Asymptotic properties of quantum dynamics in bounded domains at various time scales

I. V. Volovich ; A. S. Trushechkin

Izvestiya. Mathematics , Tome 76 (2012) no. 1, pp. 39-78

Résumé

We study a peculiar semiclassical limit of the dynamics of quantum states on a circle and in a box (infinitely deep potential well with rigid walls) as the Planck constant tends to zero and time tends to infinity. Our results describe the dynamics of coherent states on the circle and in the box at all time scales in semiclassical approximation. They give detailed information about all stages of quantum evolution in the semiclassical limit. In particular, we rigorously justify the fact that the spatial distribution of a wave packet is most often close to a uniform distribution. This fact was previously known only from numerical experiments. We apply the results obtained to a problem of classical mechanics: deciding whether the recently suggested functional formulation of classical mechanics is preferable to the traditional one. To do this, we study the semiclassical limit of Husimi functions of quantum states. Both formulations of classical mechanics are shown to adequately describe the system when time is not arbitrarily large. But the functional formulation remains valid at larger time scales than the traditional one and, therefore, is preferable from this point of view. We show that, although quantum dynamics in finite volume is commonly believed to be almost periodic, the probability distribution of the position of a quantum particle in a box has a limit distribution at infinite time if we take into account the inaccuracy in measuring the size of the box.

Détail

Keywords: dynamics of quantum systems, semiclassical limit, weak limit.