Quantum states of two- and three-dimensional coupled oscillators with high symmetry. I. Energy

Abstract

The nature of the eigenstates of two highly symmetrical two- and three-dimensional quantum oscillators is compared on the basis of their projections on one- and two-dimensional manifolds spanned by separable zero-order states and on the basis of the second derivative E″ i of the eigenvalues E i with respect to a mode coupling parameter λ. Both methods quantitatively lead to the same results in classifying regular states, isolated avoided crossing states and irregular states. The two-dimensional system is the familiar Hénon-Heiles model while in three dimensions an oscillator with tetrahedral symmetry was taken for which the potential energy along the four dissociation channels is identical with the corresponding curves in the Hénon-Heiles model. It is found that there is nearly no irregularity in the spectrum of the 3D model up to dissociation although 2 3 of the classical trajectories in this energy range propagate chaotically. The 2D system shows a drastic change of the | E″ i / E i | behaviour at the dissociation energy E D indicating a transition from the low-energy regular spectrum to irregularity. In the 3D system no such transition is observed up to energies well above E D. The missing of quantum irregularity when both the dimension and the symmetry of the system are raised, is related to an increase of localization of even the chaotic trajectories found in classical studies. This localization reflects the stabilization of toroids and vague toroids, and possibly the occurrence of low-dimensional classical chaos.