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arXiv:quant-ph/0309038 · Physical Review A

Coherent states for exactly solvable potentials

T. Shreecharan, Prasanta Kumar Panigrahi, J. Banerji

Published Jan 12, 2004• 38 citations
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Abstract

A general algebraic procedure for constructing coherent states of a wide class of exactly solvable potentials, e.g., Morse and P\"oschl-Teller, is given. The method, a priori, is potential independent and connects with earlier developed ones, including the oscillator-based approaches for coherent states and their generalizations. This approach can be straightforwardly extended to construct more general coherent states for the quantum-mechanical potential problems, such as the nonlinear coherent states for the oscillators. The time evolution properties of some of these coherent states show revival and fractional revival, as manifested in the autocorrelation functions, as well as, in the quantum carpet structures.

DOI
10.1103/physreva.69.012102

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Comment on “Coherent states for exactly solvable potentials”

Mohamad M. Toutounji

In Phys. Rev. A 69, 012102 (2004), Schreecharan, Panigrahi, and Banerji present a general algebraic procedure for constructing coherent states of a wide class of exactly solvable potentials, e.g., Morse and Poschl-Teller. I have examined coherent states of the Morse oscillator presented by the authors and have found them to have incorrect mathematical substitutions. As such, this Comment looks at the correctness of two important equations that have led to their Morse oscillator coherent states.

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arXiv:quant-ph/0407160•Journal of Physics A Mathematical and General•Aug 18, 2004•36 citations
A general algebraic procedure for constructing coherent states of a wide class of exactly solvable potentials, e.g., Morse and P"oschl-Teller, is given. The method, a priori, is potential independent and connects with earlier developed ones, including the oscillator-based approaches for coherent states and their generalizations. This approach can be straightforwardly extended to construct more general coherent states for the quantum-mechanical potential problems, such as the nonlinear coherent states for the oscillators. The time evolution properties of some of these coherent states show revival and fractional revival, as manifested in the autocorrelation functions, as well as, in the quantum carpet structures.
A general procedure for constructing coherent states, which are eigenstates of annihilation operators, related to quantum mechanical potential problems, is presented. These coherent states, by construction are not potential specific and rely on the properties of the orthogonal polynomials, for their derivation. The information about a given quantum mechanical potential enters into these states, through the orthogonal polynomials associated with it and also through its ground state wave function. The time evolution of some of these states exhibit fractional revivals, having relevance to the factorization problem.
Coherent states in the harmonic oscillator may be defined in several equivalent ways. One definition describes coherent states as special states satisfying a minimum-uncertainty requirement in position and momentum spaces. This definition is generalized for other potentials according to a method developed by Nieto et al. [Phys. Rev. D 20, 1321 (1979)] and is herein applied to the P\"oschl-Teller potential. A classical limit based on the harmonic-oscillator coherent-state classical limit is then developed and applied to the P\"oschl-Teller minimum-uncertainty states. In this limit, classical behavior may be obtained from these quantum states. Together with a completeness argument for the generalized coherent states, this result provides insight into quantum classical correspondence through the statistical interpretation of quantum mechanics.
Generalized coherent states for shape invariant potentials are constructed using an algebraic approach based on supersymmetric quantum mechanics. We show this generalized formalism is able to: a) supply the essential requirements necessary to establish a connection between classical and quantum formulations of a given system (continuity of labeling, resolution of unity, temporal stability, and action identity); b) reproduce results already known for shape-invariant systems, like harmonic oscillator, double anharmonic, Poschl-Teller and self-similar potentials and; c) point to a formalism that provides an unified description of the different kind of coherent states for quantum systems.
Generalized coherent states for shape invariant potentials are constructed using an algebraic approach based on supersymmetric quantum mechanics. We show this generalized formalism is able to: a) supply the essential requirements necessary to establish a connection between classical and quantum formulations of a given system (continuity of labeling, resolution of unity, temporal stability, and action identity); b) reproduce results already known for shape-invariant systems, like harmonic oscillator, double anharmonic, Poschl-Teller and self-similar potentials and; c) point to a formalism that provides an unified description of the different kind of coherent states for quantum systems.