  The dynamics after a quantum quench is determined by the weights of the
initial state in the eigenspectrum of the final Hamiltonian, i.e., by the
distribution of overlaps in the energy spectrum. We present an analysis of such
overlap distributions for quenches of the anisotropy parameter in the
one-dimensional anisotropic spin-1/2 Heisenberg model (XXZ chain). We provide
an overview of the form of the overlap distribution for quenches from various
initial anisotropies to various final ones, using numerical exact
diagonalization. We show that if the system is prepared in the
antiferromagnetic N\'eel state (infinite anisotropy) and released into a
non-interacting setup (zero anisotropy, XX point) only a small fraction of the
final eigenstates gives contributions to the post-quench dynamics, and that
these eigenstates have identical overlap magnitudes. We derive expressions for
the overlaps, and present the selection rules that determine the final
eigenstates having nonzero overlap. We use these results to derive concise
expressions for time-dependent quantities (Loschmidt echo, longitudinal and
transverse correlators) after the quench. We use perturbative analyses to
understand the overlap distribution for quenches from infinite to small nonzero
anisotropies, and for quenches from large to zero anisotropy.
