  We determine an exact formula for the transition amplitude between any two
arbitrary eigenstates of the local $z$-magnetization operators in the quantum
XY chain. We further use this formula to obtain an analytical expression for
the return amplitude of fully polarized states and the N\'eel state on a ring
of length $L$. Then, we investigate finite-size effects in the return
amplitude: in particular quasi-particle interference halfway along the ring, a
phenomenon that has been dubbed traversal~\cite{FE2016}. We show that the
traversal time and the features of the return amplitude at the traversal time
depend on the initial state and on the parity of $L$. Finally, we briefly
discuss non-analyticities in time of the decay rates in the thermodynamic limit
$L\rightarrow\infty$, which are known as dynamical phase transitions.
