Does Quantum Mechanics Select Out Regularity and Local Mode Behaviour in Nonlinearly Coupled Vibrational Systems?

Abstract

AbstractA two dimensional strongly nonharmonic vibrational system with nonlinear intermode coupling is studied both classically and quantum mechanically. The system was chosen such that there is a low lying transition (in energy) from a region where almost all trajectories move regularly to a region where chaotic dynamics strongly dominates. The corresponding quantum system is far away from the semiclassical limit. The eigenfunctions are calculated with high precision according to a linear variational scheme using conveniently chosen basis functions. It is the aim of this paper to check whether the prediction from semiclassical theory, namely that the measure of classically chaotic trajectories in phase space approaches the measure of irregular states in corresponding energy ranges, holds when the system is not close to the classical limit. It is also the aim to identify individual eigenfunctions with respect to regularity and to differentiate between local and normal vibrational states. It is found that there are quantitative and also qualitative differences between the quantum results and the semiclassical predictions. We found a relatively high amount of regularity in the quantum system for energies where almost all classical trajectories move chaotically. This high regularity is manifested via level spacing statistics as well as by a careful study of the nodal behaviour of the eigenfunctions. Another result was not expected from semiclassical predictions: Some of the wave functions are localized and irregular, a behaviour which was not observed in the corresponding classical system.