  Quantum weak measurements with states both pre- and postselected offer a
window into a hitherto neglected sector of quantum mechanics. A class of such
systems involves time dependent evolution with transitions possible. In this
paper we explore two very simple systems in this class. The first is a toy
model representing the decay of an excited atom. The second is the tunneling of
a particle through a barrier. The postselection criteria are chosen as follows:
at the final time, the "atom" remains in its initial excited state for the
first example and the particle remains behind the barrier for the second. We
then ask what weak values are predicted in the physical environment of the
"atom" (to which no net energy has been transferred) and in the region beyond
the barrier (to which the particle has not tunneled). Previous work suggests
that very large weak values might arise in these regions for long durations
between pre- and postselection times. Our calculations reveal some distinct
differences between the two model systems.
