Relativity and Probability, Classical and Quantal

O. Costa de Beauregard

O. Costa de Beauregard

J G Leopold, D Richards

The authors examine the effect of the Coulomb singularity on the classical and semiclassical dynamics of the one-dimensional model for atomic hydrogen, sometimes called the surface-state-electron model, in a periodic electric field and determine a necessary condition, in terms of the field strength, frequency and the initial quantum number, for the violation of the uncertainty principle. They show that when this violation occurs the quantal transition probabilities are significantly less than the equivalent classical probabilities. This criterion is tested numerically and shown to provide a good estimate of the boundary to one of the regions in which classical dynamics fails. The authors also show that when the uncertainty principle is not violated the fluctuations of the quantal about the classical probability spectra can be related to structure in classical phase space. These fluctuations are due to different classical and quantal transport through cantori.

Simon W. Saunders

D. Richards, J G Leopold

We provide a brief review of the response of an excited hydrogen atom to a microwave field, and a detailed discussion of the behaviour of a one‐dimensional model in a high frequency field. In particular we show that although classical dynamics is incorrect for the parameters of experimental interest an examination of the classical behaviour is helpful in understanding the quantal behaviour. We show further that, for these parameters, the quantal probabilities fluctuate dramatically with the variation of most system parameters and attribute this non‐classical behaviour to the dominant influence of relatively few, about ten, quasi‐resonant states.