Investigations on the Relativistic Interactions in One-Electron Atoms with Modified Yukawa Potential for Spin 1/2 Particles

Energy levels of one electron atoms have been re-examined by applying an alternative perturbative scheme in solving the modified Dirac equation (m.d.e.) for the modified Yukawa potential model with an arbitrary spin-orbit quantum number       1 ~ , ~ ~ , 1 ,         l l k l l k by means Bopp’s shift method instead to solving (m.d.e.) with star product, in the framework of noncommutativity three dimensional real space (NC: 3D-RS). It is observed that the obtained corrections of energies are depended on the new discrete atomic quantum numbers   m m k k ~ , , ~ , under spin-symmetry and pseudospin symmetry and two infinitesimal parameters     k k k ij ij ij    , ,    which induced by position-position noncommutativity. Furthermore, in limit of parameters     0 , 0 ,   k k  , the new energy equations for modified Yukawa potential are consistent with the results of ordinary relativistic quantum mechanics for ordinary Yukawa potential.

by means Bopp's shift method instead to solving (m.d.e.) with star product, in the framework of noncommutativity three dimensional real space (NC: 3D-RS).It is observed that the obtained corrections of energies are depended on the new discrete atomic quantum numbers   m m k k , , , under spin-symmetry and pseudospin symmetry and two infinitesimal parameters   which induced by position-position noncommutativity.Furthermore, in limit of parameters , the new energy equations for modified Yukawa potential are consistent with the results of ordinary relativistic quantum mechanics for ordinary Yukawa potential.

Introduction
In very recent years the energy levels of atoms, nuclei, and hadrons have been studied by several analytic methods in the case the two relativistic equations Klein-Gordon and Dirac for certain potentials of the physical interest , some of these potentials are known to play important roles in many fields, one of such potential is the Yukawa potential has been a subject of many studies, it is a central potential of nuclear shell model in addition to used in nonrelativistic quantum mechanics to study the interactions between nonrelativistic particles, in atomic and molecular physics, it represents a screened Coulomb potential due to the cloud of electronic charges around the nucleus, on the other hand, it has also been used to describe the interaction between charged particles in plasma, solid and colloidal suspensions [22].It is well known that the ordinary quantum structures obey the standard Weyl-Heisenberg algebra in both Schrödinger and Heisenberg (the operators are depended on time) pictures ((HP) and (SP)), respectively, as (Throughout this paper the natural unit where the operators       t p t x i i , in (HP)are related to the corresponding operators   i i p x , in (SP) from the following projections relations:

ˆ
denote to the ordinary quantum Hamiltonian operator for Yukawa potential.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 equaltime anti-commutator for field operators as follows: Very recently, many authors have worked on solving the fundamental equations (Schrödinger, Klein-Gordon and Dirac equations) with physical potential in the new structure of quantum mechanics, known by NC quantum mechanics which known firstly by H. Snyder, it is shown a direct relationship between string theory and noncommutative geometry and to obtaining profound and new applications for different areas of matter sciences in the microscopic and nano scales within the framework of quantum mechanics and quantum field theory .It is important to noticing that, the new quantum structure of NC space based on the following NC canonical commutations relations (NCCRs) in both (HP) and (SP)), as follows : , ˆin (HP) are related to the corresponding new operators   i i p x , ˆ in (SP) from the new projections relations [27][28][29][30][31][32][33]: in ordinary three dimensional spaces : are the new function in (NC: 3D-RS), the following term ( ) is induced by (space-space) noncommutativity properties and  

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The non-vanish-commutators in (NC-3D: RS) can be determined as follows: which allow us to getting the operator 2 r on NC three dimensional spaces as follows [28][29][30][31][32][33][34]: (10) the coupling 13 23 12 Furthermore, the new equal-time anti-commutator for fermionic field operators' in NC spaces can be expressed as [28][29][30][31][32][33][34]: The purpose of the present work is to extend and present the solution of the Dirac equation with spin-1/2 particle moving in modified Yukawa potential of the new form: In (NC: 3D-RS) using the generalization 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 .The plan of recently work is as follows: In next section, we briefly review the Dirac equation with Yukawa potential on based to refs.[18][19][20][21][22].In section three, we give a description of the Bopp's shift method for (m.d.e.) with modified Yukawa potential.Then in section four, we apply standard perturbation theory to establish exact modifications at first order of infinitesimal parameters    ,  for the perturbed Dirac equation in (NC-3D: RS) for spin-orbital (pseudo-spin orbital) and the relativistic magnetic spectrum for modified Yukawa potential.Section fifth is devoted for discussion the global spectrum and corresponding NC Hamiltonian for modified Yukawa potential.Finally, we give a brief conclusion in last section.

Review of the Dirac equation for Yukawa potential in relativistic quantum mechanics
Let us start this section by the review of a relativistic particle in spherically symmetric for the potential    , r V which known by Yukawa potential on based to the main reference [22]: where  and 0  denote to the rang of nuclear force and strength of potential, respectively.The Dirac equation describing a fermionic particle (spin-1/2 particle) with scalar potentials is given by [18][19][20][21][22]: International Frontier Science Letters Vol.11 here M are E the fermions' mass and the relativistic energy while can be expressed as [18][19][20][21][22]: (15) where ~ is related to the total angular momentum quantum numbers for spin symmetry l and p-spin symmetry l ~ as [18][19][20][21][22]: ) are obtained by solving the following differential equations [18][19][20][21][22]: (18) and The exact spin symmetry corresponding , thus the radial function   r F nk satisfying the following like Schrödinger equation [22]: The relativistic energy k n E , and radial upper wave   r F nk are given by [22]:

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IFSL Volume 11 and where   The Bopp's shift method permutes to reduce the above NC equation to simplest form with usual product and translations applied to in space and phase operators: Where the new Hamiltonian operator can be expressed in three general varieties: both NC space and NC phase (NC-3D: RSP), only NC space (NC-3D: RS) and only NC phase (NC: 3D-RP) as, respectively: International Frontier Science Letters Vol.11 In recently work, we are interest with the second variety which present by eq. ( 28) and by the means of the auxiliary two variables )) where the modified Yukawa potential is given by: The Dirac equation in the presence of above interaction can be rewritten according Bopp shift method as follows: ) are obtained by solving two equations:  33) and (34), we can obtain the following two Schrödinger-like differential equations in (NC-3D: RS) symmetries as follows: (35) and It's clearly that, the first term represent the ordinary Yukawa potential while the rest parts are produced by the deformation of space, this allows writing the modified Yukawa potential in the NC case as an equation similarly to the usual Dirac equation of the commutative type with a non local potential.Furthermore, using the unit step function (also known as the Heaviside step function or simply the theta function) we can rewrite the modified Yukawa potential to the following form: (40) where We generalized the constraint for the pseudospin (p-spin) symmetry in (NC-3D: RS) and inserting the potential   r V yp ˆ in eq. ( 39) into the two Schrödinger-like differential equations ( 35) and (36), one obtains:

The exact relativistic spin-orbital Hamiltonian and the corresponding spectrum for modified Yukawa potential in (NC: 3D-RS) symmetries for
th n excited states for one-electron atoms:

The exact relativistic spin-orbital Hamiltonian for modified Yukawa potential in (NC: 3D-RS) symmetries for one-electron atoms:
In this part of the paper, again, the two perturbative terms can be rewritten to the equivalent physical form for modified Yukawa potential as follows: To the best of our knowledge, we just replace the two spin-orbital coupling L S and S L  by the expression , which allows us to form two diagonal    Our goal in this sub section is to find the modifications to the energy levels and spin-up) and ( and spin down), respectively, at first order of infinitesimal parameter  , for excited th n state, obtained by applying the standard perturbation theory, using Eqs.( 22), ( 44), ( 45), ( 46) and ( 45) as: The first parts represent the modifications to the energy levels for the spin symmetric cases while the second part represent the modifications to the energy levels ( ) for the spin spin-symmetry, then we have explicitly: Inserting the radial function   r F nk given by Eq. ( 22) into the above two Eqs.( 51) and ( 52) to obtain: Now, we introduce the following new factor   as follows: we apply the following special integration [72]: obtained from the generalized the hypergeometric function denote to the usual Gamma function.After straightforward calculations, we can obtain the explicitly results for the factor   which allow us to obtain the modifications to the energy levels for th n excited states:

The exact relativistic magnetic spectrum for modified Yukawa potential for th
n excited states for one-electron atoms in (NC: 3D-RS) symmetries: Having obtained the exact modifications to the relativistic energy levels for th n excited states which produced with relativistic NC spin-orbital Hamiltonian operator, our objective now, in this section, we consider another interested new physically meaningful phenomena, which also can be produce from the perturbative terms of Yukawa potential related to the influence of an external uniform magnetic field, it's sufficient to apply the following three replacements to describing these phenomena:

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Then, the relativistic magnetic modification of energy corresponding ground state on the (NC-3D: RS) symmetries, can be determined from the following relation: The relativistic magnetic modification of energy for pseudospin symmetry can be obtained as: where m and m ~ are represent the angular momentum quantum numbers satisfying the intervals ) values.

The main results of exact modified global spectrum for modified Yukawa potential for one-electron atoms under spin-symmetry and p-spin symmetry in (NC: 3D-RS):
This principal part of the paper is devoted to the presentation of the several results obtained in the previous sections, we resume the th n excited states eigenenergies ( , aligned spin 0 k and spin-down) and ( , un aligned spin 0 k and spin up), respectively, at first order of parameter  , for modified Yukawa potential in (NC: 3D-RS), respectively, on based to the obtained new results ( 58), ( 59), ( 63) and (64), in addition to the original results ( 21) and ( 23) of energies in commutative space, we obtain the following original results: As it is mentioned in ref. [18], in view of exact spin symmetry in commutative space ( ), we need to generalize the above translations to the case of NC three dimensional spaces, then the negative values  Allow us to deduce the important original results: every state in usually three dimensional space will be replace by   , equations ( 65), (66), (67) and (68) reduces to (21), (23) and (69) and one recovers the standard textbook results.Finally, one concludes; our obtained results are sufficiently accurate for practical purposes.

The important concluding remarks
Let us summarize our results as follows: (i) The solution procedure presented in this paper is based on the both of Bopp's shift method and standard perturbation theory, we investigate the bound state energies of th n excited states for modified Yukawa potential described on (NC: 3D-RS).which presented by Eq. ( 68).
The energy eigenvalues are in good agreement with the results previously.Finally, we point out that these exact results (65), (66) and (67) obtained for this new proposed form of the modified Yukawa potential   r V yp ˆ may have some interesting applications in the study of different quantum mechanical systems, atomic and molecular physics.


operator of the extended quantum system described on (NC: 3D-RS) symmetries.The very small parameters   (compared to the energy) are elements of antisymmetric real matrix of dimension   denote to the new star product (the Moyal-Weyl product), which is generalized between two arbitrary functions

) 3 . 3 . 1
NC relativistic Hamiltonian for modified Yukawa potential: Formalism of Bopp's shift Method In this section I first highlight in brief the basics of the concepts of the quantum noncommutative quantum mechanics in the framework of relativistic Dirac equation for modified Yukawa potential   r V ao ˆ on based to our works [28-43]: a) Ordinary Dirac Hamiltonian operator   spinor   r  replace by new spinor   r    , c) Ordinary relativistic energy nk E replaces by new relativistic energy yp nc E  and ordinary product replace by new star product  .Thus, the Dirac equation in ordinary quantum mechanics will change into the modified Dirac equation in extended quantum mechanics for modified Yukawa potential as follows: to writing the modified Yukawa potential   r V yp ˆin (NC-3D: RS) as follows: infinitesimal parameter  , thus we can considered as a perturbations terms.
to the following new equivalent form for (m.a.o.) potential: NC: 3D-RS) as:

4. 2 .
The exact relativistic spin-orbital spectrum for modified Yukawa potential symmetries for th n excited states for one-electron atoms in (NC: 3D-RSP) symmetries: here  is infinitesimal real proportional's constants, and we choose the magnetic field k B B  for simplify the calculations, which allow us to introduce the modified new magnetic Hamiltonian the (NC: 3D-RS), as:

,
two ordinary and pseudo Hamiltonians of Zeeman effect.To obtain the exact NC magnetic modifications of energy potential under spin-symmetry case which produced automatically from the effect of operator we make the following two simultaneously replacements: ordinary Hamiltonian operator in the commutative space.In this way, one can obtain the complete energy spectra for modified Yukawa potential in (NC: 3D-RS) symmetries.The following accompanying constraint relations: a) The two quantum numbers   m m, ~ satisfied the two intervals: values for these quantum numbers, b) We have also two values for p-spin symmetry under p-spin symmetry and spin symmetry, which permits to fixing the degenerated states to the NC: 3D-RS) symmetries.It is easy to see that the obtained originally results reduce to the ordinary results described on quantum mechanics when the noncommutativity of space disappears    

(
ii) It is found that the energy eigenvalues depend on the dimensionality of the problem and the new atomic quantum numbers   We have also constructing the corresponding NC Hermitian Hamiltonian operator noncommutative three dimensional spaces and phases, International Letters of Chemistry, Physics and Astronomy.60 (2015) 11-19.[36] A. Maireche, A new study to the Schrödinger equation for modified potential in nonrelativistic three dimensional real spaces and phases, International Letters of Chemistry, Physics and Astronomy.61 (2015) 38-48.

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It's clearly, that the obtained eigenvalues of energies are real; consequently, the modified quantum Hamiltonian operator