A New study to the Schrödinger equation for Modified Potential  6 r b r a r r V in

anharmonic potential, noncommutative space and noncommutative phase. ABSTRCT: In present work, by applying Boopp’s shift method and standard perturbation theory we have generated exact nonrelativistic solution     r V c r b r a r r V i s p e r t i s in both three dimensional noncommutative space and phase (NC:

, we have also derived the corresponding noncommutative Hamiltonian.

INTRODUCTION:
It is well-known that the exact solutions the non relativitivstic Schrödinger equation and relativitivstic two equations Klein-Gordon and Dirac with central potentials play principal and important roles in different fields of sciences like atomic, nuclear, molecular, an harmonic and harmonic spectroscopy . In the last few years, the provisos study were extended to the noncommutative space-phase at two, three and N generalized dimensions . 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 parameters antisymmetric   The Boopp's shift method will be apply in this paper instead of solving the (NC-3D) spaces and phases with star product, the Schrödinger equation will be treated by using directly the two commutators, in addition to usual commutator on quantum mechanics [29][30][31][32][33][34][35][36][37][38][39][40][41]: and , (3) The main goal of this work is to extend our study in reference [34]  in three dimensional spaces. In section 3, we review and applying Boopp's shift method to derive: the deformed potential and noncommutative spin-orbital Hamiltonian. In section 4, we apply perturbation theory to find the spectrum for ground stat and first excited states and then we deduce the spectrum produced automatically by the external magnetic field. In section 5, we resume the global spectrum for modified potential   , l and E represent angular momentum and the energy while Then, the complete normalized wave functions and corresponding energies for the ground state and the first existed states, respectively [23]:

International Letters of Chemistry, Physics and Astronomy Vol. 61
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 simplest the form: The new modified Hamiltonian defied as a function of i x and i p : (16) The two i x and i p operators in (NC-3D) phase and space are given by [25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42]: On based to our references [37-40], we can write the two operators 2 r and 2 p in noncommutative three dimensional spaces and phases as follows: respectively. After straightforward calculations one can obtains the different terms in noncommutative three dimensional spaces and phases as follows: a ar r a (19) Which allow us to writing the modified three dimensional studied potential   r V isˆi n noncommutative three dimensional spaces and phases as follows: It's clearly that, the first 3-terms in eq. (20) represent the ordinary potential while the rest term is produced by the deformation of space and phase. The global perturbative potential operators in both noncommutative three dimensional spaces and phases will be written as:

THE NONCOMMUTATIVE HAMILTONIAN FOR MODIFIED POTENTIAL
  r V isˆ :

THE NONCOMMUTATIVE SPIN-ORBITAL HAMILTONIAN FOR MODIFIED
Here 2 1  S denote the spin of electron; it's possible also to rewriting eq. (22) to equivalent physical form: After profound straightforward calculation, one can show that, the radial function for spin up and spin down, respectively, at first order of two parameters  and  obtained by applying the standard perturbation theory: A direct simplification gives:

International Letters of Chemistry, Physics and Astronomy Vol. 61
Where, the five terms   Applying the following special integration [46]:

IN (NC: 3D-RSP):
Now, we turn to the modifications to the energy levels for first excited states ip -u1 E and ip -d1 E corresponding 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. (29) to obtain:    , then we can make the following translation:   ) of a particle fermionic with spin up and spin down corresponding fundamental states and first excited states, respectively, for modified potential  