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arXiv:1404.5110 · International Journal of Modern Physics A

GENERALIZED COHERENT STATES FOR THE SPHERICAL HARMONICS $Y_{m}^{m}(\theta,\phi)$

Hossein Fakhri, B. Mojaveri

Published Apr 20, 2010• 9 citations
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Abstract

The associated Legendre functions [Formula: see text] for a given l-m, may be taken into account as the increasing infinite sequences with respect to both indices l and m. This allows us to construct the exponential generating functions for them in two different methods by using Rodrigues formula. As an application then we present a scheme to construct generalized coherent states corresponding to the spherical harmonics [Formula: see text].

DOI
10.1142/s0217751x10048226

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GENERALIZED COHERENT STATES FOR THE SPHERICAL HARMONICS Ymm(θ,ϕ)Y_{m}^{m}(\theta,\phi)Ymm​(θ,ϕ)
The associated Legendre functions [Formula: see text] for a given l-m, may be taken into account as the increasing infinite sequences with respect to both indices l and m. This allows us to construct the exponential generating functions for them in two different methods by using Rodrigues formula. As an application then we present a scheme to construct generalized coherent states corresponding to the spherical harmonics [Formula: see text].
In this paper, the general framework to construct generalized coherent states for the generalized Heisenberg algebra, using a [Formula: see text]-analog of exponential function and [Formula: see text]-integers, is discussed. Based on these notions, generalized Bell polynomials and Bell numbers, and so their exponential generating functions as well as a [Formula: see text]-analog of the latter and generalized Dobiński formulas are obtained. Then, the normal ordered form of exponential function of generalized number operator and its [Formula: see text]-analog exponential are investigated for general characteristic functions. In particular, the Arik–Coon [Formula: see text]-deformed quantum harmonic oscillator and the undeformed quantum physical system of infinite square well potential cases are well studied.
We explicitly construct a Hamiltonian whose exact eigenfunctions are the generalized Laguerre functions. Moreover, we present the related raising and lowering operators. We investigate the corresponding coherent states by adopting the Gazeau–Klauder approach, where resolution of unity and overlapping properties are examined. Coherent states are found to be similar to those found for a particle trapped in a Pöschl–Teller potential of the trigonometric type. Some comparisons with Barut–Girardello and Klauder–Perelomov methods are noted.
This paper is concerned with the construction of phase operators, phase states, vector phase states, and coherent states for a generalized Weyl-Heisenberg algebra. This polynomial algebra (that depends on real parameters) is briefly described. The various states are defined on a finite- or infinite-dimensional space depending on the parameters. This report constitutes an introduction to three papers published by the authors in J. Phys. A [43 (2010) 115303 and 45 (2012) 244036] and J. Math. Phys. [52 (2011) 082101]. See these three papers for the relevant references.
The aim of this work is to construct nonlinear coherent states of the pseudo-harmonic oscillator and to analyze some of its properties. The pseudo-harmonic potential can be viewed as an intermediate case between the harmonic and inharmonic potentials. The fact allows the implementation (with notable results), for inharmonic potentials, of some methods well known for harmonic potentials. An example is represented by the construction of nonlinear coherent states, which in the case of harmonic oscillator has found a lot of applications.