New exact solution of the bound states for the potential family

ABSTRCT. In present work we obtain the modified bound-states solutions for central family 

ABSTRCT. In present work we obtain the modified bound-states solutions for central family in both noncommutative three dimensional spaces and phases. It has been observed that the energy spectra in ordinary quantum mechanics was changed, and replaced degenerate new states, depending on two infinitesimals parameters  and corresponding the noncommutativity of space and phase, in addition to the discrete atomic quantum numbers: and m corresponding to the two spins states of electron by (up and down) and non polarized electron. The deformed anisotropic Hamiltonian formed by three operators: the first describes usual the usual family potential, the second describe spin-orbit interaction while the last one describes the modified Zeeman effect (containing ordinary Zeeman effect).

INTRODUCTION:
Although, the two fundamentals equations of Klein-Gordon and Dirac are satisfied very large susifall at high energy for describing scalar particle with spin zero and fermionic particle with spin 2 / 1 respectively, the Schrödinger equation rest as a big revolution like general relativity and special relativity for describing physics phenomena at microscopic (Planks scales) and macroscopic scales (the planets movements). In last few years among different forms of physical central and non central potentials which appear in the operator of Hamiltonian, those received great attention the recent years in commutative and noncommutative spaces-phases at two and three dimensional spaces and phases . In 1947 Mr.: H. Snyder introduces a new physical notion in quantum mechanics know by the noncommutative geometry at small length scales to obtains an profound interpretations to physics and chemical and another field [24]. The notions of noncommutativity of space and phase based essentially on Seiberg-Witten map and Boopp's shift method and the star product :   LTD, 2015 In this present work, we want to study the family potential (   noncommutative 3D space and phase to discover the new symmetries which satisfied by Boopp's shift method instead of solving the (NC-3D) spaces and phases with star product, the Schrödinger equation will be treated by using directly star product procedure: The star product replaced by usual product together with a Boopp's shift [31][32][33][34][35][36][37][38][39][40][41][42][43]: The rest of present search is organized as follows: In next section, we briefly review the Schrödinger equation with family potential and we find the exact quantum spectrum of the bound states in (NC-3D) space and phase for studied potential. Finally, the important found results and the conclusions are discussed in the four and last section.

IN THREE DIMENSIONAL SPACES
, this potential for 0  C has been used to describe molecular structure and interactions [7][8][9][10][11], and also this potential has raised great interest in atomic and molecular physics for 1 and 0   k A [7,11], in another hand it is applied to examine the Zeeman quadratic effect and the magnetic field effect in the hydrogen atom [7,12]. On based to the principal reference [7], we can write the following differential equation for the radial function   r R nl in ordinary 3D spcace: Where n and l are radial and orbital angular momentum quantum numbers. A , B and C are strictly positive constants [7]. By using the following ansatz: Now, we present the analytical solutions for the potential [7][8][9][10]13] With [7][8][9][10]13]:

NONCOMMUTATIVE PHASE-SPACE HAMILTONIAN FOR FAMILY POTENTIAL
Know, we shall present the fundamental principles of the quantum noncommutative Schrödinger equation which resumed in the following steps [31][32][33][34][35][36][37][38][39][40][41][42][43] Which allow us to writing the three dimensional space-phase quantum noncommutative Schrödinger equations as follows: The Boopp's shift method permutes to reduce the above equation to the form: Here the two i x and i p operators in (NC-3D) phase and space are given by [31][32][33][34][35][36][37][38][39][40][41][42][43]: It's convenient to introduce to notations: After straightforward calculations, one can derive, the two operators r1 , based, on the eq. (15), in the first order of two infinitesimal parameters  and , the four important terms which will be used to determine the noncommutative new family potential can be written explicitly, in (NC-3D) spaces and phases as: in both (NC-3D) phase and space will be written as: Now we replace As it's known, this operator traduces the coupling between spin and orbital momentum. The After profound straightforward calculation, one can show that, the radial function   r R nl satisfied the following the equation, in (NC-3D: RSP): , respectively:  (24)       It's convenient to writing the above table to the following form: Applying the following special integration [44]: (37)

ILCPA Volume 58
After straightforward calculations, we can obtain the results: With:      The first two terms in above sex equations are proportional's with infinitesimal parameter  , which represent the modifications of Spectrum for noncommutative spcace while the rest parts are proportional's to second infinitesimal parameter  which represent the modifications of Spectrum for noncommutative phase. We conclude, from Eqs. (24), (25), (26), (42), (43) and (44) that, the total energy of electron with spin up and down   in (NC-3D) phase and space:

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ILCPA Volume 58 Here 2  and 2  are just infinitesimal real proportional constants, the magnetic moment