Geodesic
The $q$-deformed harmonic oscillator, coherent states, and the
Teoretičeskaâ i matematičeskaâ fizika, Tome 147 (2006) no. 2, pp. 315-322

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For a $q$-deformed harmonic oscillator, we find explicit coordinate representations of the creation and annihilation operators, eigenfunctions, and coherent states {(}the last being defined as eigenstates of the annihilation operator{\rm)}. We calculate the product of the “coordinate–momentum” uncertainties in $q$-oscillator eigenstates and in coherent states. For the oscillator, this product is minimum in the ground state and equals $1/2$, as in the standard quantum mechanics. For coherent states, the $q$-deformation results in a violation of the standard uncertainty relation{;} the product of the coordinate- and momentum-operator uncertainties is always less than $1/2$. States with the minimum uncertainty, which tends to zero, correspond to the values of $\lambda$ near the convergence radius of the $q$-exponential.
Mots-clés : $q$-deformation
Keywords: harmonic oscillator, creation operators, annihilation operators, coherent states, uncertainty relation. 
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V. V. Eremin; A. A. Meldianov. The $q$-deformed harmonic oscillator, coherent states, and the. Teoretičeskaâ i matematičeskaâ fizika, Tome 147 (2006) no. 2, pp. 315-322. http://geodesic.mathdoc.fr/item/TMF_2006_147_2_a7/