  We derive a general quantum formula giving the mean-square displacement of a
diffusing particle as a function of time. Near {\bf 0 K} we find a universal
logarithmic behavior (valid for times longer than the relaxation time), and
deviations from classical behavior can also be significant at larger values of
time and temperature. Our derivation depends neither on the specific
composition of the heat bath nor on the strength of the coupling between the
bath and the particle. An experimental regime of microseconds and microdegrees
Kelvin would elicit the pure logarithmic diffusion.
