Periodic orbit spectroscopy: Breaching the impenetrable fortress.

摘要

The study of complex chaotic systems has long been hampered by the visual nature or dimensional limits of the historic methodologies. To overcome this boundary a method is developed which is applicable to any chaotic Hamiltonian system, even those of high dimensionality. This method relies on the propagation of the classical analog to the quantum mechanical wavefunction. A dipole-dipole autocorrelation function is calculated as the wavepacket propagates. The Fourier transform of the resulting curve is the classical equivalent to the ionization spectrum, a quantum mechanical observable. In addition the stable and unstable periodic orbits which dominate the structure in the spectra are identified.; This method is applied to several physical systems starting with the $x2y2$ potential where a stable orbit is readily found which eluded scientists for decades. The problem of the hydrogen atom in external electric and magnetic fields in both parallel and crossed orientations is considered. Here numerous results are presented which explain the dominant structures in the spectra while providing an understanding of the transitions undergone as the field strengths, and wavepacket energies are varied. The last application is to the ozone molecule in the Frank-Condon region. These calculations explain the dominant experimental spectral structures while identifying the normal mode behaviors in this unusual region. Surprising increases in the average spectra, and thus the stability, as the energy of molecule is increased are also explained in terms of the semi-classical wavepacket motion.; This method can be utilized in systems ranging from KAM to complete ergodicity and from low dimension to very high. The resulting spectra contain nearly the complete classical contribution to the experimental spectra, thus allowing the detailed study of quantum signatures of classical chaos.