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arXiv:1406.0881

Weak deformed approximate eigenfunctions for a generalized q-deformed coherent states

Sid-Ahmed Yahiaoui, M. Bentaïba

Published Jun 3, 2014• 0 citations
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Abstract

We use the weak deformed version of q-boson annihilation operator to solve the eigenvalues coherent states equation for the generalized q-deformed harmonic oscillator. We also describe the construction of their resolution of unity.

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Abstract The explicit expressions for generalized q -deformed harmonic oscillator coherent states ( q -DHO CS) obtained in terms of a weak- and strong-behavior expansions are investigated. We first employ the weak-deformed version ( s → 0 ) of q -boson annihilation operator to solve Barut-Girardello’s eigenvalues coherent states equation for generalized q -deformed harmonic oscillator, using for this some lemmas directly related to the resolution of Riccati differential equation. The similarity with the perturbation theory is established. In strong-behavior limit, ( s → ∞ ), the previous results are resumed using the variational perturbation theory, and the corresponding q -DHO CS are deduced. We also construct, in both limits, their associated completeness relations which turns out to be the elliptic Jacobi ϑ 3 -function.
We consider the q-deformed Schrödinger equation of the harmonic oscillator on the N-dimensional quantum Euclidean space. The creation and annihilation operators are found, which systematically produce all energy levels and eigenfunctions of the Schrödinger equation. In order to get the q series representation of the eigenfunction, we also give an alternative way to solve the Schrödinger equation which is based on the q analysis. We represent the Schrödinger equation by the q difference equation and solve it by using q polynomials and q exponential functions.
In this work we make use of deformed operators to construct the coherent states of some nonlinear systems by generalization of two definitions: i) As eigenstates of a deformed annihilation operator and ii) by application of a deformed displacement operator to the vacuum state. We also construct the coherent states for the same systems using the ladder operators obtained by traditional methods with the knowledge of the eigenfunctions and eigenvalues of the corresponding Schrödinger equation. We show that both methods yield coherent states with identical algebraic structure.
In the weak deformation (WD) approximation of the Weyl–Heisenberg algebra, the corresponding generalized coherent states and displacement operator are constructed. It is shown that those states, and contrary to the non-deformed Weyl-Heisenberg algebra, are not eigenstates of the annihilation operator. Moreover, and as an alternative to the Chaïchian et al. Q-deformed path integral approach (where Q is the deformation parameter), using the Bargmann Fock representation, we propose in the WD approximation, a general simple formalism. As an application, we calculate the propagator and the wave function of the harmonic oscillator.PACS Nos.: 03.65.Fd, 31.15.Kb
Adopting the framework of the generalized q -deformed Heisenberg–Weyl algebra , we present a mathematical procedure which leads us to obtain analytical expressions for a general class of q -deformed coherent states associated with the different patterns of the energy spectrum exhibited by the nonlinear f -oscillator. In particular, we establish the properties of a small group of q -deformed coherent states for α > γ > 0 with emphasis on the resolution of unity. As an application of these properties, we investigate the Robertson–Schrödinger uncertainty relation and the squeezing effect for the deformed coordinate and momentum operators, which are defined in terms of the abstract elements of this algebra. Furthermore, we also obtain the Wigner function and the correct quantum-mechanical marginal distributions in phase space.