main energies in 3D-RQM regimes.
By assuming, in Eq. ( 58), the parameters and , i.e., or one finds directly The first two parts in the right hand said of Eqs. ( 70) and ( 71) describe the ordinary relativistic energy of particle and anti-particle within the context of 3D-relativistic quantum mechanics known in the literature.The remaining terms describe the topological effect of the deformation spacespace on these main energies in 3D-RQM regimes.While the corresponding physical expectation value can be obtained by applying the limit By assuming, in Eq. ( 58), the parameters and , i.e., or and one finds directly The first two parts in the right hand said of Eqs. ( 73) and (74) describe the relativistic energy of particles and antiparticles within the context of RQM known.The remaining terms describe the topological effect of deformation space-space on these main energies in 3D-RQM regimes.
While the expectation values can be obtained by applying the limit For the s-wave which corresponds or and , Eq. ( 72) reduced to The first part in the right hand said of Eq. ( 76) describes the relativistic energy of particles for the s-wave within the framework of 3D-RQM symmetries known in the literature.The remaining terms describe the topological effect of deformation space-space on these main energies in 3D-RQM symmetries.

IMPROVED SDM COULOMB POTENTIAL PROBLEM IN 3D-NRNCS SYMMETRIES
Two crucial steps must be taken to apply the improved tially dependent mass Coulomb potential in 3-dimensional non-relativistic non-commutative space (3D-NRNCS) regimes to reach the non-relativistic limit.In standard non-relativistic quantum energy, the first step corresponds to the non-relativistic limit.Applying the steps that follow satisfies this as follows: This permuted us to get the non-relativistic energy levels under SDM Coulomb potential in 3D-NRQM regimes as 4 The second step corresponds to reporting the relativistic expectation of spin symmetry in Eq. ( 49 The first part on the right-hand side of Eq. ( 82) traduces the non-relativistic energy of fermionic particles when mass becomes constant, which we note in Ref. 5 known in the literature while the remaining terms present the topological effect of deformation space-space on these principal energies.Through our observation of the newly obtained physical values within the framework of 3D-RNCQS and 3D-NRQS regimes, we observed that the obtained energy values ( ) and ( , ) (see Eqs. ( 59) and ( 60)) are dependent on quantum numbers in addition to the potential parameters This means that the improved tensor interaction removes degeneracy between two states in the spin and pseudospin doublets.Since the total energy values have become dependent on the discrete , we can deduce that the new symmetry of 3D-RNCQS and 3D-NRQS has exact solutions in the first order of noncommutativity parameters , and also, the improved tensor interaction removes degeneracy between two states in the spin and p-spin doublets.Furthermore, it is worth mentioning that the three-simultaneous limits , recover the equations of energy for the spin symmetry and the p-spin symmetry under an SDM Coulomb potential with Coulomb-like tensor interaction which is treated explicitly in main Refs. 4,5 CONCLUSIONS teraction under p-spin and spin symmetry limits with an arbitrary spin-orbit coupling quantum number k is used to approximate an analytical solution of the 3D-DDE with the newly spatially dependent mass Coulomb potential.We have obtained the new approximate bound state energies ( ) and ( , ) for the H-atoms (He + , Li +2 , Be 3+ ) that appeared to be sensitive to the quantum numbers ( ), the potential depths ( ) of the studied potentials and non-commutativity parameters ( ).We have additionally investigated a few different potentials that are pertinent to various physical systems.This treatment of the spatially dependent mass Coulomb potential's non-relativistic limit served as the culmination of our research as well.Additionally, a few special instances that readers and researchers found to be very concerning were dealt with within the context of 3D-RNCS symmetries.Furthermore, we achieved the non-relativistic limit of the improved spatially dependent mass Coulomb potential in 3D-NRNCS symmetries, which also includes the non-relativistic limit of the typical nonrelativistic quantum energy and the influence of deformation space proportional to non-commutativity parameters and dimensionality of studied potential.In addition, we have shown that the non-commutativity of the coordinates that automatically appear, such as both perturbed spin-orbit interaction, pseudospin-orbit, new perturbed Zeeman Effect, and among others, and due to the behavior of topological properties of modified space-space, contributes to the physical effect of the four principals.These physical effects cannot be predicted using the relativistic quantum mechanics framework that has been described in the literature unless we take them into account as important concepts in the expression of the main Hamiltonian.These results have applications in chemical physics, atomic physics, and condensed matter physics.Notably, the standard physical values are recovered in all situations to produce the three simultaneous limits in Refs.
Arbitrary (k, l) States-Solutions of the Dirac and Schrödinger Equations Interacting with Improved Spatially… Yanbu Journal of Engineering and Science Since the SDMC-CLT model has spherical symmetry, allowing the spinor solutions of the known form and for spin symmetry and pseudo-spin (p-spin) symmetry, and represent the upper and lower components of the Dirac spinors while and are the spin and p-spin spherical harmonics and are the projections on the z-axis.The upper and lower components and for spin symmetry and p-spin symmetry satisfy the two uncoupled differential equations below spectively and according to the data of the studied potential expressed asThat corresponds to the p-spin symmetry, we get the following second-order Schrödinger-like equation in 3D-RQM regimes, Arbitrary (k, l) States-Solutions of the Dirac and Schrödinger Equations Interacting with Improved Spatially… Yanbu Journal of Engineering and Science and The lower component of spin symmetry and the upper component of pseudospin symmetry are obtained as follows 3. NEW APPROXIMATE BOUND STATE SOLUTIONS OF ISDM(CP-CLTI) MODEL IN 3D-RNCS REGIMES 3.1.REVISED BOPP-SHIFT LINEAR TRANSFORMATION Let's start this subsection by locating the deformed Dirac equation (DDE) in three-dimensional relativistic noncommutative space (3D-RNCS) symmetries using the ISDM (CP-CLTI) model.Applying the new ideas from the introduction, Eqs. ( while the new operators ( and ), in the 3D-RNCS symmetries, are expressed as and Substituting Eqs.(34.1) and (34.2) into Eqs.(29) and (30), we get the following two like-Schrödinger equations and with Arbitrary (k, l) States-Solutions of the Dirac and Schrödinger Equations Interacting with Improved Spatially… Yanbu Journal of Engineering and Science and It is clear to us that the difference between the set of Eqs. ( with spin-½ are expressed by the following values Respectively.Thus, the possible values for are ( case of spin-½, for example, H-atoms (He + , Li +2 , Be 3+ ) with improved SDM Coulomb potential including a Coulomb-like tensor interaction model, corresponding to the generalized excited are expressed as Here are just the ordinary relativistic energies under SDM Coulomb potential including a Coulomblike tensor interaction model obtained from equations of energy in Eqs.(22) and (18) in the context of 3D-RQM regimes.These results describe spin and pseudospin new Arbitrary (k, l) States-Solutions of the Dirac and Schrödinger Equations Interacting with Improved Spatially… Yanbu Journal of Engineering and Scienceenergies in DDE for atoms with one electron such as Hatoms (He + , Li +2 , Be 3+ ).For fermionic particles or anti-particles with spin-, we replace and by the generalized two previous values and .Allow us to obtain the total relativistic engeneralize our obtained energies ( and ) under the improved SDM Coulomb potential including a Coulomb-like tensor interaction model produced with the globally induced two potentials and for spin and pseudospin symmetries corresponding to the upper component (Uc) and lower component (Lc) (66) reduces to ½ ½ The first part in the right hand said of Eq. (69) describes the ordinary relativistic energy of particles for the s-wave within the framework of relativistic QM known in the liter-Arbitrary (k, l) States-Solutions of the Dirac and Schrödinger Equations Interacting with Improved Spatially… Yanbu Journal of Engineering and Science States-Solutions of the Dirac and Schrödinger Equations Interacting with Improved Spatially… Arbitrary (k, l) States-Solutions of the Dirac and Schrödinger Equations Interacting with Improved Spatially… ½ ½ 4,5which is treated within the framework of 3D-RQM and 3D-NRQM symmetries known in the literature.85.MairecheA.Diatomic Molecules with the Improved Deformed Generalized Deng-Fan Potential Plus Deformed Eckart Potential Model through the Solutions of the Modified Klein-Gordon and Schrödinger Equations within NCQM Symmetries.Ukr J Phys. 2022;67(3):183.doi:10.15407/ujpe67.3.18386.Maireche A. New relativistic and nonrelativistic model of diatomic molecules and fermionic particles interacting with improved modified Mobius potential in the framework of noncommutative quantum mechanics symmetries.YJES.2021;18(1):10.doi:10.5 3370/001c.2809087.Maireche A. Approximate k-state solutions of the deformed Dirac equation in spatially dependent mass for the improved Eckart potential including the improved Yukawa tensor interaction in 3D-RNCQM symmetries.Int J Geo Met Mod Phys.2022;19(06):2250085. doi:10.1142/s021988782250085 2 88.Maireche A. 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.Rev Mex Fís.2022;68(2 Mar-Apr):020801.doi:10.31349/revmexfis.68.020801 89.Maireche A. Deformed Dirac and Schrödinger Equations with Improved Mie-Type Potential for Diatomic Molecules and Fermionic Particles in the Framework of Extended Quantum Mechanics Symmetries.Ukr J Phys. 2022;67(7):485.doi:10.15407/ujpe67.7.485 90.Maireche A. A new study of relativistic and nonrelativistic for new modified Yukawa potential via the BSM in the framework of noncommutative quantum mechanics symmetries: An application to heavy-light mesons systems.YJES.2022;19(2):1-20.d oi:10.53370/001c.3961591.Maireche A. Approximate arbitrary (k,l) states solutions of deformed Dirac and Schrödinger equations with new generalized Schiöberg and Manning-Rosen potentials within the generalized tensor interactionsin 3D-EQM symmetries.Int J Geo Met Mod Phys.2023;20(02):2350028. doi:10.1142/s0219887823500287 92.Saidi A, Sedra MB.Spin-one (1 + 3)-dimensional DKP equation with modified Kratzer potential in the non-commutative space.Mod Phys Lett A. 2020;35(05):2050014. doi:10.1142/s021773232050014 5 93.Houcine A, Abdelmalek B. 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 Nuc Lett.2019;16(3):195-205.do i:10.1134/s154747711903003894.Wolfram Research.https://functions.wolfram.com/ 95.Dong SH.The Dirac equation with a Coulomb potential inDdimensions.J Phys A: Math Gen. 2003;36(18):4977-4986.doi:10.1088/0305-4470/36/18/303 96.Bencheikh K, Medjedel S. 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. 2019;383(16):1915-1921. doi:10.1016/j.physleta.2019.03.021Arbitrary (k, l) States-Solutions of the Dirac and Schrödinger Equations Interacting with Improved Spatially… Yanbu Journal of Engineering and Science