The $q$-deformed harmonic oscillator, coherent states, and the uncertainty relation

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.