  We study the intricate relationships between the dynamical scaling properties
of electron wave packets and the multifractality of the eigenstates in quantum
systems. Numerical simulations for the Harper model and the Fibonacci chain
indicate that the root mean square displacement displays the scaling behavior
$r(t)\sim t^\beta$ with $\beta=D_2^\psi$, where $D_2^\psi$ is the correlation
dimension of the multifractal eigenstates. The equality can be generalized to
$d$-dimensional systems as $\beta=D_2^\psi/d$, as long as the electron motion
is ballistic in the effective $D_2^\psi$-dimensional space. This equality
should be replaced by $\beta<D_2^\psi/d$ if the motion is non-ballistic, as
supported by all known results.
