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

Abstract

We explore the semi-classical structure of the Wigner functions Ψ(q,p) representing bound energy eigenstates | Ψ 〉 for systems with fdegrees 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 2f-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.