# Semi-Classical Mechanics in Phase Space: A Study of Wigner's Function

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 Copyright of this article belongs to Royal Society London. Others 21 Jan 2010 09:35 19 Sep 2010 05:49 http://eprints.iisc.ernet.in/id/eprint/24336