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arXiv:math-ph/0403009 · Journal of Physics A Mathematical and General

Coherent states associated with the wavefunctions and the spectrum of the isotonic oscillator

Kengatharam Thirulogasanthar, Nasser Saad

Published Apr 5, 2004• 36 citations
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Abstract

Classes of coherent states are presented by replacing the labelling parameter z of Klauder–Perelomov type coherent states by confluent hypergeometric functions with specific parameters. Temporally stable coherent states for the isotonic oscillator Hamiltonian are presented and these states are identified as a particular case of the so-called Mittag–Leffler coherent states.

DOI
10.1088/0305-4470/37/16/007

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Classes of coherent states are presented by replacing the labelling parameter z of Klauder–Perelomov type coherent states by confluent hypergeometric functions with specific parameters. Temporally stable coherent states for the isotonic oscillator Hamiltonian are presented and these states are identified as a particular case of the so-called Mittag–Leffler coherent states.
In this work, we study the physical and mathematical characteristics of Barut–Giradello coherent states for the isotonic oscillator. Some relevant optical and statistical properties are analyzed and discussed. Furthermore, the thermodynamic and temporal stabilities are obtained, respectively, from [Formula: see text]-representation of the density operator, and probability density approaches. In addition, a comparative analysis of non-classical behaviors is made between these states and those of the Gazeau–Klauder type constructed for this quantum system.
In order to examine the generalized coherent states, defined in the Barut-Girardello manner, whose normalization unction is just the four-parameter generalized Mittag-Leffler function, we used a special pair of non-linear perators which generate these states. Some characteristic properties for pure, as well as mixed (thermal) oherent states are also examined. All calculations are made using the rules of the diagonal ordering of perators (DOOT). Finally, the integral counterpart of the Mittag-Leffler coherent states are examined which is onnected with nu-function. The paper is an example of the application of a mathematical entities (Mittag-Leffler function and nu-function) in quantum mechanics (coherent states formalism)
We defined and used a pair of Hermitian annihilation and creation operators which generate the generalized coherent states, defined in the Barut-Girardello manner, whose normalization function is just the four-parameter generalized Mittag-Leffler function. We examined the characteristic properties for these pure, as well as mixed (thermal) coherent states. All calculations are made using the rules of the technique of diagonal ordering of operators. Finally, the integral counterpart of the Mittag-Leffler coherent states are examined which is connected with nu-function. The paper is an example of the application of a mathematical entity (Mittag-Leffler function) in quantum mechanics (coherent states formalism).
We introduce a right inverse for the annihilation operator which appears in the factorization of associated hypergeometric-type operators. This makes possible the definition of generalized coherent states for the associated eigenfunctions. These generalized coherent states satisfy the relevant properties, i.e., label continuity, overcompleteness, temporal stability and action identity required to establish a close connection between the quantum and classical formulations of a given physical system.
In the present scenario, coherent states of a quantum harmonic oscillator are generalized with positive Fox H auxiliary functions. The novel generalized coherent states provide canonical coherent states and Mittag-Leffler or Wright generalized coherent states, as particular cases, and resolve the identity operator, over the Fock space, with a weight function that is the product of a Fox H function and a Wright generalized hypergeometric function. The novel generalized coherent states, or the corresponding truncated generalized coherent states, are characterized by anomalous statistics for large values of the number of excitations: the corresponding decay laws exhibit, for determined values of the involved parameters, various behaviors that depart from exponential and inverse-power-law decays, or their product. The analysis of the Mandel Q factor shows that, for small values of the label, the statistics of the number of excitations becomes super-Poissonian, or sub-Poissonian, by simply choosing sufficiently large values of one of the involved parameters. The time evolution of a generalized coherent state interacting with a thermal reservoir and the purity are analyzed.
Abstract We construct a new class of nonlinear coherent states for the isotonic oscillator by replacing the factorial of the coefficients z n / n ! zn/n!{z}^{n}/\sqrt{n!}zn/n!​ of the canonical coherent states by the factorial x n γ ! = x 1 γ . x 2 γ … x n γ xnγ!=x1γ.x2γ…xnγ{x}_{n}^{\gamma }!={x}_{1}^{\gamma }.{x}_{2}^{\gamma }\dots {x}_{n}^{\gamma }xnγ​!=x1γ​.x2γ​…xnγ​ with x 0 γ = 0 x0γ=0{x}_{0}^{\gamma }=0x0γ​=0 , where x n γ xnγ{x}_{n}^{\gamma }xnγ​ is a sequence of positive numbers and γ is a positive real parameter. This also leads to the construction of a Bargmann-type integral transform which will allow us to find some integral transforms for orthogonal polynomials. The statistics of our coherent states will also be considered by the calculus of one called Mandel parameter. The squeezing phenomenon was also discussed.