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

Figures C.38-C.40.

$T=2\pi$ ($q=1$) classically generates a trivial stochastic web which is translation invariant with respect to translations by $2\pi$ in $x$-direction and arbitrary translations in $p$-direction -- cf. figure 1.9. The solution (1.26) shows that the classical dynamics is confined to vertical lines in phase space and that $p_n$ grows linearly with $n$, such that ballistic energy growth $E_n\sim n^2$ is obtained. The figures in this section display corresponding quantum pictures for several values of $\hbar$. Note the different scaling of the $x$- and $p$-axes in these pictures, as opposed to all other pictures in this appendix.

After just a few kicks, a strong stretching mechanism in $p$-direction becomes obvious, the magnitude of which increases with $\hbar$, thereby indicating a genuine quantum effect. In all three figures, after at most 100 kicks the quantum state has propagated near to the boundary of the numerically accessible region of phase space.

It is clear that for $p_0\neq 0.0$ no different kind of dynamics is to be expected; therefore such pictures are omitted here.

Similar pictures are obtained for $T=\pi$ ($q=2$).