  Adiabatic approximations break down classically when a constant-energy
contour splits into separate contours, forcing the system to choose which
daughter contour to follow; the choices often represent qualitatively different
behavior, so that slowly changing conditions induce a sudden and drastic change
in dynamics. The Kruskal-Henrard-Neishtadt theorem relates the probability of
each choice to the rates at which the phase space areas enclosed by the
different contours are changing. This represents a connection within
closed-system mechanics, and without dynamical chaos, between spontaneous
change and increase in phase space measure, as required by the Second Law of
Thermodynamics. Quantum mechanically, in contrast, dynamical tunneling allows
adiabaticity to persist, for very slow parameter change, through a classical
splitting of energy contours; the classical and adiabatic limits fail to
commute. Here we show that a quantum form of the Kruskal-Neishtadt-Henrard
theorem holds nonetheless, due to unitarity.
