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Journal of Mathematical Physics

Parametric-time coherent states for the generalized MIC-Kepler system

Nuri Ünal

Published Dec 1, 2006• 10 citations
Publisher

Abstract

In this study, we construct the parametric-time coherent states for the negative energy states of the generalized MIC-Kepler system, in which a charged particle is in a monopole vector potential, a Coulomb potential, and a Bohm-Aharonov potantial. We transform the system into four isotropic harmonic oscillators and construct the parametric-time coherent states for these oscillators. Finally, we compactify these states into the physical time coherent states for the generalized MIC-Kepler system.

DOI
10.1063/1.2399362

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In this study, we construct the parametric-time coherent states for the negative energy states of the generalized MIC-Kepler system, in which a charged particle is in a monopole vector potential, a Coulomb potential, and a Bohm-Aharonov potantial. We transform the system into four isotropic harmonic oscillators and construct the parametric-time coherent states for these oscillators. Finally, we compactify these states into the physical time coherent states for the generalized MIC-Kepler system.
Abstract In this work, we constructed parametric-time coherent states (CS) for a superintegrable Coulomb-type system using the Feynman path integral formalism. This system corresponds to a deformation-dependent extension of the Kepler–Coulomb potential and exhibits maximal superintegrability governed by an indexing parameter. Using the Duru–Kleinert transformation, we mapped the problem onto an effective four-dimensional isotropic harmonic oscillator in parametric time. This transformation allowed the explicit derivation of the propagator, energy spectrum, and coherent-state wave functions while preserving the underlying symmetry structure. The resulting parametric-time CS satisfied the minimum uncertainty relation in the unsqueezed limit and exhibited non-trivial time-dependent squeezing behaviour. We showed that the deformation parameter controlled both the effective spectral scale and the redistribution of quantum fluctuations, leading to deformation-dependent noise suppression.
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.
The coherent states for a particle in Kratzer type potentials are constructed by solving Feynman’s path integral. The action of the generalized Kratzer potential is transformed into two harmonic oscillators by Levi-Civita transformation to derive the parametric time coherent states. Green’s function, energy eigenvalues, and the corresponding wave functions for this potential are calculated. The evaluated results are reduced to the modified Kratzer potentials and Kratzer-Fues oscillators which are special cases of the generalized Kratzer potential.
The Kustaanheino-Stiefel transformation is used to transform the Coulomb problem to a four-dimensional isotropic harmonic oscillator with a constraint. Ordinary coherent states are introduced over this oscillator and are shown to evolve in a fictitious time. When projected back into the physical space, the states follow the classical Kepler orbits in physical time. The resulting quasiclassical picture is obtained without the Bohr correspondence principle of taking only contributions from high principal quantum numbers.