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arXiv:math-ph/9804015 · Journal of Physics A Mathematical and General

Quantum mechanics on theh-deformed quantum plane

Sunggoo Cho

Published Jan 1, 1999• 11 citations
arXivHTMLPDF

Abstract

We find the covariant deformed Heisenberg algebra and the Laplace-Beltrami operator on the extended h -deformed quantum plane and solve the Schrödinger equations explicitly for some physical systems on the quantum plane. In the commutative limit the behaviour of a quantum particle on the quantum plane becomes that of the quantum particle on the Poincaré half-plane, a surface of constant negative Gaussian curvature. We show that the bound state energy spectra for particles under specific potentials depend explicitly on the deformation parameter h . Moreover, it is shown that bound states can survive on the quantum plane in a limiting case where bound states on the Poincaré half-plane disappear.

DOI
10.1088/0305-4470/32/11/005

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We find the covariant deformed Heisenberg algebra and the Laplace-Beltrami operator on the extended h -deformed quantum plane and solve the Schrödinger equations explicitly for some physical systems on the quantum plane. In the commutative limit the behaviour of a quantum particle on the quantum plane becomes that of the quantum particle on the Poincaré half-plane, a surface of constant negative Gaussian curvature. We show that the bound state energy spectra for particles under specific potentials depend explicitly on the deformation parameter h . Moreover, it is shown that bound states can survive on the quantum plane in a limiting case where bound states on the Poincaré half-plane disappear.
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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
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