Regular and irregular semiclassical wavefunctions

The form of the wavefunction $ for a semiclassical regular quantum state (associated with classical motion on an N-dimensional torus in the 2N-dimensional phase space) is very different from the form of + for an irregular state (associated with stochastic classical motion on all or part of the (2N1)-dimensional energy surface in phase space). For regular states the local average probability density Il rises to large values on caustics at the boundaries of the classically allowed region in coordinate space, and $ exhibits strong anisotropic interference oscillations. For irregular states Il falls to zero (or in two dimensions stays constant) on ‘anticaustin’ at the boundary of the classically allowed region, and $ appears to be a Gaussian random function exhibiting more moderate interference oscillations which for ergodic classical motion are statistically isotropic with the autocorrelation of $ given by a Bessel function.