Abstract
We explore the semi-classical structure of the Wigner functions
Ψ(
q,p
)
representing bound energy eigenstates | Ψ 〉 for systems with
f
degrees of freedom. If the classical motion is integrable, the classical limit of Ψ is a delta function on the
f
-dimensional torus to which classical trajectories corresponding to |Ψ〉 are confined in the 2
f
-dimensional phase space. In the semi-classical limit of Ψ (
ℏ
small but not zero) the delta function softens to a peak of order Ψ−
f
and the torus develops fringes of a characteristic ‘Airy’ form. Away from the torus,Ψ 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 is calculated in a representation based on operators derived from angle coordinates and their conjugate momenta. When Ψ 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
⋲
,the system passes through three semi-classical régimes as
ℏ
diminishes.
(b)
For states |Ψ〉 associated with regions in phase space filled with irregular trajectories,
Ψ
will be a random function confined near that region of the ‘energy shell’ explored by these trajectories (this region has more thanks dimensions).
(c)
For
⋲
≠ 0,
ℏ
blurs the infinitely fine classical path structure, in contrast to the integrable case
⋲
= 0, where
ℏ
imposes oscillatory quantum detail on a smooth classical path structure.
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