This article is cited in 1 scientific paper (total in 1 paper)
Asymptotic properties of quantum dynamics in bounded domains at various time scales
I. V. Volovich, A. S. Trushechkin
Steklov Mathematical Institute, Russian Academy of Sciences
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.
dynamics of quantum systems, semiclassical limit, weak limit.
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Izvestiya: Mathematics, 2012, 76:1, 39–78
MSC: 81Q20, 81R30, 70A05, 81P05
I. V. Volovich, A. S. Trushechkin, “Asymptotic properties of quantum dynamics in bounded domains at various time scales”, Izv. RAN. Ser. Mat., 76:1 (2012), 43–84; Izv. Math., 76:1 (2012), 39–78
Citation in format AMSBIB
\by I.~V.~Volovich, A.~S.~Trushechkin
\paper Asymptotic properties of quantum dynamics in bounded domains at various time scales
\jour Izv. RAN. Ser. Mat.
\jour Izv. Math.
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This publication is cited in the following articles:
A. Trushechkin, “Semiclassical evolution of quantum wave packets on the torus beyond the Ehrenfest time in terms of Husimi distributions”, J. Math. Phys., 58:6 (2017), 062102, 16 pp.
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