  The classical and quantum evolution of a generic probability distribution is
analyzed. To that end, a formalism based on the decomposition of the
distribution in terms of its statistical moments is used, which makes explicit
the differences between the classical and quantum dynamics. In particular,
there are two different sources of quantum effects. Distributional effects,
which are also present in the classical evolution of an extended distribution,
are due to the fact that all moments can not be vanishing because of the
Heisenberg uncertainty principle. In addition, the non-commutativity of the
basic quantum operators add some terms to the quantum equations of motion that
explicitly depend on the Planck constant and are not present in the classical
setting. These are thus purely-quantum effects. Some particular Hamiltonians
are analyzed that have very special properties regarding the evolution they
generate in the classical and quantum sector. In addition, a large class of
inequalities obeyed by high-order statistical moments, and in particular
uncertainty relations that bound the information that is possible to obtain
from a quantum system, are derived.
