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.