(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


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).



 For hydrogen atoms, phenomena such as periodic modulations of the quantum spectra and quantum scars were extensively analysed in the 1990's using Gutzwiller Periodic Orbit (PO) theory. Such quantum chaos effects can be related accurately to the actions and stabilities of a set of isolated, unstable classical trajectories. However, other 'Rydberg' atoms could not be quantitatively well described with the same approach : relative to hydrogen they showed new kinds of spectral modulations, different types of quantum scarring and different statistics for their energy level distribution. We thus had to modify the theory to allow for Core-Scattering near the nucleus where the usual stationary-phase approximations used by Periodic Orbit theory fail: over a very small region different orbits become connected to each other. Our work showed that the periodic modulations seen in the spectra and the atomic energy-level statistics (the rigidities) could be accurately described by including in the theory a new type of Periodic Orbit, the classically forbidden diffractive-Periodic Orbits.  The observed spectra were explained by a close variant called Closed-Orbit theory.

Steve Owen got his PhD  in 1999 and  went to work in Morgan Stanley Bank. Paul Dando now works for the ECMWF.