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Journal of Nano- and Electronic Physics

A Nonrelativistic Subatomic Model to Describe a Behavior of 2D Anharmonic Sextic Potential for Atomic Nucleus in the Symmetries of Extended Quantum Mechanics

Abdelmadjid Maireche

Published Jan 1, 2019• 1 citation
Publisher

Abstract

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 𝜃. 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 parameters allowed us to consider the perturbation terms due to (space-phase) noncommutativity.

DOI
10.21272/jnep.11(1).01024

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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 and 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 two-dimensional system are investigated. We have shown that, the new energy of 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) reduce to new sub-group symmetry of NC two-dimensional real space (NC: 2D-RSP) under new modified Kratzer-type interactions.
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A theoretical analytical investigation for the exact solvability of non-relativistic quantum spectrum systems, at low energy for modified two-dimensional purely sextic double-well potential (2D-modified sextic DWAO potential) is discussed, by means Bopp's shift method, instead to solving deformed Schrödinger equation with star product, in the framework of both noncommutativite two dimensional real space and phase (NC: 2D-RSP), the exact corrections for lowest excitations are found straightforwardly for interactions in one-electron atoms by means of the standard perturbation theory for three special cases. Furthermore, the obtained corrections of energies are depended on the four infinitesimals parameters (θ, χ) and (θ,σ), which are induced by position-position and momentum-momentum noncommutativity, respectively, in addition to the discreet atomic quantum numbers (j=1±1/2,s=±1/2,1) and the magnetic quantum number m. We have also → generalized our obtained results to include the other atoms with spin S ≠ 1/2.
Modified Schrodinger equation plays a fundamental role for describing the behavior of a particle at microscopic scale. In this context, we have studied the non-relativistic energy spectra of the modified quantum mechanical gravitational potential plus the harmonic oscillator (MQMGHO) potential using the generalized Bopp’s shift method and standard perturbation theory within the framework of non-commutative 3-dimensional phase space (NC: 3D-RSP) symmetries. Furthermore, we have shown that the modified Hamiltonian operator containing a perturbed Coulomb potential, which physically means the global Hamiltonian of MQMGHO potential, can be described as Hydrogenic atoms interacting with strong potential composed of quantum mechanical gravitational potential plus the harmonic oscillator (QMGHO) potential and an auxiliary part. The bound state energy eigenvalues, in terms of discreet atomic quantum numbers ( j , n, landm ), four infinitesimal parameters , , ,  induced by positionposition and phase-phase non-commutativity, in addition to the dimensional parameters of MQMGHO potential and the corresponding noncommutative Hamiltonian operator were obtained for Hydrogenic atoms. Moreover, we have shown that, the total complete degeneracy of energy
The new exact energy eigenvalues for new modified Kratzer-type potentials (NMKP) are calculated for a few typical (N2, CO, NO, CH) molecules through the Bopp’s shift method in the global quantum group (GQG) of noncommutative three-dimensional real space-phase symmetries (NC: 3D-RSP) in the framework of two infinitesimal parameters  and  due to (space-phase) noncommutativity, by means of the solution of the noncommutative Schrödinger equation (NCSE). The perturbation property of the spin-orbital Hamiltonian operator and new Zeeman effect of 3D system are investigated and the corresponding energy eigenvalues Ekp(n, j ,l ,s ) and nc -kpE (n, j ,l ,s,m) are easily calculated from standard perturbation theory. We have shown also that, the GQG of (NC: 3D-RSP) reduce to new sub-group symmetry of NC threedimensional real space (NC: 3D-RS) under new three-dimensional NMKP interactions.