Classical and quantum mechanics of unstable dynamical systems

A. Carati, L. Galgani

Heinz Dietrich Doebner, V. K. Dobrev, P. Nattermann

Intrinsically irreversible representations of unstable quantum systems, I. Antoniou quantum mechanical arrows of time, A. Bohm isotropic harmonic oscillator on the quantum euclidean space, G. Fiore relevant and irrelevant nonlinear schrodinger equation, N. Gisin diffeomorphism groups and nonlinear quantum mechanics, G.A. Goldin from Hamiltonian mechanics to nonlinear Schrodinger equations, J.-D. Hennig nonlinear Schrodinger dynamics and nonlinear observables, W. Lucke nonlinear gauge transformations and the Doebner-Goldin equation, P. Nattermann chaos and the limit of classical and quantum mechanics, T. Petrosky why irreversibility? the formulation of classical and quantum mechanics for unstable dynamical systems, I. Prigogine harmonic oscillators on quantum space, S. Watamura. (Part Contents).

Jean Bellissard

Juha Javanainen

We study theoretically an atomic Bose-Einstein condensate in a double-well trap, both quantum-mechanically and classically, under conditions such that in the classical model an unstable equilibrium dissolves into large-scale oscillations of the atoms between the potential wells. Quantum mechanics alone does not exhibit such nonlinear dynamics, but measurements of the atom numbers in the potential wells may nevertheless cause the condensate to behave essentially classically.

Pavel Bóna

D. G. Levkov, A. G. Panin

Quantum transitions are described semiclassically as motions of systems along (complex) trajectories. We consider the cases when the semiclassical trajectories are unstable and find that durations of the corresponding transitions are large. In addition, we show that the probability distributions over transition times have unusual asymmetric form in cases of unstable trajectories. We investigate in detail three types of processes related to unstable semiclassical dynamics. First, we analyze recently proposed mechanism of multidimensional tunneling where transitions proceed by formation and subsequent decay of classically unstable ``states.'' The second class of processes includes one-dimensional activation transitions due to energy dispersion. In this case the semiclassical transition-time distributions have universal form. Third, we investigate long-time asymptotics of transition-time distributions in the case of overbarrier wave packet transmissions. We show that behavior of these asymptotics is controlled by unstable semiclassical trajectories which linger near the barrier top.