A Novel Exactly Theoretical Solvable of Bound States of the Dirac-Kratzer-Fues Problem with Spin and Pseudo-Spin Symmetry

New exact bound state solutions of the deformed radial upper and lower components of Dirac equation and corresponding Hermitian anisotropic Hamiltonian operator are studied for the modified Kratzer-Fues potential (m.k.f.) potential by using Bopp’s shift method instead to solving deformed Dirac equation with star product. The corrections of energy eigenvalues are obtained by applying standard perturbation theory for interactions in one-electron atoms. Moreover, the obtained corrections of energies are depended on two infinitesimal parameters    ,  ,which induced by position-position noncommutativity, in addition to the discreet nonrelativistic atomic quantum numbers: ( l s l j ~ , 2 / 1 , 1 / 1 ~     and m~ ) and we have also shown that, the usual relativistic states in ordinary three dimensional spaces are canceled and has been replaced by new degenerated   1 ~ 2 2  l sub-states in the extended quantum symmetries (NC: 3D-RS).


Introduction
In relativistic quantum mechanics, one of the interesting problems is to obtain exact solutions of the Klein-Gordon equation (spin zero particle) and Dirac equation (spin ½ particle) at high energy, much interest in providing analytic solutions to the relativistic equations in many fields of Physics and Chemistry for different central and non central potentials [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]. The quantum structure based to the ordinary canonical commutations relations in both Schrödinger and Heisenberg (the operators are depended on time) pictures, respectively (Natural units 1    c are employed throughout this paper): and where the two operators       here Ĥ denote to the ordinary quantum Hamiltonian operator. In addition, for spin ½ particles described by the Dirac equation, experiment tells us that must satisfy Fermi Dirac statistics obey the restriction of Pauli, which imply to gives the only non-null equal-time anti-commutator for field operators as follows:

International Frontier Science Letters
Submitted: 2016-09-06 ISSN: 2349-4484, Vol. 10, pp [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22] Revised: 2016-11-16 doi: 10.18052/www.scipress.com/IFSL. 10.8 Accepted: 201610.8 Accepted: -11-30 2016  . It is important to notice that, the noncommutativity idea was introduced firstly by H. Snyder andA. Connes in 1946 and1986, respectively [17,18]. Very recently, much considerable effort has been expanded on the solutions of Schrödinger, Dirac and Klein-Gordon equations to noncommutative quantum mechanics , the new quantum structure of noncommutative space based on the following noncommutative canonical commutations relations in both Schrödinger and Heisenberg pictures, respectively, as follows: , and , , , 0 , and , , , The above new quantum structure allow us to realize the non-commutative Moyal spaces, the two new operators in Heisenberg picture are related to the corresponding operators in Schrödinger picture from the two projections relations: the first term on the right side gives the ordinary product, the term ( ) is induced by (space-space) noncommutativity properties and   2  O stands for the second and higher order terms of  , a Bopp's shift method can be used, instead of solving any deformed quantum systems by using directly star product procedure: The new three-generalized coordinates   The non-vanish -commutators in (NC-3D: RS) can be determined as follows: International Frontier Science Letters Vol. 10 9 which allow us to getting the operator 2 r on noncommutative three dimensional spaces as follows: (11) where the coupling  L is given by with: Furthermore, the non-null equal-time anti-commutator for fermionic field operators in noncommutative spaces can be expressed in the following postulate relations: Moreover, the noncommutative fermions propagator can be expressed as: where T denotes to the time-ordered product. In particularly, the study of Kratzer-Fues potential has now become a very interest field due to their applications in different fields [16]. The main motivation of this work is to study and obtaining an analytic expression for the eigenenergies of a (m.k.f.) potential in (NC: 3D-RS) using the Bopp's shift method to discover the new symmetries and a possibility to obtain another applications to this potential in different fields. This work based essentially on our previously works . This work is organized as follows: In section 2, we briefly review the Dirac equation with Kratzer-Fues potential on based to Refs. [14][15][16]. The Section 3, devoted to studying the three deformed Dirac equation by applying Bopp's shift method. In the fourth section, by applying standard perturbation theory we find the quantum spectrum of the th n excited states in (NC-3D: RS) symmetries for relativistic spin-orbital interaction with (m.k.f.) potential and then, we derive the corresponding magnetic spectrum. In the section 5, we resume the global spectrum and corresponding noncommutative Hamiltonian operator for (m.k.f.) potential. Finally, in section 6 we present our conclusions.

Review of Dirac equation for Kratzer-Fues potential in ordinary commutative spaces
We briefly review the differential Dirac equation of a nucleon with mass M moving in both two potentials: attractive scalar potential ) (r S and a repulsive vector potential   r V [16]: The Kratzer-Fues potential is given in the slightly modified form: where 0 r and 0 V are the equilibrium separation and is the dissociation energy between diatomic molecules, respectively, which can be simply rewritten in the form

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IFSL Volume 10 are the fermions' mass, the relativistic energy and the usual Dirac matrices while is related to the total angular momentum quantum numbers for spin symmetry l and pspin symmetry l~ as [14][15][16]: The two radial functions (   r F nk ,   r G nk ) are the upper and lower components of the Dirac spinor which obtained by solving the following differential equations [14,15,16]: According to The Laplace transform approach (LTA), which applied in refs. [14,15,16] the normalization constant and the confluent hyper-geometric functions, respectively. Furthermore, the energy eigenvalues of the Dirac equation with Kratzer-Fues potential for the spin symmetry condition is given by [16]: where Ñ denote to the normalization constant and energy eigenvalues of the Dirac equation with Kratzer-Fues potential for the pseudo spin symmetry condition is given by [16]: . On the other hand, the generalized Laguerre polynomials can be expressed as a function of the confluent hyper-geometric functions as [53][54][55]: this leads to rewritten Eqs. (24) and (26) as follows, respectively: 3. Theory of Noncommutative relativistic Hamiltonian operator for (m.k.f.) potential in (NC-3D: RS) symmetries:

Formalism of Bopp's shift Method
Now, we shall review some fundamental principles of the quantum noncommutative Naively, to get a physical quantity on a noncommutative space, we simply take this quantity on the corresponding commutative space and replace all products by the star products [44], thus we can write the noncommutative relativistic Dirac equation for (m.k.f.) potential as follows: The Bopp's shift method permutes to reduce the above equation using old product with simultaneously translations applied to the operators i x and i p as follows: where the (m.k.f.) potential   r Vˆ is given by: In the light of the above considerations, it could be interesting to solve exactly modified Dirac equation for the (m.k.f.) potential   r Vˆ using Bopp's shift method as follows:  (40), we can obtain the following two Schrödinger-like differential equations as follows in (NC-3D: RS): On based to results of eq. (11), we can easily obtain the two terms Substituting (43) into equation (37), we obtain the new potential as: in (NC-3D: RS) and inserting the potential   r Vˆ in eq. (44) into the two Schrödinger-like differential equations (41) and (42), one obtains:  is proportional with infinitesimal parameter  , thus we can considered as a perturbations terms.

The exact relativistic spin-orbital Hamiltonian for (m.k.f.) potential in (NC: 3D-RS) symmetries for one-electron atoms:
In this sub-section, we are going to rewriting the perturbative terms  (49) As we know, we just replaced the coupling pseudo spin-orbital (exact spin-orbital) L S by the expression forms a complete of conserved physics quantities and the spin-orbit (pseudo spin-orbit) quantum number k ( k ) is related to the quantum numbers for spin symmetry l and p-spin symmetry l~ as follows [14][15][16]: A direct simplification gives: , we apply the following special integral [60,61]:    for th n exited states, produced with relativistic spin-orbital induced by noncommutative p-spinorbital Hamiltonian operator, we now turn our attention to the study another interested physically

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IFSL Volume 10 meaningful phenomena, which also produced from the perturbative terms of modified Kratzer-Fues potential related to the influence of an external uniform magnetic field, it's sufficient to apply the following two replacements to describing these phenomena:  , which allow us to fixing ( 1 2  l ) values for the orbital angular momentum quantum numbers, thus we can obtain the second class of solutions for modified Kratzer-Fues potential.

Results
Following the discussions of the Sect. 4, we proceed now to find the eigenvalues for this problem for th n excited states (  This is exactly the spectrum of three-dimensional Kratzer-Fues potential. As it is montionated in ref. [14], in view of exact spin symmetry in commutative space (