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 θ and θ¯ 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.

In present work, the exact analytical bound-state solutions of modified Schrödinger equation with Modified central potential consisting of a Cornellmodified plus pseudoharmonic harmonic potential (MCMpH) have been presented using both Boopp's shift method and standard perturbation theory, we have also constructed the corresponding noncommutative Hamiltonian which containing two new terms, the first one is modified Zeeman effect and the second is new spin-orbital interaction. The theoretical results show that the automatically appearance for both spin-orbital interaction and modified Zeeman Effect leads to the degenerate to energy levels to ( ) 1 2 2 + l sub states.

In present work we obtain the modified bound-states solutions for central family V(r)=A/r2-B/r+Crk (k=0,-1,-2) 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 Θ and θ corresponding the noncommutativity of space and phase, in addition to the discrete atomic quantum numbers: j, l, sz=+-1/2 and 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).

In this paper, we present a novel theoretical analytical perform further investigation for the exact solvability of relativistic quantum spectrum systems for modified Mie-type potential (m.m.t.) potential is discussed for spin-1/2 particles by means Boopp's shift method instead to solving deformed Dirac equation with star product, in the framework of noncommutativity three dimensional real space (NC: 3D-RS). The exact corrections for excited th n states are found straightforwardly for interactions in one-electron atoms by means of the standard perturbation theory. Furthermore, the obtained corrections of energies are depended on four infinitesimal parameter  ,which induced by position-position noncommutativity, in addition to the discreet atomic quantum numbers: 1 / 2, 1 / 2, j l s l     and m (the angular momentum quantum number) and we have also shown that, the usual states in ordinary two and three dimensional spaces are canceled and has been replaced by new degenerated   2 2 1 l  sub-states in the new quantum symmetries of (NC: 3D-RS) and we have also applied our obtained results to the case of modified Krazer-Futes potential.

In present work we study the 3-dimensional non relativistic and noncommutative space-phase Schrödinger equation for modified potential (see formula in paper) 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:n, l, j and s=±1/2 . 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.

The Klein–Gordon equation with equal scalar and vector potentials [Formula: see text] describing the dynamics of a three-dimensional under the modified Coulomb plus inverse-square potential is considered, in the symmetries of noncommutative quantum mechanics (NCQM), using Bopp’s shift method. The new energy of [Formula: see text]th excited state [Formula: see text] is obtained as a function of the shift energy [Formula: see text] and [Formula: see text] is obtained via first-order perturbation theory in the three-dimensional noncommutative real space (NC: 3D-RS) symmetries instead of solving modified Klein–Gordon equation (MKGE) with the Weyl–Moyal star product. It is found that the perturbative solutions of discrete spectrum depended by the Gamma function, the discreet atomic quantum numbers [Formula: see text] and the potential parameters (A and B), in addition to noncommutativity parameters ([Formula: see text] and [Formula: see text]), which are induced with the effect of (space–space) noncommutativity properties.

A novel study for the exact solvability of relativistic quantum spectrum systems for extended Cornell potential is discussed used both Boopp's shift method and standard perturbation theory in noncommutativity three dimensional real space (NC-3DS), furthermore the exact corrections for the spectrum of studied potential was depended on infinitesimal parameter  and a new discreet quantum numbers and we have also found the corresponding noncommutative Hamiltonian.

Abstract. Energy levels of one electron atoms have been re-examined by applying an alternative perturbative scheme in solving the modified Dirac equation (m.d.e.) for the modified Yukawa potential model with an arbitrary spin-orbit quantum number