A New Approach to the Non Relativistic Schrödinger equation for an Energy-Depended Potential

: In present work we study the 3-dimensional non relativistic and noncommutative space-phase Schrödinger equation for modified potential (

ABSTRaCT: In present work we study the 3-dimensional non relativistic and noncommutative space-phase Schrödinger equation for modified potential (     depends on energy and quadratic on the relative distance, we have obtained the exact modified bound-states solutions. It has been observed that, the energy spectra in ordinary quantum mechanics was changed, and replaced by degenerate new states, depending on new discreet quantum numbers: m , l , j and 2 1   s . We show the noncommutative new anisotropic Hamiltonian containing two new important terms, the first new term describe the spin-orbit interaction while the second describes the modified Zeeman effect.

INTRODUCTION
During the past several decades and few years, must effort has gone into studying the stationary non relativistic Schrödinger equation with central and non central potentials, in commutative and noncommutative two and three dimensional space and phase . In recent years, the study of central and non central potentials were prolonged to the relativistic stationary Schrödinger equation and the relativistic Dirac equation, in particularly, the spherical symmetry have received great attention because of their wide applications, a good example, we give the potential [7]. The purpose of this work is to present a new study of the previously potential in both noncommutative two and three dimensional spaces and phases, to discover the new spectrum. The notions of noncommutativity of space and phase based essentially on Seiberg-Witten map and Boopp's shift method and the star product, defined on the first order of two infinitesimal  as : As a direct principal's result of the above equation, the two new non nulls commutators in the notion of star product: The 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 [31][32][33][34][35][36][37][38][39][40][41][42][43]: The study of energy-depended potential have yielded acceptable results in the annals of particle and nuclear physics in addition to used in modification the scalar product to have a conserved norm  [7,8]. This study is based on our previous work [33][34][35][36][37][38][39]. The rest of present search is organized as follows: In next section, we briefly review the Schrödinger equation with an energy-depended potential in three dimensional spaces. In section 3, we review and applying Boopp's shift method to derive the deformed Hamiltonian for an energy-depended potential in noncommutative three dimensional space-phase ordinary three dimensional spaces. In the forth section we apply standard perturbation theory to find the exact quantum spectrum of the bound states in (NC-3D) space and phase for studied potential in first order of two infinitesimal parameters'  and . Finally, the important found results and the conclusions are discussed in the last section.

REVIEW OF ENERGY-DEPENDED POTENTIAL IN THREE DIMENSIONAL SPAPACES
In this section, we shall review the eigenvalues and eigenvalues for spherically symmetric quadratic depended on the special coordinates and linear one for the energy potential l n E , Is the energy of a fermionic particle moving in this potential, the parameters 0 V and  are constants [7], the complex eignenfunctions  . From reference [7], one can deduce immediately, that, the hyperradial part in 3-dimensions:

FORMALISM OF BOOPP'S SHIFT METOD
In this sub-section, we shall review some fundamental principles of the quantum noncommutative Schrödinger equation which resumed in the following steps for an energydepended potential [31][32][33][34][35][36][37][38][39][40][41][42][43][44]: 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][44]: n based to our reference [39], we written the two operators 2 r and 2 p in (NC-3D) spaces and phases as follows: edpˆ for an energy-depended potential in both (NC-3D) phase and space will be written as: It's clearly that, the first term in above equation are given the ordinary potential while the rest terms are proportional's with two infinitesimals parameters (  and ) and then gives the terms of perturbations for an energy-depended potential in (NC-3D) real space and phase as:

THE SECOND PART OF NONCOMMUTATIVE PHASE-SPACE HAMILTONIAN FOR ENERGY -DEPENDED POTENTIAL
Let us now to replace  , which allow us to writing the perturbative terms for an energy-depended potential:

THE THERID PART OF NONCOMMUTATIVE PHASE-SPACE HAMILTONIAN FOR ENERGY -DEPENDED POTENTIAL
On another hand, it's possible to consider the two at infinitesimals parameters (  and ) are the sum of two infinitesimals parameters to each one as [33][34][35][36][37][38] Then, the second part of noncommutative magnetic Hamiltonian operator for an energy-depended potential  

THE GLOBAL NONCOMMUTATIVE PHASE-SPACE HAMILTONIAN FOR ENERGY -DEPENDED
After profound straightforward calculation, one can show that, the radial function   r R l n, satisfied the following differential equation, in (NC-3D: RSP) for an energy-depended potential:

NONCOMMUTATIVE SPECTRUM FOR ENERGY DEPENDED -POTENTIAL
The purpose of this fundamental sub section is to draw the strategy of total exact energies: ( ) for an energy-depended potential in both (NC-3D) phase and space, respectively: are the exact spin-orbital (up-down) and magnetic modifications for the studied potential.   2 and 16

NONCOMMUTATIVE MAGENETIC SPECTRUM FOR ENERGY DEPENDED -POTENTIAL
To obtain the exact noncommutative magnetic modifications of energy  (31) Where m denote to eigenvalue of operator z L and can be take (