Reflections on my own research

I don't normally talk about my own research on this blog. Mainly this is because my research is in a fairly esoteric field (quantum chaos), so it's pretty technical and doesn't really fit the general theme of the blog. But I just finished writing a new paper and while writing it I was struck by the vast difference between the way the research is laid out in the paper and the way it actually happened. So I thought it would be interesting to tell the story of our "discovery" and then contrast it with how I wrote the paper. (I'm not claiming that it is a discovery that is in any way important, even within my field of specialty - although I hope it is considered important enough for the paper to be published - but nonetheless we did find and explain a new phenomenon that has not previously been discussed in the literature.)

The project was one I worked on with an undergraduate student. The aim of the project was to examine the relationship between classical mechanics and quantum mechanics in a particular model system. In quantum mechanics a particle trapped in a small region of space can only have certain specific values of energy. These are known as the energy eigenvalues of the system. It turns out that the statistical properties of the differences between consecutive eigenvalues depend on whether or not the dynamics in the classical version of the system is regular or chaotic . For one-dimensional systems like the one we were studying (which are always regular according to Newtonian physics) the general expectation is that the eigenvalues will be uniformly spaced. But our system has an unusual feature: it has non-Newtonian orbits. I won't go into the details, but these are basically paths that the particle follows in the classical limit of quantum mechanics, but not in Newtonian mechanics. It turns out our system has a lot of these non-Newtonian orbits and we were interested to see if that might have some affect on the spacings between consecutive quantum eigenvalues.

It turns out that they do have an effect. My student and I set up the necessary numerical calculations and she did all the hard work. The figure below shows the spacings she calculated.



Without the non-Newtonian orbits in the classical system we would expect all of the quantum eigenvalue spacings to equal one. Clearly they do not. So we found what I had hoped we would find. We also expected the spacings to get closer to one as we went to higher and higher energies, which they do. By my student's calculations had uncovered something else. Notice that at certain values of n the seemingly random scatter of spacings coalesces into a few distinct curves. In particular, around n=525 or so the sequence of spacings forms three distinct curves. This was totally unexpected and as far as I know it's never been seen before (it's also probably not very important, but let's not dwell on that now).

So what were these unexpected features in the spacings sequence? As soon as I saw it I had an idea of what it MUST be. I was sure it was a resonance phenomenon. Resonances show up all over the place in physics, and to me this looked like a resonance. But there can be many different kinds of resonance phenomena - what kind was this? I immediately focused in on two possibilities: either it was a resonance between the periods of two non-Newtonian periodic orbits (so that the ratio of their two periods would form a rational number with a small denominator) or it was a resonance between the actions of two such orbits (action is an important quantity in classical physics, but it's hard to describe without a lot of math). My gut told me it was the period resonance, but I knew that classical actions can play an important role in determining quantum eigenvalues so I couldn't neglect that choice. I was able to derive a formula for each of these options that would allow me to predict exactly where these resonance features should appear. I checked the formula against the results shown in the figure above and they BOTH matched - for the particular set of parameters we were using in our calculations they gave the same predictions. However, I could tell that for a different set of parameters the two formulas would give different predictions, so my student did the calculations with the new parameters: the period resonance formula fit perfectly and the action resonance formula failed.

At first glance this seems to fit a Popperian model of science. I came up with two competing theories to explain the observed data and subjected both theories to rigorous testing. One theory was falsified, the other theory survived. Now I can publish the successful theory and move on, right? But that's not really the whole story. First of all, how did I come up with these two theories? How did I recognize this odd behavior in a sequence of numbers as a resonance phenomenon? How did I know before doing any testing which of the two theories was better? Was it just a lucky guess? I don't think so, but I can't really articulate how I formed the idea. It just LOOKED like a resonance, and the period resonance idea made more physical sense to me. Obviously my training played a part in this (as I said above, resonances are all over physics - so resonances are the kind of thing physicists are trained to spot). But I can't really think of another example of a resonance that looks just like this one. And I find this fascinating, in part because I'm reading Michael Polanyi's Personal Knowledge right now and he makes a big deal of the inarticulate, tacit component of science. I feel as though I saw Polanyi's ideas in action in my own work.

I should point out that although some aspects of this work do seem to fit Popper's model, none of it fits the empiricist/positivist induction model. I used nothing more than a qualitative knowledge about the resonance features to derive my period resonance theory (and the competing action resonance theory). I didn't pay any attention at all to the actual energies at which these resonance occur until AFTER I had devised a formula (or, rather, two formulas) to predict the energies at which they SHOULD occur (if my theory was right).

But that's not all that struck me about this research. As I said, I just finished writing the paper and in the paper you won't see a discussion of these two competing theories. It became unnecessary. You see, once I was absolutely certain that my period resonance theory was correct I figured out how to DERIVE it from something called semiclassical theory. Now this was not a trivial derivation and I honestly don't believe I could possibly have done it without knowing exactly where I wanted to end up. It is not at all obvious (to me, anyway) from the semiclassical theory itself that resonance features should show up in the sequence of eigenvalue spacings. It took a lot of work to bring those features out, and I wouldn't have know how to do that work (or even that the work should be done) if I hadn't already figured out what the result would be.

In my paper, though, you will just find a presentation of the numerical data and some discussion pointing out the unusual features. Then you will find a derivation of the period resonance theory from the semiclassical theory. You'd never know from my paper that I figured out the period resonance theory BEFORE I even started working with the semiclassical theory. I think that this is fairly typical of scientific papers (I know it is typical of my own). Scientific papers rarely describe the actual process of discovery. Some of what is left out is a description of errors and dead ends (that's the case for my current paper too - we spent a long time incorrectly calculating the spacings before we realized what we were doing wrong, but you won't find any discussion of THAT in our paper). But often there are some really interesting aspects of the process of science that get left out. And I think that's a shame. I guess that's why I decided to write this blog entry!