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Modern Physics Letters A

COHERENT STATE OF THE EXTENDED SCARF I POTENTIAL AND SOME OF ITS PROPERTIES

N. Aizawa, Pinaki Roy

Published Aug 30, 2013• 3 citations
Publisher

Abstract

Using shape invariance property we construct coherent state for a class of potentials containing Scarf I and its extensions, the solutions of the latter being given in terms of the recently discovered Jacobi-Xl type exceptional polynomials. It is shown that the coherent state possesses the property of resolution of unity and exhibits sub-Poissonian behavior. We then investigate the coherent state of Scarf I potential and its l = 1 extension in some detail to understand the similarities (and differences) between the exceptional orthogonal polynomials and their classical counterparts.

DOI
10.1142/s021773231350123x

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Using shape invariance property we construct coherent state for a class of potentials containing Scarf I and its extensions, the solutions of the latter being given in terms of the recently discovered Jacobi-Xl type exceptional polynomials. It is shown that the coherent state possesses the property of resolution of unity and exhibits sub-Poissonian behavior. We then investigate the coherent state of Scarf I potential and its l = 1 extension in some detail to understand the similarities (and differences) between the exceptional orthogonal polynomials and their classical counterparts.
We show that the Scarf I potential problem in a position-dependent mass background of the type m(α;x)=(1+αsin⁡x)−2m(α;x) = (1 + α\sin x)^{-2}m(α;x)=(1+αsinx)−2, 0<α<10<α<10<α<1, can be solved by using a point canonical transformation mapping the corresponding Schr\" odinger equation onto that of the Scarf I potential with constant mass. The inverse point canonical transformation then provides some exactly-solvable rational extensions of the Scarf I potential with positive-dependent mass associated with XmX_mXm​-Jacobi exceptional orthogonal polynomials of type I, II, or III. The Scarf I potential problem with position-dependent mass is shown to exhibit a deformed shape invariance property in a deformed supersymmetric framework. Such a property is also valid for extended potentials of type I and II. The results are illustrated with a simple example.
The revival structures for the X_m exceptional orthogonal polynomials of the Scarf I potential endowed with position-dependent effective mass is studied in the context of the generalized Gazeau-Klauder coherent states. It is shown that in the case of the constant mass, the deduced coherent states mimic full and fractional revivals phenomena. However in the case of position-dependent effective mass, although full revivals take place during their time evolution, there is no fractional revivals as defined in the common sense. These properties are illustrated numerically by means of some specific profile mass functions, with and without singularities. We have also observed a close connection between the coherence time τ_coh^m? and the mass parameter ?.
Gazeau-Klauder coherent states of the extended trigonometric Scarf potential, underlying the quadratic energy spectrum and associated with Jacobi-type Xm exceptional orthogonal polynomials P(a,b,m) n (x), are constructed. The temporal evolution of wave-packet coherent states are performed by means of an autocorrelation function and the full revival properties are investigated in the usual time-domain analysis. This latter seems to be less useful for describing the fractional revivals due to the complicated nature of coherent wave-packet. Fortunately the autocorrelation function revels a little signature of fractional revivals at the vicinity of quarters of the revival time Trev due to the quadratic energy spectrum and the use of the wavelet-based time-frequency analysis of the autocorrelation function provides an analytical and numerical observation of the fractional revivals at different orders of the system.
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.