  In delocalized systems, particle number fluctuations, also known as quantum
surface roughness, and the mean-square displacement exhibit a temporal
power-law growth followed by a saturation to a system-size-dependent value. We
use simple scaling arguments to show that these quantities satisfy the
Family-Vicsek scaling law and derive a dynamic scaling relation between the
dynamical exponents, assuming that the saturation times of both quantities
scale similarly with the system size. This relation clarifies the mechanism
behind quantum surface roughness growth and suggests that diffusive quantum
many-body systems belong to the Edwards-Wilkinson universality class. Moreover,
it provides a convenient way to assess quantum transport in cold-atom
experiments. We numerically verify our results by studying two non-interacting
models and one interacting model having regimes with distinct dynamical
exponents.
