  We investigate dynamical scaling properties of the 1D tight-binding Anderson
model with a weak diagonal disorder, by means of the spreading of a wave
packet. In the absence of disorder, and more generally in the ballistic regime,
the wavefunction exhibits sharp fronts. These ballistic fronts yield an
anomalous time dependence of the $q$-th moment of the local probability
density, or dynamical participation number of order $q$, with a non-trivial
exponent $\tau(q)$ for $q>2$. This striking feature is interpreted as
bifractality. A heuristic treatment of the localised regime demonstrates a
similar anomalous scaling, but with the correlation length $\xi_0$ near the
band center replacing time. The moments of the position of the particle are not
affected by the fronts, and they exhibit normal scaling. The crossover
behaviour of all these quantities between the ballistic and the localised
regime is described by scaling functions of one single variable, $x=t/\xi_0$.
These predictions are confirmed by accurate numerical data, both in the normal
and in the anomalous case.
