What Determines the Spreading of a Wave Packet?

The multifractal dimensions $D_2^{\mu }$ and $D_2^{\psi }$ of the energy spectrum and eigenfunctions, respectively, are shown to determine the asymptotic scaling of the width of a spreading wave packet. For systems where the shape of the wave packet is preserved, the $k$th moment increases as $t^{k\beta }$ with $\beta {=D}_2^{\mu }/D_2^{\psi }$, while, in general, $t^{k\beta }$ is an optimal lower bound. Furthermore, we show that in $d$ dimensions asymptotically in time the center of any wave packet decreases spatially as a power law with exponent $D_2^{\psi }-d$, and present numerical support for these results.