  We explore the role of the initial state on the onset of thermalization in
isolated quantum many-body systems after a quench. The initial state is an
eigenstate of an initial Hamiltonian $\hat{H}_I$ and it evolves according to a
different final Hamiltonian $\hat{H}_F$. If the initial state has a chaotic
structure with respect to $\hat{H}_F$, i.e., if it fills the energy shell
ergodically, thermalization is certain to occur. This happens when $\hat{H}_I$
is a full random matrix, because its states projected onto $\hat{H}_F$ are
fully delocalized. The results for the observables then agree with those
obtained with thermal states at infinite temperature. However, finite real
systems with few-body interactions, as the ones considered here, are deprived
of fully extended eigenstates, even when described by a nonintegrable
Hamiltonian. We examine how the initial state delocalizes as it gets closer to
the middle of the spectrum of $\hat{H}_F$, causing the observables to approach
thermal averages, be the models integrable or chaotic. Our numerical studies
are based on initial states with energies that cover the entire lower half of
the spectrum of one-dimensional Heisenberg spin-1/2 systems.
