In my last few years of practice, I did a fair amount of expert witness work. One thing you learn very quickly, is that attorneys at trial never ask a question they don’t already know the answer to. The following questions don’t fall into that category.

Question #1: We’ve been doing perturbation theory for the past week or so in the QM course. The underlying assumption is that a small change in the Hamiltonian (usually in the potential) produces a small effect in the set of wavefunctions and their eigenvalues. This was certainly the state of affairs when I first took QM in the spring of ’61. Since then we know this isn’t always (or even usually) so for many differential equations. What about chaos? How do we know the perturbation didn’t put us into the chaotic regime? Is it the fact that the derivatives in the Schrodinger equation are to the first power hence linear? I got the following from the QM professor — “Perturbation theory is always approximate and will usually work but sometimes not. If chaos raises its beautiful head, we just have to deal with it.” I asked another computational chemist — the response was “Interesting question”. Any thoughts?

Question #2: Presently we must use perturbation theory and variational principles to do QM on systems too complicated to solve in closed form. Will this always be the case? Could advances in mathematics increase the number of solvable Schrodinger equations? John Gribbin in one of his excellent writings (I don’t recall the actual source) said “It’s important to appreciate, though, that the lack of solutions to the three-body problem is not caused by our human deficiencies as mathematicians; it is built into the laws of mathematics.” Is this correct? If so it’s sort of a Godel’s theorem on equations (and any chemistry more complicated than the hydrogen atom). Is Gribbin correct? Any thoughts?

Question #3: I shudder at bringing this one up. See The Curious Wavefunction’s post of 4 November “A wrong kind of religion, Freeman Dyson, Superfreakonomics, and Global Warming” for just why. Nonetheless, in the 2 October ’09 Science on pp. 28 – 29 the data that there has been no change whatsoever in global temperatures for the past decade is presented, along with a reply of the climate modelers.

Modelers reran their simulations 10 times for a total of 700 years and found 17 episodes of stagnating temperatures lasting a decade or more. The longest period was 15 years, so we’ll have an idea of how good the present models are in another 5 years. The modelers would have more credibility if they had published this sort of thing 10 years earlier before the data became available (did they publish this sort of thing when the models first saw the light of day? — they would have more credibility if they did).

The press is full of stories about retreating glaciers, diminishing artic sea ice, the march of temperate species northward, endangered polar bears etc. etc. An ice cube will melt given enough time if you set it outside the fridge. Is this what is causing the above — is global temperature already too high and causing these changes? Or are they in fact due to something else? If so what? No polemics please.

## Comments

I am far from “the cognoscenti” when it comes to chaos, but I would think we are light years from calculating quantum chaotic phenomena for things like the application of PT to real chemical systems since we already have to bend over backwards to calculate even the simplest of systems with non-chaotic PT.

It’s very interesting that you raise the analogy with Godel. A post is in order and it will be up soon. But yes, it does seem that there are fundamental limitations in applying the Schrodinger equation for anything more complicated than hydrogen. The question is; for what chemical systems does the error really matter?