  We study the distance of two wave functions under chaotic time evolution. The
two initial states are differed only by a local perturbation. To be entitled
"chaos" the distance should have a rapid growth afterwards. Instead of focusing
on the entire wave function, we measure the distance $d^2(t)$ by investigating
the difference of two reduced density matrices of the subsystem $A$ that is
spatially separated from the local perturbation. This distance $d^2(t)$ grows
with time and eventually saturates to a small constant. We interpret the
distance growth in terms of operator scrambling picture, which relates $d^2(t)$
to the square of commutator $C(t)$ (out-of-time-order correlator) and shows
that both these quantities measure the area of the operator wave front in
subsystem $A$. Among various one-dimensional spin-$\frac{1}{2}$ models, we
numerically show that the models with non-local power-law interaction can have
an exponentially growing regime in $d^2(t)$ when the local perturbation and
subsystem $A$ are well separated. This regime is absent in the
spin-$\frac{1}{2}$ chain with local interaction only. After sufficiently long
time evolution, $d^2(t)$ relaxes to a small constant, which decays
exponentially as we increase the system size and is consistent with eigenstate
thermalization hypothesis. Based on these results, we demonstrate that $d^2(t)$
is a useful quantity to characterize both quantum chaos and quantum
thermalization in many-body wave functions.
