Semiclassical amplitudes: Supercaustics and the whisker map - art. no. 012107

Citation
Nt. Maitra et Ej. Heller, Semiclassical amplitudes: Supercaustics and the whisker map - art. no. 012107, PHYS REV A, 6001(1), 2000, pp. 2107
Citations number
16
Categorie Soggetti
Physics
Journal title
PHYSICAL REVIEW A
ISSN journal
10502947 → ACNP
Volume
6001
Issue
1
Year of publication
2000
Database
ISI
SICI code
1050-2947(200001)6001:1<2107:SASATW>2.0.ZU;2-D
Abstract
The semiclassical approximation fur a quantum amplitude is given by the sum of contributions from intersections of the appropriate manifolds in classi cal phase space. The intersection overlaps are just the Van Vleck, determin ants multiplied by a phase given by a classical action. Here we consider tw o nonstandard instances of this semiclassical prescription which would appe ar to be on shaky ground, yet the corresponding physical situations are not unusual. The first case involves momentum-space WKB theory fur scattering potentials; the second is a propagator for the whisker map that arises in g eneric two-dimensional systems. In the former case two manifolds become asy mptotically tangent, and the semiclassical formula needs to he uniformized in order to give a meaningful wave function. We give a uniformization proce dure. In the latter case, there are an infinite number of intersections in phase space within a zone with the area of Planck's constant (the limit of resolution for quantum mechanics), yet the semiclassical sum over all contr ibutions is shown to be correct.