M. MAIRECHE Abdelmadjid

In the present work, a nonrelativistic analytical model is presented and used to modify the Schrödinger equation for subatomic scales with two-dimensional new anharmonic sextic potential(2DNASP) for atomic nuclei in noncommutative two-dimensional real space-phase (NC: 2D-RSP). We applied the generalized Bopp’s shift method to obtain 2DNASP, which is composed of ordinary anharmonic sextic potential and new additive part proportional to the infinitesimal parameter ?. We have also observed the kinetic term containing a new additive part proportional to the infinitesimal parameter . Thus, the global potential proportional to two infinitesimal parame￾ters allowed us to consider the perturbation terms due to (space-phase) noncommutativity. Furthermore, the standard perturbation theory allowed us to find the corrections   , , , ,  E n m l b c u d as  and
E n m b c mag- sa  , , , corresponding to spin-orbital interaction and modified Zeeman effect, respectively. These corrections allowed us to find global energy levels 
  nc - u-d clh E n b c j l s m , , , , , ,, which depend on the discrete atomic quantum numbers (j, l, s, m) and the parameters of the studied potential (a, b, c), and we constricted the corresponding Hamiltonian operator p x nc as i i   ˆ ˆ,  . We have also generalized our obtained results to include the other nuclei’s atoms with spin Sdiffers from1 / 2 
. This study has allowed us to obtain the energy spectrum in the extended quantum mechanics, which interpreted three physical phenomena. The first one is an ordinary two dimensional new anharmonic sextic potential, while the second and third are the automatic appearance of spin-orbit interaction and modified magnetic influence, these phenomena were generated from the property of noncommutativity of space and phase. The obtained analytical results (the energy eigenvalues and the corresponding Hamiltonian operator)
are in good agreement with the existing results. The previous results, in ordinary quantum mechanics, become special cases when we make simultaneously the limits  , 0,0    .

In this work, an analytical expression for the nonrelativistic energy spectrum of some diatomic molecules was obtained through the Bopp’s shift method in the noncommutative (NC)
two-dimensional real space-phase symmetries (NC: 2D-RSP) with a new modified Kratzer-type potential (NMKP) in the framework of two infinitesimal parameters θ and θ due to (space-phase) noncommutativity, by means of the solution of the noncommutative Schrödinger equation. The perturbation property of the spin-orbital Hamiltonian operator and new Zeeman effect of the two-dimensional system are investigated. We have shown that the new energy of the diatomic molecule is the sum of ordinary energy of modified Kratzer-type potential, in commutative space, and new additive terms due to the contribution of the additive part of the NMKP. We have shown also that, the group symmetry of (NC: 2D-RSP) reduces to new sub-group symmetry of NC two-dimensional real space (NC: 2D-RSP) under new modified Kratzer-type interactions.

In this study, three-dimensional modified time-independent Schrödinger equation of modified vibrational-rotational structure of supersingular plus Coulomb (v.r.s.c) potential was solved using Boopp’s shift method instead to apply star product, in the framework of both noncommutativities three-dimensional real space and phase (NC: 3D-RSP). We have obtained the explicit energy eigenvalues for ground state and first excited state for interactions in one-electron atoms. Furthermore, the obtained corrections of energies are dependent on infinitesimal parameters , and  , which are induced by position-position and momentum-momentum noncommutativity, respectively, in addition to the discreet atomic quantum numbers: j  l 1/ 2,s  1/ 2,l and the angular momentum quantum number. We have also shown that the usual states in ordinary three-dimensional spaces for the ordinary vibrational-rotational structure of supersingular plus Coulomb potential are canceled and has been replaced by new degenerated
22l 1sub-states in the extended new quantum symmetries of (NC: 3D-RSP).

In present work, by applying Boopp’s shift method and standard perturbation theory we have generated exact nonrelativistic bound states solution for a modified potential
ˆ  , ,2 4 6     V r ar br cr V r is part in both three-dimensional noncommutative space and phase (NC: 3D-RSP) at first order of two infinitesimal parameters antisymmetric    ,, we have also derived the corresponding noncommutative Hamiltonian.

In present work, we study the 3-dimensional non-relativistic and noncommutative space-phase Schrödinger equation for modified potential (
    20 1,20 ,LL V  En lr V  En l  ) 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 , jand21s   . We show the noncommutative new anisotropic Hamiltonian containing two new important terms, the first new term describes the spin-orbit interaction while the second describes the modified Zeeman effect.

In present work we obtain the modified bound-states solutions for central family    0, 1, 22  Cr k   rBrV rkin 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  andcorresponding the noncommutativity of space and phase, in addition to the discrete atomic quantum numbers: j , l , sz 1/ 2and corresponding to the two spins states of electron by (up and ) and nonpolarized electron. The deformed anisotropic Hamiltonian formed by three operators: the first describes usually the usual family potential, the second describes spin-orbit interaction while the last one describes the modified Zeeman effect (containing ordinaryZeeman ).

A novel study for the exact solvability of nonrelativistic quantum spectrum systems
for companied Harmonic oscillator potential and its inverse (the isotropic harmonic oscillator plus
inverse quadratic potential) is discussed used both Boopp’s shift method and standard perturbation
theory in both noncommutativity two dimensional real space and phase (NC-2D: RSP), furthermore
the exact corrections for the spectrum of studied potential was depended on two infinitesimals
parameters and which plays an opposite rolls, this permits us to introduce a new fixing gauge
condition and we have also found the corresponding noncommutative anisotropic Hamiltonian.