  The physics of many closed, conservative systems can be described by both
classical and quantum theories. The dynamics according to classical theory is
symplectic and admits linear instabilities which would initially seem at odds
with a unitary quantum description. Using the example of three-wave
interactions, we describe how a time-independent, finite-dimensional quantum
system, which is Hermitian with all real eigenvalues, can give rise to a linear
instability corresponding to that in the classical system. We show that the
instability is realized in the quantum theory as a cascade of the wave function
in the space of occupation number states, and an unstable quantum system has a
richer spectrum and a much longer recurrence time than a stable quantum system.
The conditions for quantum instability are described.
