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Journal of Mathematical Physics

Coherent states for Kratzer-type potentials

Nalan Kandırmaz

Published Jun 1, 2018• 6 citations
Publisher

Abstract

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.

DOI
10.1063/1.4999704

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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 coherent states for the Kratzer-Fues potential, which are eigenstates of the annihilation operator and minimize the generalized position-momentum uncertainty relation, are constructed using the algebraic approach. The method is extended to include the rotating Kratzer-Fues oscillator. Construction of such states is important in the theory of molecular interactions with coherent electromagnetic fields, for example, in calculation of the dynamic alternation of the refractive index due to the interactions of the laser pulse with the coherent vibration-rotational states of the diatomic molecule.
The Feynman path integral for the generalized harmonic oscillator is reviewed, and it is shown that the path integral can be used to find a complete set of wave functions for the oscillator. Harmonic oscillators with different (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) {\em 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.
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.
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.
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.