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Physics of Atomic Nuclei

D-dimensional Smorodinsky-Winternitz potential: Coherent state approach

Nuri Ünal

Published Oct 1, 2013• 0 citations
Publisher

Abstract

In this study, we construct the coherent states for a particle in the D -dimensional maximally superintegrable Smorodinsky-Winternitz potential. We, first, map the system into 2 D harmonic oscillators, second, construct the coherent states of them by evaluating the transition amplitudes. Third, in the Cartesian and the hyperspherical coordinates, we find the coherent states and the stationary states of the original sytem by reduction.

DOI
10.1134/s1063778813090202

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In this study, we construct the coherent states for a particle in the D -dimensional maximally superintegrable Smorodinsky-Winternitz potential. We, first, map the system into 2 D harmonic oscillators, second, construct the coherent states of them by evaluating the transition amplitudes. Third, in the Cartesian and the hyperspherical coordinates, we find the coherent states and the stationary states of the original sytem by reduction.
In this study, we construct the coherent states for a particle in the Smorodinsky-Winternitz potentials, which are the generalizations of the two-dimensional harmonic oscillator problem and the Kepler-Coulomb problem. In the first case we find the nonspreading wave packets by transforming the system into four oscillators in Cartesian, and also polar, coordinates. In the second case, the coherent states are constructed in Cartesian coordinates by transforming the system into three nonisotropic harmonic oscillators. All of these states evolve in physical-time. In the third case, the system is transformed into four oscillators and the parametric-time coherent states are constructed in two coordinate frames. In the fourth case, the system is transformed into two oscillators with the reflection symmetry and the parametrictime coherent states are constructed in two coordinate frames.
Abstract In this study, we construct the coherent states for a particle in the Smorodinsky-Winternitz potentials, which are the generalizations of the two-dimensional harmonic oscillator problem. In the first case, we find the non-spreading wave packets by transforming the system into four oscillators in Cartesian, and also polar, coordinates. In the second case, the coherent states are constructed in Cartesian coordinates by transforming the system into three non-isotropic harmonic oscillators. All of these states evolve in physical-time. We also show that in parametric-time, the second case can be transformed to the first one with vanishing eigenvalues.
An alternative method for the description of quantum superintegrable systems in two dimensions through the use of quantum algebraic techniques is introduced. It is suggested that such systems can be described in terms of a generalized deformed oscillator, characterized by a structure function specific to the system. The energy eigenvalues corresponding to a state with finite-dimensional degeneracy can then be determined directly from the properties of the relevant structure function. The validity of the method is demonstrated in the case of the isotropic harmonic oscillator in a space with constant curvature. The method can be used for constructing the quantum versions of several classical superintegrable systems, the Holt potential being given as an example.
In this paper a set of generalized coherent states for the d-dimensional Coulomb problem in coordinate representation are constructed. A coordinate transformation in hyperspherical space is used that maps the d-dimensional Coulomb problem into the D-dimensional harmonic oscillator and the generalized coherent states for the d-dimensional Coulomb problem are then obtained. This exactly soluble model can provide an adequate means for a quantum coherency description of the Coulomb problem in arbitrary dimensions, specifically in the special case of the hydrogen atom, in many theoretical and applied related fields such as in coherent scattering.