Approach of the continuous q-Hermite polynomials to x-representation of q-oscillator algebra and its coherent states

We use the recursion relations of the continuous [Formula: see text]-Hermite polynomials and obtain the [Formula: see text]-difference realizations of the ladder operators of a [Formula: see text]-oscillator algebra in terms of the Askey–Wilson operator. For [Formula: see text]-deformed coherent states associated with a disc in the radius [Formula: see text], we obtain a compact form in [Formula: see text]-representation by using the generating function of the continuous [Formula: see text]-Hermite polynomials, too. In this way, we obtain a [Formula: see text]-difference realization for the [Formula: see text]-oscillator algebra in the finite interval [Formula: see text] as a [Formula: see text]-generalization of known differential formalism with respect to [Formula: see text] in the interval [Formula: see text] of the simple harmonic oscillator.

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Nonclassicality of photon-modulated spin coherent states in the Holstein-Primakoff realization

Chinese Physics B ◽

10.1088/1674-1056/ac40f5 ◽

2021 ◽

Author(s):

Xiaoyan Zhang ◽

Jisuo Wang ◽

Lei Wang ◽

Xiangguo Meng ◽

Baolong Liang

Keyword(s):

Coherent States ◽

Wigner Distribution ◽

Hermite Polynomials ◽

Photon Number ◽

Spin Ladder ◽

Bernoulli Distribution ◽

Ladder Operators ◽

Spin Number ◽

Photon Number Distribution ◽

Spin Coherent States

Abstract Two new photon-modulated spin coherent states (SCSs) are introduced by operating the spin ladder operators J ± on the ordinary SCS in the Holstein-Primakoff realization and the nonclassicality is exhibited via their photon number distribution, second-order correlation function, photocount distribution and negativity of Wigner distribution. Analytical results show that the photocount distribution is a Bernoulli distribution and the Wigner functions are only associated with two-variable Hermite polynomials. Compared with the ordinary SCS, the photon-modulated SCSs exhibit more stronger nonclassicality in certain regions of the photon modulated number k and spin number j, which means that the nonclassicality can be enhanced by selecting suitable parameters.

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EXACT SOLUTIONS, LADDER OPERATORS AND BARUT–GIRARDELLO COHERENT STATES FOR A HARMONIC OSCILLATOR PLUS AN INVERSE SQUARE POTENTIAL

International Journal of Modern Physics B ◽

10.1142/s0217979205032735 ◽

2005 ◽

Vol 19 (28) ◽

pp. 4219-4227 ◽

Cited By ~ 15

Author(s):

SHI-HAI DONG ◽

M. LOZADA-CASSOU

Keyword(s):

Harmonic Oscillator ◽

Exact Solutions ◽

Coherent States ◽

Factorization Method ◽

Ladder Operators ◽

Dynamical Group ◽

One Dimensional ◽

Hidden Symmetry ◽

The One ◽

Analytical Expressions

We present exact solutions of the one-dimensional Schrödinger equation with a harmonic oscillator plus an inverse square potential. The ladder operators are constructed by the factorization method. We find that these operators satisfy the commutation relations of the generators of the dynamical group SU(1, 1). Based on those ladder operators, we obtain the analytical expressions of matrix elements for some related functions ρ and [Formula: see text] with ρ=x2. Finally, we make some comments on the Barut–Girardello coherent states and the hidden symmetry between E(x) and E(ix) by substituting x→ix.

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The symmetric q-oscillator algebra: q-coherent states, q-Bargmann–Fock realization and continuous q-Hermite polynomials with 0 < q < 1

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887816500286 ◽

2016 ◽

Vol 13 (03) ◽

pp. 1650028 ◽

Cited By ~ 5

Author(s):

H. Fakhri ◽

A. Hashemi

Keyword(s):

Text Analysis ◽

Coherent States ◽

Hermite Polynomials ◽

Representation Space ◽

Oscillator Algebra ◽

Fock Representation ◽

Minimum Uncertainty

The symmetric [Formula: see text]-analysis is used to construct a type of minimum-uncertainty [Formula: see text]-coherent states in the Fock representation space of the symmetric [Formula: see text]-oscillator ∗-algebra with [Formula: see text]. Then, its corresponding [Formula: see text]-Hermite polynomials are derived by using the [Formula: see text]-Bargmann–Fock realization of the symmetric [Formula: see text]-oscillator algebra.

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Coherent States for the Isotropic and Anisotropic 2D Harmonic Oscillators

Quantum Reports ◽

10.3390/quantum1020023 ◽

2019 ◽

Vol 1 (2) ◽

pp. 260-270 ◽

Cited By ~ 4

Author(s):

James Moran ◽

Véronique Hussin

Keyword(s):

Harmonic Oscillator ◽

Configuration Space ◽

Coherent States ◽

Probability Density Functions ◽

Harmonic Oscillators ◽

Uncertainty Relations ◽

Density Functions ◽

Ladder Operators ◽

New States ◽

2D System

In this paper we introduce a new method for constructing coherent states for 2D harmonic oscillators. In particular, we focus on both the isotropic and commensurate anisotropic instances of the 2D harmonic oscillator. We define a new set of ladder operators for the 2D system as a linear combination of the x and y ladder operators and construct the S U ( 2 ) coherent states, where these are then used as the basis of expansion for Schrödinger-type coherent states of the 2D oscillators. We discuss the uncertainty relations for the new states and study the behaviour of their probability density functions in configuration space.

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A New Class of Higher-Order Hypergeometric Bernoulli Polynomials Associated with Lagrange–Hermite Polynomials

Symmetry ◽

10.3390/sym13040648 ◽

2021 ◽

Vol 13 (4) ◽

pp. 648

Author(s):

Ghulam Muhiuddin ◽

Waseem Ahmad Khan ◽

Ugur Duran ◽

Deena Al-Kadi

Keyword(s):

Generating Function ◽

Functional Equations ◽

Laguerre Polynomials ◽

Hermite Polynomials ◽

Bernoulli Polynomials ◽

Higher Order ◽

New Class

The purpose of this paper is to construct a unified generating function involving the families of the higher-order hypergeometric Bernoulli polynomials and Lagrange–Hermite polynomials. Using the generating function and their functional equations, we investigate some properties of these polynomials. Moreover, we derive several connected formulas and relations including the Miller–Lee polynomials, the Laguerre polynomials, and the Lagrange Hermite–Miller–Lee polynomials.

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NON-HERMITIAN OSCILLATOR-LIKE HAMILTONIANS AND λ-COHERENT STATES REVISITED

Modern Physics Letters A ◽

10.1142/s021773230100295x ◽

2001 ◽

Vol 16 (02) ◽

pp. 91-98 ◽

Cited By ~ 9

Author(s):

JULES BECKERS ◽

NATHALIE DEBERGH ◽

JOSÉ F. CARIÑENA ◽

GIUSEPPE MARMO

Keyword(s):

Harmonic Oscillator ◽

Hilbert Spaces ◽

Coherent States ◽

Scalar Product ◽

Points Of View ◽

Usual Scalar ◽

Usual Scalar Product

Previous λ-deformed non-Hermitian Hamiltonians with respect to the usual scalar product of Hilbert spaces dealing with harmonic oscillator-like developments are (re)considered with respect to a new scalar product in order to take into account their property of self-adjointness. The corresponding deformed λ-states lead to new families of coherent states according to the DOCS, AOCS and MUCS points of view.

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