  In order to perform quantum Hamiltonian dynamics minimizing localization
effects, we introduce a quasi-one dimensional tight-binding model whose mean
free path is smaller than the size of the sample. This one, in turn, is smaller
than the localization length. We study the return probability to the starting
layer of the system by means of direct diagonalization of the Hamiltonian. We
create a one dimensional excitation and observe sub-diffusive behavior for
times larger than the Debye time but shorter than the Heisenberg time. The
exponent corresponds to the fractal dimension $d^{*} \sim 0.72$ which is
compared to that calculated from the eigenstates by means of the inverse
participation number.
