Semiclassical behaviour of expectation values in time evolved Lagrangian states for large times

Abstract

We study the behaviour of time evolved quantum mechanical expectation values in Lagrangian states in the limit $\hbar\to 0$ and $t\to\infty$. We show that it depends strongly on the dynamical properties of the corresponding classical system. If the classical system is strongly chaotic, i.e. Anosov, then the expectation values tend to a universal limit. This can be viewed as an analogue of mixing in the classical system. If the classical system is integrable, then the expectation values need not converge, and if they converge their limit depends on the initial state. An additional difference occurs in the timescales for which we can prove this behaviour, in the chaotic case we get up to Ehrenfest time, $t\sim \ln(1/\hbar)$, whereas for integrable system we have a much larger time range.