Quantum Brownian motion on a lattice

For a quantum particle on a one-dimensional lattice, the dynamics under the influence of a time-dependent, white-noise potential is shown to differ significantly from the classical diffusive behavior. In particular, the square of the distance displaced from origin is proportional to ${\mathit{t}}^{0.5–0.6}$. The origin of this dispersive behavior is traced to the special properties of the force-force correlation function, which integrates to zero and has a magnitude varying as ${\mathit{t}}^{\mathrm{-}0.5}$. These characteristics are attributed to the continuous spread of the quantum wave packet as well as its interaction with the discrete lattice.