Lie algebraic approach to quantization of nonseparable systems with internal nonlinear resonance

Abstract

Nonlinear resonances play an important role in a wide variety of molecular systems. Since they can lead to dramatic changes in the topology of the classical trajectories, resonances may give rise to problems in the application of semiclassical quantization schemes. In particular, the presence of resonances complicates the determination of the most appropriate action-angle variables in which to effect quantization. The best set of action-angle variables should ideally be determined by the physical system rather than by convenience (as is often the case), and, in fact, an unphysical choice of actions may give rise to unphysical singularities in the quantization procedure. In this article a new perspective is presented to the problem of defining physical action-angle variables for nonseparable coupled harmonic oscillator systems displaying internal nonlinear resonance. For the sake of illustration, the major emphasis is directed to a system exhibiting Fermi resonance. The choice of action-angle variables is based on a direct consideration of the Lie symmetries associated with the zero order (i.e., uncoupled) system, which are then used to quantize the full problem. In conjunction with the appropriate semiclassical quantization rules the method provides excellent agreement with accurate quantum results for the Fermi resonant system studied, even when the dynamics is chaotic. In addition, the treatment provides considerable insight into the application of classical perturbation theory to semiclassical quantization, and provides a consistent framework for the treatment of resonant systems. Although presented in the context of classical perturbation theory, the choice of good action variables is a central issue in most semiclassical methods (e.g., adiabatic switching), and the usefulness of the approach to these methods is briefly discussed.