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Physica Scripta

Parametric-time coherent states in a superintegrable Coulomb-type system

Nalan Kandırmaz

Published Sep 10, 2026• 0 citations

Abstract

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.

DOI: 10.1088/1402-4896/aea5fa

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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.

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.

A one-dimensional wave function is assumed whose logarithm is a quadratic form in the configuration variable with time-dependent coefficients. This trial function allows for general time-dependent solutions both of the harmonic oscillator (HO) and the reversed harmonic oscillator (RO). For the HO, apart from the standard coherent states, a further class of solutions is derived with a time-dependent width parameter. The width of the corresponding probability density fluctuates, or "breathes" periodically with the oscillator frequency. In the case of the RO, one also obtains normalized wave packets which, however, show diffusion through exponential broadening with time. At the initial time, the integration constants give rise to complete sets of coherent states in the three cases considered. The results are applicable to the quantum mechanics of the Kepler-Coulomb problem when transformed to the model of a four-dimensional harmonic oscillator with a constraint. In the classical limit, as was shown recently, the wave packets of the RO basis generate the hyperbolic Kepler orbits, and, by means of analytic continuation, the elliptic orbits are also obtained quantum mechanically.

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.

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 Feynman path integral for the generalized harmonic oscillator is reviewed, and it is shown that the path integral can be used to nd a complete set of wave functions for the oscillator. Harmonic oscillators with dierent (time-dependent) parameters can be related through unitary transformations. The existence of generalized coherent states for a simple harmonic oscillator can then be interpreted as the result of a (formal) invariance under a unitary transformation which relates the same harmonic oscillator. In the path integral formalism, the invariance is reflected in that the kernels do not depend on the choice of classical solutions.