  We consider the radially vibrating spherical quantum billiard as a
representative example of vibrating quantum billiards. We derive necessary
conditions for quantum chaos in $d$-term superposition states. These conditions
are symmetry relations corresponding to the relative quantum numbers of
eigenstates considered pairwise. In this discussion, we give special attention
to eigenstates with null angular momentum (for which the aforementioned
condition is automatically satisfied). Moreover, when these necessary
conditions are met, we observe numerically that there always exist parameter
values that correspond to chaotic configurations. We focus our numerical
studies on the ground and first excited states of the radially vibrating
spherical billiard with null angular momentum eigenstates. We observe chaotic
behavior in this configuration, which dispels the common belief that one is
required to pass to the semiclassical ($\hbar \longrightarrow 0$) or high
quantum number limits in order to meaningfully discuss quantum chaos. The
results in the present paper are also of practical import, as the radially
vibrating spherical quantum billiard may be used as a model for the quantum dot
nanostructure, the Fermi accelerating sphere, and intra-nuclear particle
behavior.
