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Acta Physica Sinica

A STUDY OF SUq(2) COHERENT STATE BY MAKING USE OF q-DEFORMATION HARMONIC OSCILLATORS REALIZATION OF SUq(2) QUANTUM ALGEBRA

Hao San-Ru, 长沙水利电力师范学院物理系,长沙410077

Published Jan 1, 1993• 1 citation
Publisher

Abstract

The SUq(2) coherent state is constructed by making use of the q- deformation harmonic oscillators realization of the SUq(2) quantum algebra. It is shown that the basis functions of the SU, (2) quantum algebra is orthogonal. The completeness and normalizing property of the SUq(2) coherent state is studied. We point out that the coherent property of the SUq(2) co-herent state is affected by q parameter, and show more generalility than the usual SU(2) co-herent state.

DOI
10.7498/aps.42.691

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The SUq(2) coherent state is constructed by making use of the q- deformation harmonic oscillators realization of the SUq(2) quantum algebra. It is shown that the basis functions of the SU, (2) quantum algebra is orthogonal. The completeness and normalizing property of the SUq(2) coherent state is studied. We point out that the coherent property of the SUq(2) co-herent state is affected by q parameter, and show more generalility than the usual SU(2) co-herent state.
A deformation of quantum mechanics obtained by replacing commutators by q-commutators (where q is close enough to unity to be compatible with experiment) is studied. This idea is explored by studying the q-harmonic oscillator in both configuration space and momentum space. The complete set of states in both representations as well as the transformation between x and p space is obtained. The states as well as the matrix elements lie in the SUq(2) algebra. To obtain expectation values and transition probabilities one must average over the SUq(2) algebra.
By making use of the Fock States of the q-deformed quantum oscillator which possesses the symmetry of quantum group, we construct the Glauber coherent state of the q-deformed quantum oscillator (the q-Glauber coherent state ㄧα>q). We discuss the of completeness properties, particle number distribution, oscillator strength distribution and the minimal uncertainty principle of the q-Glauber coherent states, and point out that the coherent degree of the q-Glauber coherent state can be described by the parameter q.
It is shown that the generalization of the quantum harmonic oscillator to the case of the relativistic configurational space is a q-oscillator. The corresponding group of dynamical symmetry is the quantum group SUq(1,1). The deformation parameter being q=e- omega omega 0/ where omega 0=4mc2 and omega is a frequency of the oscillator. The deformed creation and annihilation operators are finite difference ones. The corresponding deformation of the Heisenberg-Weyl group and new coherent states are also considered.
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.