  In this work we provide an overview of our recent results about the quench
dynamics of one-dimensional many-body quantum systems described by spin-1/2
models. To illustrate those general results, here we employ a particular and
experimentally accessible initial state, namely the N\'eel state. Both cases
are considered: clean chains without any disorder and disordered systems with
static random on-site magnetic fields. The quantity used for the analysis is
the probability for finding the initial state later in time, the so-called
survival probability. At short times, the survival probability may decay faster
than exponentially, Gaussian behaviors and even the limit established by the
energy-time uncertainty relation are displayed. The dynamics at long times
slows down significantly and shows a powerlaw behavior. For both scenarios, we
provide analytical expressions that agree very well with our numerical results.
