Investigation of Ergodic Character of Quantized Vibrational Motion

Abstract

The concept of quantum ergodicity and the degree of ergodic behavior reflected by the bound energy eigenstates are studied for some vibrational systems in two and three dimensions. Different approaches are attempted in order to be able to classify and quantify ergodicity in a given system by investigating the energy eigenfunctions. It is argued that the concept of quantum ergodicity is fundamentally connected to the similarity between eigenstates close in energy and to their globality. Previous investigations and definitions of quantum ergodicity can be seen to connect to this theme; they provide different measures of similarity between eigenstates. Here we propose two practical measures to investigate quantum ergodicity. The systems treated include the famous two-dimensional Henon-Heiles and Barbanis systems, which have previously been investigated both classically and quantum mechanically. As a more realistic three-dimensional example, we consider the vibrations of nonrotating NO2 close to dissociation.

Ergodicity and rapid electron delocalization – The dynamical mechanism of atomic reactivity and covalent bonding

“…From our point of view, the spectral energy gap distributions certainly do relate to the ergodic properties of the system in much the way proposed in the Wigner analysis but there are exceptions when the analysis does not apply and when it applies it is, in our view, the consequence of a more fundamental analysis of quantum ergodicity in terms of the eigenfunctions rather than the corresponding eigenvalues alone [49,50]. Let us recall the spectral formulation of quantum dynamics where the time development of a wavepacket ψ(t) is obtained by expansion in terms of the energy eigenvalues {φ k } as…”

“…In an earlier work [49,50], we have studied the apparent degree of ergodicity of coupled systems of two and three harmonic oscillators within the basis set of vibrational energy eigenfunctions in the absence of anharmonic coupling. In molecular systems, the atomic orbitals are the natural basis functions within which to study apparent ergodicity.…”

From Electronegativity towards Reactivity—Searching for a Measure of Atomic Reactivity

“…The strains of the atomic ground state structures discussed are related to what in classical mechanics would be referred to as “nonergodic effects”, i.e., effects related to the inability of dynamics to take the trajectory uniformly over all states of the given initial energy [ 34 , 35 ]. In quantum mechanics, such effects are associated with degeneracy or near-degeneracy of the energy levels in the spectrum of energy eigenstates and eigenvalues of the system [ 36 , 37 ], i.e., the atom in our case.…”

Eigenstates of Thiophosgene Near the Dissociation Threshold: Deviations From Ergodicity

“…On the other hand, existence of localized eigenstates at the reaction threshold would suggest strong deviations from the RRKM regime. Consequently, there have been many studies [12][13][14][15][16][17][18][19][20] aiming to characterize the nature of highly excited eigenstates in different systems. Recent reviews [21,22] highlight the relationship between eigenstate assignments based on classical phase space structures and deviations from the RRKM regime.…”