  Measurable quantities that have positive values in classical dynamical
systems need not to be positive in quantum theory. For example, consider a free
quantum mechanical particle in one dimension. There are quantum states in which
the particle's velocity is positive with probability 1, but where the
probability flux for its position is locally negative; that is, while its
velocity points to the right, the particle travels to the left. These effects
are however small and limited in space and time by certain lower bounds, which
are called "quantum inequalities". Similar effects also appear for a particle
whose motion is governed by a Schroedinger equation with a certain class of
potentials.
