$T=\pi /2$, $p_0=0.0$

Figures 4.2 and C.1-C.8. (Figures from chapters 4 and 5 are not repeated in this appendix.)

Figures 4.2 and C.1 nicely show how the quantum web develops with increasing number $n$ of kicks. Obviously, for $\left\vert \psi_0 \right>$ centered in a regular region of phase space, $\left\vert \psi_n \right>$ also tends to concentrate in the meshes of the web, rather than in the channels, where the phase space density gets transported away along the classical separatrices rapidly.

In figure C.2, $\hbar$ and $V_0$ are large enough to avoid the formation of a web-like structure, and the algorithm has to be stopped after only around 500 kicks, because the norm of the computed state has decayed considerably already.

In the remaining figures of this subsection, the initial states are localized enough -- due to the smaller values of $\hbar$ -- to yield localized states even for a very large number of kicks. An exception to this rule is displayed in figure C.5, where the large value of $V_0=30.0$ leads to the state flowing apart quickly; note that for $n{ {\protect\begin{array}{c} >\protect\\ [-0.3cm]\sim \protect\end{array}} }10000$ the HUSIMI distribution has developed a periodic structure that is quite different from the quantum web of figures 4.2 and C.1.

It is important to keep in mind that although a localized initial state, together with small enough $\hbar$ and $V_0$, leads to localized dynamics this does not rule out the existence of a global stochastic web: clearly, the complete web can be obtained by using an initial state with nonzero contributions to $F^{\rm H}(x,p,-0;1)$ in each of the meshes of the classical web.