  We show that the isotropic harmonic oscillator in the ordinary euclidean
space ${\bf R}^N$ ($N\ge 3$) admits a natural q-deformation into a new quantum
mechanical model having a q-deformed symmetry (in the sense of quantum groups),
$SO_q(N,{\bf R})$. The q-deformation is the consequence of replacing $ R^N$ by
${\bf R}^N_q$ (the corresponding quantum space). This provides an example of
quantum mechanics on a noncommutative geometrical space. To reach the goal, we
also have to deal with a sensible definition of integration over ${\bf R}^N_q$,
which we use for the definition of the scalar product of states.
