 At best, quantum chaology in isolated systems has produced results that have interesting relationships with integrable and non-integrable classical systems and some important experimental results (e.g., Bayfield and Koch 1974; Casati, Chirikov, Izrailev and Ford 1979; Fishman, Grempel and Prange 1982; Casati, Chirikov and Shepelyanski 1984; Berry 2001). These relationships are all statistical as indicated. One issue with studying isolated, closed quantum systems is that the state spaces of these systems do not allow the formation of the state-space structures typically associated with classical chaotic systems. There are some exceptions discussed in the literature, but it is actually ambiguous if these are genuine cases of chaos. One example discussed in the literature is a quantum Hamiltonian operator for an \(N\)-dimensional torus: \(\bfrac{1}{2}(g_{k} n_{k} + n_{k} g_{k})\), where \(n_{k} = -i\partial / \partial \theta_{k}, \theta_{k}\)  is an angle variable, and \(d\theta_{i} /dt = g_{i}(\theta_{k})\) for \(i, k = 1, 2,3,..., N\) (Chirikov, Izrailev and Shepelyanski 1988, p. 79). The probability density for momentum grows exponentially fast, which seems to parallel SDIC for trajectories in the classical case. Again, it is far from clear that this is chaos; there is no principled reason for considering the exponential growth in some quantity as a mark of chaos (recall the example in the first paragraph of §1.2.6 above). 