Coherent states of Krawtchouk oscillator and beyond

Borzov, V. V.; Damaskinsky, E. V.

doi:10.1109/dd.2006.348175

Cited by 3 publications

(5 citation statements)

References 15 publications

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“…In this paper, we have studied the different patterns of energy spectrum of the nonlinear f -oscillator associated with the q-deformed Heisenberg-Weyl algebra U (α,β,γ ) q (h 4 ) where, in particular, inequidistant spectra of energy were detected and analysed in details. This fact has motivated the construction of a general class of generalized q-deformed coherent states related to the nonlinear f -oscillator which encompasses a wide class of q-deformed coherent states proposed in the literature [6][7][8][9][10][11][12][13][14][15][16][17]. Next, we have obtained analytical expressions for the scalar product, the completeness relation, the displacement operator, and the time evolution of a small group of q-deformed coherent states defined for q ∈ (0, 1) and restricted to specific values of parameters α and γ , namely α > γ > 0.…”

Section: Discussionmentioning

confidence: 99%

“…In fact, one of the most common states studied in literature are Klauder's coherent states [5] defined by application of a unitary displacement operator D(z) = exp(za † − z * a) acting on the vacuum state |0 (i.e., |z = D(z)|0 ), with {a, a † , n} satisfying the Heisenberg-Weyl algebra h 4 . Many authors have also investigated the q-deformed coherent states, which are associated with a generalization of the harmonic oscillator commutation relation, in different contexts and predicted new interesting results [6][7][8][9][10][11][12][13][14][15][16][17][18] For instance, Man'ko et al [13] have extended the theory of the usual coherent states to nonlinear coherent states using as examples the deformation relative to the dark states of trapped ions [12]. El Baz and Hassouni [14] have discussed the construction of coherent states associated to deformed bosons oscillators with emphasis on the normalizability, continuity and resolution of unity properties.…”

Section: Introductionmentioning

confidence: 99%

“…In addition, the author has shown that the resolution of unity can be expressed in the form of an ordinary integral with a positive weight function obtained through the analytic solution of the associated Stieltjes power-moment problem. On the other hand, Borzov and Damaskinsky [16] have defined, for oscillator-like systems, coherent states of Barut-Girardello type in connection with q-Hermite polynomials. Finally, Atakishiyev et al [18] have constructed a model, ruled by the dynamical symmetry of the quantum algebra su q (2), for a finite q-oscillator which has lower-and upper-bound spectra of equidistant energies.…”

Section: Introductionmentioning

confidence: 99%

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On theq-deformed coherent states of a generalizedf-oscillator

Marchiolli

2005

Phys. Scr.

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Adopting the framework of the generalized q-deformed Heisenberg-Weyl algebra U (α,β,γ ) q (h 4 ), 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.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On theq-deformed coherent states of a generalizedf-oscillator

Marchiolli

2005

Phys. Scr.

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“…In that connection, recently we have shown in Ref. [17] that definite 3-, 4-, and 5-parameter deformed extensions of the p,q-oscillator considered in [18,19,20,21] also belong to the Fibonacci class, i.e., possess the FP. In that same paper, we studied a principally different, socalled µ-deformed oscillator proposed earlier in [22], and shown that it does not possess the FP.…”

Section: Introductionmentioning

confidence: 95%

Polynomially deformed oscillators ask-bonacci oscillators

Gavrilik¹,

Rebesh²

2010

J. Phys. A: Math. Theor.

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A family of multi-parameter, polynomially deformed oscillators (PDOs) given by polynomial structure function ϕ(n) is studied from the viewpoint of being (or not) in the class of Fibonacci oscillators. These obey the Fibonacci relation/property (FR/FP) meaning that the n-th level energy E n is given linearly, with real coefficients, by the two preceding ones E n−1 , E n−2 . We first prove that the PDOs do not fall in the Fibonacci class. Then, three different paths of generalizing the usual FP are developed for these oscillators: we prove that the PDOs satisfy respective k-term generalized Fibonacci (or "k-bonacci") relations; for these same oscillators we examine two other generalizations of the FR, the inhomogeneous FR and the "quasi-Fibonacci" relation. Extended families of deformed oscillators are studied too: the (q; µ)-oscillator with ϕ(n) quadratic in the basic q-number [n] q is shown to be Tribonacci one, while the (p, q; µ)-oscillators with ϕ(n) quadratic (cubic) in the p,q-number [n] p,q are proven to obey the Pentanacci (Nine-bonacci) relations. Oscillators with general ϕ(n), polynomial in [n] q or [n] p,q , are also studied.1 The famous Fibonacci numbers stem from the relation1 extensions of the p,q-oscillator considered in [18,19,20,21] also belong to the Fibonacci class, i.e., possess the FP. In that same paper, we studied a principally different, socalled µ-deformed oscillator proposed earlier in [22], and shown that it does not possess the FP. For that reason, a new concept has been developed for this µ-oscillator. Namely, it was demonstrated that the µ-oscillator belongs to the more general, than Fibonacci, class of so-called "quasi-Fibonacci" oscillators [17].The goal of the present paper is to study yet another classes of nonlinear deformed oscillators which do not belong to the Fibonacci class. We treat, from the viewpoint of three possible ways of generalizing the FP, a class of polynomially deformed oscillators. It is proven, using the notion of deformed oscillator structure function [23,24,25], that those oscillators are principally of non-Fibonacci nature. Then we develop the generalization of FP for these oscillators along three completely different paths: (i) as oscillators with k-term generalized Fibonacci property; (ii) as oscillators obeying inhomogeneous Fibonacci relation; (iii) as quasi-Fibonacci oscillators. Besides, we study a family of (q; µ)-oscillators which is, in a sense, a mix of the quadratic and the AC type q-deformed oscillators, and demonstrate its Tribonacci property. This result is extended to a general r-th order polynomial in the AC-type of q-oscillator bracket [N] q , naturally leading to k-bonacci relations. In this respect, let us mention that similar k-bonacci relations were treated in [26] in connection with generalized Heisenberg algebras [27]. Likewise, for the (q; {µ})-oscillators with {µ} = (µ 1 , µ 2 , ..., µ r ), combining the polynomial and the q-deformed AC features, the general statement on their k-bonacci property is proven. In a similar manner, the th...

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“…For example, the widely studied harmonic oscillator system has several q-deformed descriptions [4,5]. These are obtained from each other by transformation, as shown in [6,7]. Some of the main problems in obtaining the qoscillator are the spectrum, Hamiltonian, position and time operator.…”

Section: Introductionmentioning

confidence: 99%

Thermal properties of a solid through q-deformed algebra

Marinho,

Brito,

Chesman

2011

Preprint

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We study the thermodynamics of a crystalline solid by applying q-deformed algebras. We based part of our study on both Einstein and Debye models, exploring primarily q-deformed thermal and electric conductivities as a function of Debye specific heat. The results revealed that q-deformation acts as a factor of disorder or impurity, modifying the characteristics of a crystalline structure, as for example, in the case of semiconductors.

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Coherent states of Krawtchouk oscillator and beyond

Cited by 3 publications

References 15 publications

On theq-deformed coherent states of a generalizedf-oscillator

On theq-deformed coherent states of a generalizedf-oscillator

Polynomially deformed oscillators ask-bonacci oscillators

Thermal properties of a solid through q-deformed algebra

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