大时间演化拉格朗日态期望值的半经典行为
Semiclassical behaviour of expectation values in time evolved Lagrangian states for large times
摘要：
研究了拉格朗日态在极限$\hbar\to 0$和$t\to\infty$下随时间演化的量子力学期望值的行为。我们表明，它强烈依赖于相应的经典系统的动力学性质。如果经典系统是强混沌的，即Anosov，则期望值趋于一个普适极限。这可以看作是经典系统中混合的模拟。如果经典系统是可积的，那么期望值不需要收敛，如果它们收敛，它们的极限取决于初始状态。在我们可以证明这种行为的时间尺度上出现了另一个差异，在混沌情况下，我们得到了Eklfest时间，$t\sim \ln(1/\hbar)$，而对于可积系统，我们有一个更大的时间范围。
英文摘要：
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.
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