  Recently probabilistic hysteresis in isolated Hamiltonian systems of
ultracold atoms has been studied in the limit of large particle numbers, where
a semiclassical treatment is adequate. The origin of irreversibility in these
sweep experiments, where a control parameter is slowly (adiabatically) tuned
back and forth, turned out to be a passage blue back and forth across a
separatrix (integrable case) or a passage in and out of a chaotic sea in phase
space (chaotic case). Here we focus on the full quantum mechanical description
of the integrable system and show how the semiclassical results emerge in the
limit of large particle numbers. Instead of the crossing of a separatrix in
phase space, where classical adiabaticity fails, the origin of irreversibility
in the quantum system is a series of avoided crossings of the adiabatic energy
levels: they become so close that already for modest particle numbers the
change of the external parameter has to be unrealistically slow to reach the
quantum adiabatic limit of perfectly reversible evolution. For a slow but
finite sweep rate we find a broad regime where the quantum results agree with
the semiclassical results, but only if besides the limit $N\to \infty$ an
initial ensemble of states is considered, with sufficient initial energy width.
For a single initial energy eigenstate we find in contrast that the backward
sweep reveals strong quantum effects even for very large particle numbers.
