Berry, MV (1977) Semi-Classical Mechanics in Phase Space: A Study of Wigner's Function. In: Philosophical Transactions of the Royal Society of London series A-Mathematical Physical and Engineering Sciences, 287 (1343). pp. 237-271.
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Abstract
We explore the semi-classical structure of the Wigner functions ($\Psi $(q, p)) representing bound energy eigenstates $|\psi \rangle $ for systems with f degrees of freedom. If the classical motion is integrable, the classical limit of $\Psi $ is a delta function on the f-dimensional torus to which classical trajectories corresponding to ($|\psi \rangle $) are confined in the 2f-dimensional phase space. In the semi-classical limit of ($\Psi $ ($\hslash $) small but not zero) the delta function softens to a peak of order ($\hslash ^{-\frac{2}{3}f}$) and the torus develops fringes of a characteristic 'Airy' form. Away from the torus, $\Psi $ can have semi-classical singularities that are not delta functions; these are discussed (in full detail when f = 1) using Thom's theory of catastrophes. Brief consideration is given to problems raised when ($\Psi $) is calculated in a representation based on operators derived from angle coordinates and their conjugate momenta. When the classical motion is non-integrable, the phase space is not filled with tori and existing semi-classical methods fail. We conjecture that (a) For a given value of non-integrability parameter ($\epsilon $), the system passes through three semi-classical regimes as ($\hslash $) diminishes. (b) For states ($|\psi \rangle $) associated with regions in phase space filled with irregular trajectories, ($\Psi $) will be a random function confined near that region of the 'energy shell' explored by these trajectories (this region has more than f dimensions). (c) For ($\epsilon \neq $0, $\hslash $) blurs the infinitely fine classical path structure, in contrast to the integrable case ($\epsilon $ = 0, where $\hslash $ )imposes oscillatory quantum detail on a smooth classical path structure.
Item Type: | Journal Article |
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Publication: | Philosophical Transactions of the Royal Society of London series A-Mathematical Physical and Engineering Sciences |
Publisher: | Royal Society London |
Additional Information: | Copyright of this article belongs to Royal Society London. |
Department/Centre: | Division of Biological Sciences > Molecular Biophysics Unit |
Date Deposited: | 21 Jan 2010 09:35 |
Last Modified: | 30 Jan 2019 09:24 |
URI: | http://eprints.iisc.ac.in/id/eprint/24336 |
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