  The mean square displacement $\langle\left(x(t)-x(0)\right)^2\rangle$ of the
position $x$ of a free particle of mass $m$ at thermal equilibrium is evaluated
quantum mechanically. An analytical expression is obtained which shows an
initial quadratic increase of the mean square displacement with time and later
on a linear growth, with the slope $\hbar/m$, quite at variance with the result
from classical statistical mechanics. Results are discussed in relation to
observables from helium scattering or spin-echo experiments, and their possible
interpretation in terms of the classical and quantum mechanical expression for
the mean square displacement of an essentially free particle.
