Probing the classical/quantal boundary using the Stark effect

T. P. Hezel, Charles E. Burkhardt, M. Ciocca +1 more

The unique properties of the hydrogen atom, especially those that result from the dynamic symmetry of the 1/r potential, are presented from both classical and quantal points of view. Using these properties as a starting point, the response of a classical hydrogen atom, an electron executing a Keplerian ellipse about a proton, to a weak electric field is described. Because the field is weak, the elliptical orbit may be treated as a dynamical element itself, variations in its shape and orientation depicting the classical Stark effect. The behavior of this orbit is then correlated with the quantum mechanical description of the first-order Stark effect in hydrogen, a subject that is included in most introductory courses in quantum mechanics.

D. Vrinceanu, M. Raymond Flannery

An exact expression for the probability P(n)l'l of Stark mixing transitions between arbitrary angular momentum states l and l' within the same energy shell n, as a result of a collision with a slow charged projectile at large impact parameters is presented. The formula obtained is compact and easy to use for numerical evaluations even for very large quantum numbers (n~100). A classical approximation is directly obtained and compared with the exact quantal result in the limit of large n. Two distinct sets of quantal oscillations are predicted.

В. В. Колосов

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.

Denis A. W. White