"Quantization of a free particle interacting linearly with a harmonic oscillator"

As just promised, the full-length article that I wrote with my former SURF student Tom Mainiero just came out. The proofing stage took a little longer than usual; in fact, the journal was apparently waiting on our final approval after several proof iterations (with a couple of mess-ups on our part and a rather large number of mess-ups on their part --- including forgetting to capitalize the title of the article, changing our grammatically-correct corrections to grammatically incorrect statements [which I found irksome], and a comedy of errors in the display style for one of the equations) in order to finish posting the final articles in their December 2007 issue. I definitely managed to annoy the publishers a bit for this particular article (they gave this away with a couple of statements in the notes they conveyed to us along with the second set of page proofs), but if they're not going to implement one of my corrections (or if they adjust how they do so), I expect an explanation of why or else I'm just going to ask them to correct it again when I get the subsequent round of page proofs. As anybody who knows me should know, I am extremely anal about things like this; just look at how I mark up the pages of any paper drafts that my students have submitted to me! Of course, the publishers and typesetters were very much attempting to get things right -- they were just a little sloppier than I would have liked on this particular occasion.

The article is titled, "Quantization of a free particle interacting linearly with a harmonic oscillator."

The abstract reads as follows:

We investigate the quantization of a free particle coupled linearly to a harmonic oscillator. This system, whose classical counterpart has clearly separated regular and chaotic regions, provides an ideal framework for studying the quantization of mixed systems. We identify key signatures of the classically chaotic and regular portions in the quantum system by constructing Husimi distributions and investigating avoided level crossings of eigenvalues as functions of the strength and range of the interaction between the system’s two components. We show, in particular, that the Husimi structure becomes mixed and delocalized as the classical dynamics becomes more chaotic.


I think the best thing to add to convey which this is interesting, let me also reproduce the bold introductory paragraph from the article:

Typical classical Hamiltonians systems are neither fully integrable nor fully chaotic, but instead possess mixed dynamics, with islands of stability situated in a chaotic sea. In this paper, we investigate the quantization of a recently-studied system with mixed dynamics [1]. This example consists of a free particle that moves around a ring that is divided into two regions. At the boundaries between these regions, the particle is kicked impulsively by a harmonic oscillator (in a manner that conserves the system’s total energy), but the particle and oscillator otherwise evolve freely. Although the system is not generic, its separation into regular and chaotic components also allows more precise investigations (both classically and quantum-mechanically) than is typically possible, making this an ideal example to achieve a better understanding of the quantization of mixed systems. By examining avoided level crossings and Husimi distributions in the quantum system, we investigate the quantum signatures of mixed dynamics, demonstrating that the Husimi structures of nearby states become mixed and delocalized as chaos becomes a more prominent feature in the classical phase space.