,

For larger

, the figures C.20 and C.31, using the same parameter values but

for the former and $p_0=2\pi /\sqrt {3}$ for the later, exhibit very similar phase portraits, despite the different initial conditions. For

, figure C.31 shows a (numerical) state the norm of which has already decayed considerably (to less than $10^{-10}$ ). Nevertheless, the most prominent feature of the phase portrait, its hexagonal structure, is still clearly visible.

The sequence of figures in the present subsection also nicely demonstrates some aspects of the transition from genuinely quantum behaviour (for larger values of $\hbar$ ) to more classical behaviour (for smaller $\hbar$ ): compare figure C.32 with figure C.35 and figure C.34 with figure C.36, for example. The smaller value of $\hbar$ leads to smaller structures in phase space, and to a closer resemblance with the classical hexagonal stochastic web as displayed in figure 1.3a.

What is more, the figures exhibit some fingerprints of the classical POINCARÉ map (1.21) in the quantum phase space dynamics generated by the quantum map (2.37). In particular the figures C.35 (

) and C.36 (

) apparently demonstrate the way in which the classical heteroclinic connections of the periodic points forming the skeleton of the web act as separatrices, separating an incident (part of the) quantum wave packet into two parts that are driven away from the corresponding fixed point along the unstable manifold of that fixed point.

The same observations, but for $T=\pi /2$ rather than $T=2\pi /3$ , can be made with respect to figures C.12-C.17 and 4.3. Especially the figures 4.3 (

) and C.12 (

) show the effect quite clearly. See also figure 4.5 (

) for the same pattern in the quantum $T=\pi /3$ -dynamics.

It should be kept in mind, though, that these are just phenomenological observations; the transition from quantum to classical behaviour cannot be described in these terms in a mathematically sound way. This is reflected by the fact that for smaller $\hbar$ a (possibly much) larger basis is needed to describe quantum states in the same region of phase space -- cf. table C.1.

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