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Chinese Physics Letters

Even and Odd Q -coherent State Representations of the Q -deformed Heisenberg-Weyl Algebra

Wang Fabo, Kuang Le-Man

Published Dec 1, 1992• 1 citation
Publisher

Abstract

We construct explicitly even and odd q -coherent states. These q -coherent states are introduced in terms of the q -functions defined in the paper. It is shown that the even and odd q -coherent states form a kind of representations of the q -deformed Heisenberg-Weyl algebra which is realized in the form of matrix q -differential operators in the even and odd q -coherent state space. We also analyse some different between the even and odd q -CSs and the usual even and odd CSs.

DOI
10.1088/0256-307x/9/12/001

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We construct explicitly even and odd q -coherent states. These q -coherent states are introduced in terms of the q -functions defined in the paper. It is shown that the even and odd q -coherent states form a kind of representations of the q -deformed Heisenberg-Weyl algebra which is realized in the form of matrix q -differential operators in the even and odd q -coherent state space. We also analyse some different between the even and odd q -CSs and the usual even and odd CSs.
At first, we introduce α-deformed algebra as a kind of generalization of the Weyl–Heisenberg algebra so that we get the su(2)- and su(1, 1)-algebras whenever α has specific values. After that, we construct coherent states of this algebra. Third, a realization of this algebra is given in the system of a harmonic oscillator confined at the center of a potential well. Then, we introduce two-boson realization of the α-deformed Weyl–Heisenberg algebra and use this representation to write α-deformed coherent states in terms of the two modes number states. Following these points, we consider mean number of excitations (we call them in general photons) and Mandel parameter as statistical properties of the α-deformed coherent states. Finally, the Fubini–Study metric is calculated for the α-coherent states manifold.
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.