Exact Non-relativistic Energy Eigen Values for Modified Inversely Quadratic Hellmann Plus Inversely Quadratic Potential

In this current research, the solutions of modified Schrödinger equation (MSE) are presented for two companied potentials namely: modified inversely quadratic Hellmann potential and modified inversely quadratic potential (MIHQP), using generalization of Bopp’s shift method (instead to solving MSE with star product) and standard perturbation theory in extended quantum mechanics (EQM), we obtained modified Hamiltonian operator and corresponding modified eigenvalues in both three dimensional noncommutative space and phase (NC-3D: RSP) symmetries.


Introduction
It is well known that, the ordinary Schrodinger equation is one of the fundamental wave equations in physics. Recently, considerable efforts have been made towards obtaining exact analytic solution of MSE for central potentials in two and three dimensional space in different fields of nuclear physics, spectroscopy, quantum chemistry and many fields of matter sciences to search an profound physical and chemical interpretations at Nano and Plank's scales [1][2][3][4][5][6][7][8][9], and in particularly our works in this context [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29]. The algebraic physical structure of EQM based on the following fundamental four NC canonical commutations relations (NCCRs), in both Schrödinger and Heisenberg pictures (SP and HP), respectively, as ( ) are determined from two projection relations, respectively, as follows [16], By differentiating equation (3), we find that general two operators obey the Heisenberg equations of motion [16]: The formalism of star product, Bopp's shift method and the Seiberg-Witten map were played crucial roles in EQM. The Bopp's shift method will be apply in this paper instead of solving MSE in global group symmetry (GGS) (NC-3D:RSP), the MSE will be treated by using directly the two new commutators, in addition to usual ordinary commutators on quantum mechanics, in the both SP and HP representation, respectively [17][18][19][20][21][22], And a similarly non-vanish 9-commutators in HP .The aim of this work is to study the MIHQP in NC 3-D space and phase to discover the new spectrum in this new symmetries from which other modified potentials are deduced as special cases, this potential plays an important role in many fields of physics such as molecular physics, solid state and chemical physics on based to the main reference [30,31] in QM and our previously works [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29] in EQM. The rest of this search paper is organized as follows: in the next section we briefly present and review the basic of eigenvalues and eignenfunctions for IHQP in ordinary 3-D spaces. In section 3, we give a brief review of Bopp's' shift method, then, we derive the spin-orbital NC Hamiltonians ˆs o ih H − for MIHQP in GGS (NC-3D: RSP), we find the exact spectrum produced by ˆs o ih H − by applying ordinary perturbation theory and then we deduce the exact spectrum produced by NC magnetic Hamiltonians ˆm ih H − for MIHQP in GGS (NC: 3D-RSP). In section four we resume the global spectrum for MIHQP and the main results. Finally, section 5 is kept for conclusive remarks.

The IHQP in Ordinary 3-Dimensional Spaces
The purpose of this section is to give a briefly review of eigenvalues and eignenfunctions for ordinary IHQP on based to the main reference [30], where r represents the internuclear distance, a and δ are the strengths of the coulomb and Yukawa potentials, respectively, while δ is the screening parameter. The ordinary SE with potential ( ) Where E ordinary energies corresponding ( ) r V in ordinary 3-D spaces, the method of separation of variable has been applied in reference [30], The radial function ( ) nl R r for IHQP satisfying the following differential equation [30], Where 2 r z = , n N is the normalization constant and the factor ( )

Overview of the Formalism of Bopp's shift method
We have been given a brief description of the MSE in GGS (NC-3D: RSP) on based to our previously works [22][23][24][25][26][27]. To achieve this goal, we apply the important 4-steps on the ordinary SE: While, the last step corresponds to replace the ordinary old product by new star product ( ) * , which allow us to constructing the MSE in GGS (NC-3D: RSP) as: The Bopp's shift method allows finding the reduced above MSE without star product as: Thus, we have: On based to our references [26][27][28][29], we can write the two operators . Now, after straightforward calculations one can obtains the different terms for MIHQP in GGS (NC-3D: RSP) as follows: Which allow us to writing the modified potential ( ) ih V r in GGS (NC-3D: RSP) as follows: Where the additive operator We can observe that the above operator is proportional with two infinitesimal parameters θ andθ , thus, we considering as a perturbative term.

The Spin-Orbital NC Hamiltonian Operator for MIHQP in GGS (NC-3D: RSP)
In order to discover the new contribution of perturbative term for MIHQP, we turn to the case of spin ½ particles described by the MSE, we make the following two simultaneously transformations: Here S denote to the spin of a fermionic particle (like electron in Hydrogen atom). Now, it is possible to replace the spin-orbital to obtain directly the corresponding eigenvalues, and then new physical As it well known, the 4-operators ( respectively [23][24][25][26][27][28][29]. Then, one can form a diagonal matrixˆs o ih After profound calculation, one can show that, the radial function ( ) nl R r for MIHQP satisfying the following differential equation, in EQM structure of GGS (NC-3D: RSP): The aim of this subsection is to obtain the modifications to the energy levels for th n excited states u-ih E and d-ih E corresponding a fermionic particle with two polarizations spin up and spin down, respectively, at first order of two infinitesimal parameters Θ andθ . In order to achieve this goal, we apply the standard perturbation theory using eq. (14) for MIHQP: It is possible to write both u-ih E and d-ih The explicit mathematical forms of 3-factors Applying the following special integration [32], Inserting the above obtained expressions into equations (28), gives the following results for exact modifications u-ih E and d-ih E produced by new spin-orbital operator effect for MIHQP:  Having found out how to calculate the corrections of energies for the automatically produced spin-orbital, we can discover a second symmetry produced by the effect and influence of the noncommutativity of space-phase, known by modified Zeeman Effect for MIHQP, to found this physical symmetry we apply the same strategy in our previously works as follows [25][26][27][28][29],

The Exact Spectrum Produced by NC Magnetic Hamiltonians
The two parameters χ and σ are just only infinitesimal real J Nanosci Curr Res proportional's constants and B is a uniform external magnetic field, we orient it to ( ) Oz axis and then we can make the following two translations for MIHQP: As follows:  s ≠ ), we replace the one of two factors And the corresponding NC Hamiltonian operatorsˆn c ih H − can be fixed by the following results: We now look at some special cases and relationships between our recently results and some other existing results in our previously works.      (21)  , , , It is important to notice that, the appearance of the polarization states of a fermionic particle for MIHQP in the non-relativistic MSE indicates a validity of obtained results at high energy where the two relativistic equations Klein-Gordon and Dirac are applied; this gives a positive indication of the possibility to apply these results of various Nano-particles at nano scales. Finally, if we make the two simultaneously limits ( ) ( )

Conclusion
In this article, the Bopp's' shift method has been studied and applied to MIHQP and the corresponding new energy eigenvalues of MSE are successfully investigated by applying the standard perturbation theory in GGS (NC-3D: RSP), we showed the obtained degenerated spectrum depended by ordinary discrete atomic quantum numbers ( m , 1 2 j l = ± and 1 2 z s = ± ). Furthermore, the validity of obtained corrections can be prolonged to Nano-particles at Nano and Plank's scales. In addition, we recover the ordinary commutative spectrums when, we make the two simultaneously limits: ( ) ( ) , 0, 0 θ θ → for MIHQP in GGS (NC-3D: RSP). The results are in excellent agreement with our reference no. [16].