 One difference between classical chaotic dynamics and quantum dynamics is that the state space of the former supports fractal structure while the state space of the latter does not. A second difference is that classical chaotic dynamics has a continuous energy spectrum associated with its motion. As previously noted, classical chaos is considered to be a property of bounded macroscopic systems. In comparison, the quantum dynamics in bounded, isolated systems has a discrete energy spectrum associated with its motion. Moreover, phenomena such as SDIC could only be possible in quantum systems that appropriately mirror classical system behaviors. From semi-classical considerations, Berry et al. (1979) showed that semi-classical quantum systems (see below for how such systems are constructed) could be expected to mirror the behavior of their corresponding classical systems only up to the Ehrenfest time \(t_{E}\), of the order \(\ln(2\pi/h)\) secs,  an estimate also known as the log time reflecting the exponential instability of classical chaotic trajectories. In these semi-classical studies, \(h/2\pi\) often is treated as a parameter that is reduced in magnitude as the classical domain is approached. On this view, the smaller \(h/2\pi\), the more “classical” the system’s behavior becomes. For instance, assuming the value of Planck’s constant in KMS units, \(t_{E} \sim 80\) secs. As Planck’s constant decreases, \(t_{E}\) grows. In nonchaotic classical systems the orbits in state space are well isolated and everything well behaved for very long times. In contrast for bound chaotic systems the orbits start coalescing in increasing numbers on the scale of \(t_{E}\), implying that the semi-classical approximation fails by \(t_{E}\). On the other hand, a Gaussian wave packet centered on a classical trajectory is thought to be able to shadow that trajectory up to \(t_{E}\) before becoming too spread out over the energy surface since \(t_{E}\) is a measure of when quantum wave packets have spread too much to mimic classical trajectories and the Ehrenfest theorem breaks down. So there are two effects at work in semi-classical systems over time: (1) the coalescing of classical chaotic trajectories and (2) the spreading of quantum wave packets. Between the lack of nonlinearity in quantum mechanics and the latter two effects, things look rather bleak for finding close quantum analogs of classical chaos. 