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arXiv:quant-ph/0411056 · Journal of Physics A Mathematical and General

Coherent states of the Pöschl–Teller potential and their revival dynamics

Utpal Roy, J Banerji, P K Panigrahi

Published Sep 28, 2005• 24 citations
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Abstract

A recently developed algebraic approach for constructing coherent states for solvable potentials is used to obtain the displacement operator coherent state of the Pöschl–Teller potential. We establish the connection between this and the annihilation operator coherent state and compare their properties. We study the details of the revival structure arising from different time scales underlying the quadratic energy spectrum of this system.

DOI
10.1088/0305-4470/38/41/019

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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.

Physical Review A•Oct 16, 2013•6 citations
A recently developed algebraic approach for constructing coherent states for solvable potentials is used to obtain the displacement operator coherent state of the Pöschl–Teller potential. We establish the connection between this and the annihilation operator coherent state and compare their properties. We study the details of the revival structure arising from different time scales underlying the quadratic energy spectrum of this system.
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.
We analyze the time evolution of two kinds of coherent states for a particle in a P\"oschl-Teller potential. We find a pair of canonically conjugate operators and compare the behavior of their time evolution for both coherent states. The nonlinear ones are more localized. The trajectory in the phase space of the mean values of these two operators is a kind of generalization of the Rose algebraic curves. The new pair of canonically conjugate variables leads to a fourth-order Schr\"odinger equation which has the same energy spectrum as the P\"oschl-Teller system.
This work prolongs recent investigations by Bergeron et al [see 2012 J. Phys. A: Math. Theo. 45 244028] on new SUSYQM coherent states for Pöschl-Teller potentials. It mainly addresses explicit computations of eigenfunctions and spectrum associated to the higher order hierarchic supersymmetric Hamiltonian. Analysis of relevant properties and normal and anti-normal forms is performed and discussed. Coherent states of the hierarchic first order differential operator Am,ν,βA_{m,ν,β}Am,ν,β​ of the Pöschl-Teller Hamiltonian Hν,β(m)H_{ν,β}^{(m)}Hν,β(m)​ and their characteristics are studied.
We find the existence of sub-Planck scale structures in the P\"oschl-Teller potential, which is an exactly solvable potential with both symmetric and asymmetric features. We analyze these structures in both cases by looking at the Wigner distribution of the state evolved from an initial coherent state up to various fractional revival times. We also investigate the sensitivity to perturbations of the P\"oschl-Teller potential and we verify that, similar to the harmonic oscillator, the presence of sub-Planck structure in phase space is responsible for a high sensitivity to phase-space displacements.
In a recent short note (Bergeron et al 2010 Europhys. Lett. 92 60003), we have presented the good properties of a new family of semi-classical states for Pöschl–Teller potentials. These states are built from a supersymmetric quantum mechanics (SUSYQM) approach and the parameters of these 'coherent' states are points in the classical phase space. In this paper, we develop all the mathematical aspects that have been left out of the previous paper (proof of the resolution of unity, detailed calculations of the quantized version of classical observables and mathematical study of the resulting operators: problems of domains, self-adjointness or self-adjoint extensions). Some additional questions such as asymptotic behavior are also studied. Moreover, the framework is extended to a larger class of Pöschl–Teller potentials. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to 'Coherent states: mathematical and physical aspects'.