Roderick V. Jensen

Chaotic price behavior in a non-linear cobweb model

Abstract The cobweb model for the evolution of prices is considered with backward bending supply curves and multi-valued demand curves. These deterministic, non-linear dynamical systems are shown to exhibit a broad spectrum of complex time behavior including stable periodic cycles of arbitrary order and chaos.

Chaotic ionization of highly excited hydrogen atoms

Recent experimental measurements of the microwave ionization of highly excited hydrogen atoms with principal quantum numbers ranging from n = 30 to 90 are well described by a classical treatment of the nonlinear electron dynamics. In particular, the predictions of the threshold field for the onset of significant ionization is found to coincide with the onset of classical chaos in a one-dimesional model of the experiment. In this brief note I emphasize that this excellent agreement between the theoretical and experimental ionization thresholds requires a proper theoretical treatment of the slow, adiabatic turn-on of the microwave perturbation in which the persistence on nonlinear resonances in the chaotic phase space plays a crucial role.

Microwave ionization of highly excited hydrogen atoms: Experiment and theory

This article elaborates on a talk delivered by the first author at the First International Conference on the Physics of Phase Space (University of Maryland, 20–23 May 1986). It reviews briefly our still limited, but rapidly growing understanding of a dynamical process, the ionization of highly-excited hydrogen atoms by a microwave electric field. Classical dynamics explains surprisingly well many recent experimental results from Stony Brook, on which the article focusses. Some experimental results not well explained, however, appear to be essentially quantal in origin. These are just now beginning to be understood. New, detailed questions continue to arise as older questions are answered.

CHAOS IN ATOMIC PHYSICS

Since 1974 Bayfield and Koch have been studying the microwave ionization of highly excited hydrogen atoms. Until recently, their remarkable measurements of ionization which depends strongly on the microwave intensity and relatively weakly on the frequency has defied quantum analysis. However, the threshold fields for significant ionization are well described by the onset of chaos in a purely classical model of the bound electron in an oscillating electric field. I will describe the manifestations of the effects of classical chaos in the ionization of these quantum systems and I will review recent progress in the understanding of the quantum mechanisms which account for this chaotic ionization.

Stochastic Ionization of Bound Electrons

Recent experimental measurments of the ionization of Rydberg atoms in low frequency electromagnetic fields are discused. These observations of ionization rates which depend strongly on the intensity of the oscillating fields and only weakly on the frequency provide clear evidence for chaotic behavior in a quantum system. In an attempt to understand this ionization mechanism an analagous, one dimensional system is considered consisting of a surface state electron bound to the surface of liquid helium by its image charge. A classical analysis of the behavior of this nonlinear oscillator in a microwave field shows that, above a critical field, the electron diffuses in energy until it ionizes. Since the microwave frequencies and field strengths required for stochastic ionization are readily available, this system provides an opportunity to thoroughly explore the manifestations of classical chaos in a quantum system.

High-frequency microwave ionisation of excited hydrogen atoms

The authors analyse the effect of a high-frequency electric field on an excited hydrogen atom. Using classical perturbation theory and correspondence principle methods they show that the quantal system can be described by a severely truncated basis comprising the photon states, which are relatively few in number. Further, the problem may be transformed, locally, to the standard map, but quantised with an effective Plancks constant which is of order unity; this implies that, for large scaled frequencies and for all initial principal quantum numbers accessible to current experiments, the quantal ionisation mechanism is unrelated to classical diffusion.

Microwave Excitation and Ionization of H Atoms at High Scaled Frequencies: Comparisons of Experiments and Theories

A hydrogen atom with principal quantum number n0»1 in a strong microwave electric field A(t)Fsin(ωt+ o) is a periodically driven, deterministic, Hamiltonian system that exhibits a transition to classical chaos. (Here F is the electric amplitude, A(t) is a slowly varying envelope function, ω is the angular frequency, and o is a phase averaged by all experiments described below) Because experiments, analytical theory, and numerical simulations can be carried out,1-11 it is an important testing ground for quantum chaos (the study of a quantal system whose classical counterpart is chaotic.) The values of scaled frequency ω and scaled fieldF largely govern the behavior of the dynamics. (Relative to the state with initial principal actionI 0 = n0ħ, the first is the ratio of ω) and the Kepler orbital frequency, and the second is the ratio of F and the mean, initial Coulomb binding field.)