Semi-classical mechanics in phase space: A study of Wigner’s function

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.