Dependence on Initial Conditions

Two different mechanisms of localization of a quantum state have to be distinguished. First, there are the quantum manifestations of classical properties leading to localization, such as invariant lines in classical phase space not yet having broken up und thus being obstacles to global dynamics in phase space. This is the dominant feature of the quantum dynamics displayed in figures 5.9 and 5.13; from a classical point of view it is to be expected and therefore less interesting. Second, there are the genuinely quantum effects that are leading to localization as visible in the remaining figures in subsection 5.2.1 (and in section C.4 of the appendix) and that are to be explained in section 5.3. Quite naturally, these quantum effects are in the focus of attention in the present chapter.

The quantum dynamics for $V_0=1.0$, while at first sight dominated by classical localization only, is also affected by the quantum localization mechanism in a crucial way. This can be seen by starting the dynamics with different initial conditions. For figures 5.9 and 5.13, the $\left\vert \psi_0 \right>=\left\vert m \right>$ have been chosen to be localized within the closed invariant lines, leading to the classical localization result. For figure 5.16,

on the other hand, the same initial states have been chosen (here for $\hbar =1.0$ and $\hbar =0.1$ only), but shifted to the point $(0,12.5)^t$, well beyond the outermost invariant line: $\left\vert \psi_0 \right>=\hat{D}(0,12.5)\left\vert m \right>$. (Note that the topmost classical initial point in figure 1.4 is $(0,12.5)^t$, too, making comparison between the classical and quantum cases easier.) The resulting dynamics obviously differs from the case with initial conditions near $(0,0)^t$: the quantum dynamics remains localized, while the classical dynamics becomes diffusive for large enough $n$.

To summarize, depending on the initial condition, one may have quantum localization for classical or quantum reasons, or for a combination of both.

The localization phenomena that have been demonstrated in section 5.2 using numerical means are given a theoretical foundation in the following section 5.3.