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arXiv:1410.0399

2D Schrödinger Equation with Mixed Potential in Noncommutaive Complex space

Slimane Zaim, Hakim Guelmamene

Published Oct 1, 2014• 1 citation
arXivHTMLPDF

Abstract

We obtain exact solutions of the 2D Schrödinger equation for Hydrogen atom with the lenear and Harmonic Potentials in noncommutative complex space, using the Power-series expansion method. Hence we can say that the Schrödinger equation in noncommutative complex space describes to the particles with spin (1/2)in an external uniform magnitic field. Where the noncommutativity play the role of magnetic field with created the total magnetic moment of particle with spin 1/2, who in turn shifted the spectrum of energy. Such effects are similar to the Zeeman splitting in a commutative space.

DOI
10.48550/arxiv.1410.0399

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We obtain exact solutions of the 2D Schrödinger equation for Hydrogen atom with the lenear and Harmonic Potentials in noncommutative complex space, using the Power-series expansion method. Hence we can say that the Schrödinger equation in noncommutative complex space describes to the particles with spin (1/2)in an external uniform magnitic field. Where the noncommutativity play the role of magnetic field with created the total magnetic moment of particle with spin 1/2, who in turn shifted the spectrum of energy. Such effects are similar to the Zeeman splitting in a commutative space.
We obtain exact solutions of the 2D Schrödinger equation with the Singular Even-Power and Inverse-Power Potentials in non-commutative complex space, using the Power-series expansion method. Hence we can say that the Schrödinger equation in non-commutative complex space describes to the particles with spin (1/2)in an external uniform magnitic field. Where the noncommutativity play the role of magnetic field with created the total magnetic moment of particle with spin 1/2, who in turn shifted the spectrum of energy. Such effects are similar to the Zeeman splitting in a commutative space.
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.
The two-dimensional hydrogen atom in an external magnetic field is considered in the context of phase space. Using the solution of the Schrödinger equation in phase space, the Wigner function related to the Zeeman effect is calculated. For this purpose, the Bohlin mapping is used to transform the Coulomb potential into a harmonic oscillator problem. Then, it is possible to solve the Schrödinger equation easier by using the perturbation theory. The negativity parameter for this system is realised.
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 paper described the exact solutions of Schrödinger equation for diatomic molecules with pseudoharmonic potential (r) = D(-), it was solved by applying the Boopp's shift method to first order in the non-commutativity parameter and standard perturbation theory, the bound state energy (eigen values) found to be degenerate. It's shown that the noncommutative Hamiltonian (NC) described automatically a particle with spin (1/2) interacted with an external magnetic field, including the usual Zeeman effect in commutative space. Furthermore, I observed the NC Hamiltonian can be described automatically the spin-orbit interaction, I justified my results by calculate the absolute incertitude.
We study the noncommutative nonrelativistic quantum dynamics of a neutral particle, which possesses an electric qaudrupole moment, in the presence of an external magnetic field. First, by introducing a shift for the magnetic field, we give the Schrödinger equations in the presence of an external magnetic field both on a noncommutative space and a noncommutative phase space, respectively. Then by solving the Schrödinger equations both on a noncommutative space and a noncommutative phase space, we obtain quantum phases of the electric quadrupole moment, respectively. We demonstrate that these phases are geometric and dispersive.