  We study the commutation relations, uncertainty relations and spectra of
position and momentum operators within the framework of quantum group %
symmetric Heisenberg algebras and their (Bargmann-) Fock representations. As an
effect of the underlying noncommutative geometry, a length and a momentum scale
appear, leading to the existence of minimal nonzero uncertainties in the
positions and momenta. The usual quantum mechanical behaviour is recovered as a
limiting case for not too small and not too large distances and momenta.
