What Determines the Spreading of a Wave Packet?

Abstract

The multifractal dimensions ${D}_{2}^{\ensuremath{\mu}}$ and ${D}_{2}^{\ensuremath{\psi}}$ 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 ${t}^{k\ensuremath{\beta}}$ with $\ensuremath{\beta}{\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}D}_{2}^{\ensuremath{\mu}}/{D}_{2}^{\ensuremath{\psi}}$, while, in general, ${t}^{k\ensuremath{\beta}}$ 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 ${D}_{2}^{\ensuremath{\psi}}\ensuremath{-}d$, and present numerical support for these results.