(1)   Periodic Orbit theory of non-hydrogenic atoms      Dando P., Monteiro T.S.,Owen S., Phys.Rev Lett 80 2797 (1998)

(1)      Beyond periodic orbits : core-scattering in diamagnetic atoms ,

Dando P., Monteiro T.S., Delande D., Taylor K T Phys.Rev Lett 74 1099(1995).

(2)    Chaos in atoms, molecules and quantum wells (gzipped .8Mb)) ,   Monteiro T S.,Owen S M, Saraga D S Phil.Trans.Roy.Soc.A 357 1359 (1999

POPULAR INTRODUCTION TO CHAOS  AND PERIODIC ORBITS
A hallmark of classical chaos is that an infinitely fine level of detail is needed to describe a particle's motion, since a typical chaotic trajectory meanders forever in a convoluted track, never retracing itself.  Also at a given energy there is an infinite `sea' of possible chaotic trajectories. The quantum world, however, is bound by Heisenberg's Uncertainty Principle which sets a limit to how much detail  we can know.  A quantum distribution does not permit the fine detail seen in popular illustrations of eg fractals.

A  subject of great interest in the 19980’s and 1990’s was the discovery that the quantum probability density can be strongly concentrated along the paths of a few isolated short classical trajectories which survive as `periodic orbits' - in other words paths that retrace themselves.  Chaotic spectra were thus analysed using “Periodic Orbit Theory” a variant of Feynman’s path integral theory developed by Martin Gutzwiller and others. Although these short periodic orbits are rare -  they provide the best tool we have to analyse messy quantum spectra which arise in the chaotic regime, where there are no longer any good quantum numbers to classify the quantum states.

The leading paradigm of chaos in atomics physics and periodic orbits was the Hydrogen atom in a strong magnetic field (strong enough so that the so-called Quadratic Zeeman effect is important). The classical orbits of  the electron become chaotic and the usual quantum numbers (n,l) are no longer useful. It was extensively studied experimentally. Concentrations of probability along periodic orbits are sometimes called `quantum scars' .Usually this refers only to unstable/chaotic orbits. But below we show examples of quantum localization in the regime intermediate between regularity and chaos. The figure shows a comparison between the classical dynamics in the mixed phase space regime and the corresponding quantum behaviour for a hydrogen atom in a magnetic field (Wigner functions of 3 eigenstates, scaled energy=-0.4) There is clear localization on classical tori. But the quantum phase-space distribution can be negative (in the black regions).