A New Nonrelativistic Investigation for the Lowest Excitations States of Interactions in One-Electron Atoms, Muonic, Hadronic and Rydberg Atoms with Modified Inverse Power Potential

A new theoretical analytical investigation for the exact solvability of non-relativistic quantum spectrum systems at low energy for modified inverse power potential (m.i.p.) is discussed by means Boopp’s shift method instead to solving deformed Schrödinger equation with star product, in the framework of both noncommutativite two dimensional real space and phase (NC: 2D-RSP), the exact corrections for lowest excitations are found straightforwardly for interactions in oneelectron atoms, muonic, hadronic and Rydberg atoms by means of the standard perturbation theory. Furthermore, the obtained corrections of energies are depended on the four infinitesimals parameters ( ,  ) and ( , ), which are induced by position-position and momentum-momentum noncommutativity, in addition to the discreet atomic quantum numbers ( l s l j , 2 / 1 , 1 / 1     andm ) and we have also shown that, the old states are canceled and has been replaced by new degenerated   1 2 4  l sub-states.


Introduction
Over the past few years, many efforts have been produced to study quantum systems on based to the Dirac equation, Klein-Gordon equation and Schrödinger equation, which is undoubtedly the most widely studied equation of modern physics, for spherical and non spherical potentials that are used in different fields of physics and materials sciences by using various methods such as the Nikiforov-Uvarov, super-symmetric quantum mechanics, numerical calculations, asymptotic iteration, the path integral approach, asymptotic iteration method, exact method and many others . Recently, physicists have developed previously works by applying the noncommutativity properties on space and phase , the two ordinary generalized coordinates of space and momentums respectively, in natural units : The symbol   * and the two parameters are star product and antisymmetric real tensors induced by position-position and momentum-momentum noncommutativity, respectively [48], it's important to notice that, the above two fundamental commutation relations are satisfied in particulars' cases from the general star product     The first term is the usual product in commutative space while the rest two parts are represents the effects of the noncommutativity of (space-space) and (phase-phase), respectively; a Boopp's shift method can be used, instead of solving any quantum systems by using directly star product procedure: The generalized positions and momentum coordinates in the noncommutative quantum mechanics   Which allow us to getting the two operators 2 r and 2 p on a noncommutative two dimensional space-phase as follows [26][27][28][29][30][31][32][33][34][35][36][37][38][39]: , this work is aimed at obtaining an analytic expression for the eigenenergies of a inverse power potential in (NC: 2D-RSP) using the generalization Boopp's shift method based on mentioned formalisms on above equations to discover the new symmetries and a possibility to obtain another applications to this potential in different fields, it is important to notice that, this potential was studied, in ordinary two dimensional spaces, by authors Shi-Hai Dong, Zhoung-Qi Ma, Giam pieero Esposito, E. Vogt and G. H. Wannier of the Refs. [23,24,25]. This potential become useful for describes interactions in one-electron atoms, muonic, hadronic and Rydberg atoms; photo decay of excited states, when gaseous ions or electrons move through a gas whose molecules are not too large and we have also for Dirac equation for a spin 2  [23,24,25]. The contents of the rest paper are as follows: In next section, we briefly review the Schrödinger equation with inverse power potential. The Section 3, reserved to derive the deformed potential and perturbative terms by applying both Boopp's shift method to the inverse power potential. In the fourth section and by applying standard perturbation theory we find the quantum spectrum of the lowest excitations in (NC: 2D RSP) for spin-orbital interaction. In the next section, we derive the magnetic spectrum for studied potential. In the sixth section, we resume the global spectrum and corresponding noncommutative Hamiltonian for inverse-power potential. In the seventh section, we extended our study to include the energy spectra for muonic, hadronic and Rydberg atoms in (NC: 2D RSP). Finally, the important results and the conclusions are discussed in last section.

Review the eignenfunctions and the energy eigenvalues for inverse-power potential in ordinary two dimensional spaces
Let's present a brief review of time independent Schrödinger equation for a fermionic particle like electron of rest mass 0 m and its energy E moving in inverse power potential [23,25]: is the solution in the 2-dimensional polar coordinates, the complete wave function    , r  separated as follows: Substituting eq. (8) into eq. (7), we obtain the radial function   r R m satisfied the following equation, in ordinary two dimensional spaces (2D) [23,25]: (9) here m denote to the orbital angular momentum quantum numbers. The Schrödinger equation given in Eq. (9) has been solved by means of the Nikiforov-Uvarov (NU) method [25], in 2D spaces for inverse power potential for ground state and first excited state, respectively: It has the two eigenvalues  0 E and  1 E (two values for every state), respectively [25]:  The two normalizations constants 0 N and 1 N are given by [25]: And the last step corresponds to replace the ordinary old product by new star product    , which allow us to constructing the modified two dimensional Schrödinger equation in both (NC-2D: RSP) as for (m.i.p.) potential: It's important to notice that, the two obtained results in eq. (5) conserved the symmetry between , as mentioned before, we apply the Boopp's shift method on the above equation (11) to obtain the reduced Schrödinger equation for (m.i.p.) potential (without star product): Where the new operator of Hamiltonian  

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In recently work, we are interest with the first variety (19), after straightforward calculations, we can obtain the five important terms, which will be use to determine the (m.i.p.) potential potential in (NC: 2D-RSP): From above relations, one can write the deformed operator   Which allow us to obtaining the global potential operator   r H-ip nc for inverse power potential in both (NC: 2D-RSP) as: It's clearly, that the four first terms are given the ordinary inverse power potential in 2D space, while the rest terms are proportional's with two infinitesimals parameters ( and ) and then gives the terms of perturbations The result (18) is logical for two reasons: the first one To the best of our knowledge, we just replace the coupling spin-orbital L S by the Substituting eq.
Ii is clearly that the above equation including equation (19), the perturbative terms of Hamiltonian operator, which we are subject of discussion in next sub-section, we look for a possibility to getting an exact solution of equation (29) (26).

4-1 The exact spin-orbital spectrum for (m.i.p.) potential in both (NC: 2D-RSP) for ground states in one-electron atoms:
In this sub section, we are going to study the modifications to the energy levels for ground states ( -ip u0 E and -ip d0 E ) for spin up and spin down, respectively, at first order of two parameters ( and  ) obtained by applying the standard perturbation theory, using eqs. (13) and (29)

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Know we apply the special integral [51]: Which allow us to obtaining the exact modifications of fundamental states ( -ip u0 E and -ip d0 E ) produced by spin-orbital effect:  for spin up and spin down, respectively, at first order of two parameters ( and  ), which obtained by applying the standard perturbation theory: Now we apply the special integral which represents by eq. (35) to obtain the explicitly results:

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Which allow us to obtaining the exact modifications ( ip -u1 E and ip -d1 E ) of degenerated first excited states produced for spin-orbital effect:  On the other hand, we consider interested physically meaningful phenomena, which produced automatically from the perturbative terms of modified inverse-power potential related to the influence of an external uniform magnetic field, it's sufficient to apply the following three replacements to describing these phenomena: We insert the values of the parameters given in Eqs. (58), (59) and (60) into Eqs. (37) and (51)   , which allow us to fixing ( 1 2  l ) values for the orbital angular momentum quantum numbers.

Obtained results
Let us now resume the eigenenergies of the modified Schrödinger equation obtained in this paper, the total modified energies (

Conclusions
To summarize, this work has been devoted to find the solutions of modified Schrodinger equation for the modified inverse power potential, we have obtained the exact energy spectrum for ground states and first excited states in (NC: 2D-RSP). We shown that the old states were changed radically and replaced by degenerated new states, describing two new original spectrums, the first one, produced by the of spin-orbital interaction while the second new spectrum produced by an external magnetic field, determined by the new results (37), (38), (51), (52), (61) and (62) in addition to the usual spectrum produced by ordinary potential. And we have also shown, every state in usually two dimensional spaces will be   1 2 4  l sub-states in the symmetries of (NC: 2D-RSP). Furthermore, our recently study can be generalized to including the interactions for muonic, hadronic and Rydberg atoms.