  We show how a potential that is well-defined everywhere on the positive
half-line, but diverges to $-\infty$ as $x\rightarrow 0^+$, may still be able
to dynamically confine a particle to the (positive) half-line. We shall call
this effect quantum bungee jumping: in the familiar quantum tunneling scenario
one heading towards a wall is expected to bounce off it but instead passes
through, while in the present scenario one jumping off a cliff is expected to
fall but instead is pulled back. We discuss a particular example of a potential
displaying this property, and study its energy eigenstates and spectrum.
