What Determines the Spreading of a Wave Packet?

The multifractal dimensions D^μ₂ and D^ψ₂ 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 kth moment increases as t^kβ with β = D^μ₂/D^ψ₂, while, in general, t^kβ 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^ψ₂-d, and present numerical support for these results.