a period ($T_{cl}/2$) of phase. In other words, the CS will be situated on the other side of the potential well, but with the same amplitude. The wave-packet splits into two parts at time $t=T_{rev}/4$ with a relative phase of half a period $T_{cl}/2$. Four way break-up occurs instead at time $t=T_{rev}/8$, and one can see that the state at this fractional revival time becomes

$$\label{oneeightcl}
\chi_{\beta}(x,T_{rev}/8)=\frac{1}{2}\left[e^{-i\pi/4}\chi_{\beta}^{cl}(x,t)
+\chi_{\beta}^{cl}(x,t+T_{cl}/4)-e^{-i\pi/4}\chi_{\beta}^{cl}(x,t+T_{cl}/2)+
\chi_{\beta}^{cl}(x,t+3T_{cl}/4)\right].$$

One can generalize in a straightforward way to a generic fractional revival time. In fact, in this case the state can be written as $$\chi_{\beta}(x,t=\frac{r}{s}T_{rev})=\sum_{p=0}^{l-1}a_{p}\chi_{\beta}^{cl}(x,t-\frac{p}{l}T_{cl}),$$ where $a_{p}$ gives the amplitude with probability $|a_{p}|^{2}$ of each $l$ number of sub-wave-packets or clones of the original wave-packet. Each clone differs in phase from the original one by $T_{cl}/l$. As illustrated above, the six way break-up can be observed at the time $T_{rev}/12$.

![image](fig2.eps){width="7.in"}

Wigner distribution and sub-Planck scale structure
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The Wigner quasi-probability distribution provides a clear phase-space description of a quantum state [@schleich]. It is particularly suited for displaying quantum interference phenomena, which are associated to oscillations and to the presence of regions where the Wigner function attains negative values. From its definition, the Wigner function of the CS evolved up to 