This is not a bad Zen koan - just a walk through some very useless and very fascinating mathematics that I took this weekend. I was looking to find out about constraints imposed on the adiabatic theorem by symmetry (and its breaking) - while I did find an interesting article, more on that some other time.
For today however, a ramble into isoperimetric problems. Someone in some book told me this story about a lady called Dido who founded Carthage
: when she arrived on the African continent, she bargained with the apparently shrewd natives who sold her as much as land as could be covered by an ox-hide. Being well versed in optimization problems, she first cut up the hide into thin strips and made a long rope. Then she went on to encircle a large strategically located hill and ruled happily ever after. I don't know if Carthage had circular (or semi-circular, if she included a straight section of the coast) borders, but this fact that an n-sphere has the minimum surface area for a given volume leads to some nifty results. If you are the kind who needs profound statements like these to have rigorous proofs, here's one which works in n=3 (or perhaps 10, if you talk to the wrong people): soap bubbles are spherical
. Which has as corollary a fact that my grandmother knows: bubbles stick to walls.
In 1894 that master of most things vibrational, Lord Rayleigh, conjectured thusly: given a fixed area of ox-hide to make a drum, the ground tone is lowest if you make your drum circular
. About 30 years later, this was proved independently by Georg Faber and Edgar Krahn. For the quantum mechanics out there, the following insight must then be readily obvious: For a particle in an infinite potential well of fixed (n-dimensional) volume, the ground state energy is lowest for a spherical well
. I bumped into this whole business in:
Marko Robnik, An extremum property of the n-dimensional sphere,
J. Phys. A 13
, L349 (1980)
The Faber-Krahn proofs use a neat idea, which today goes by the terribly illuminating name of Schwarz symmetrization
. The basic idea being as follows: draw a lot of iso-probability contours on the (real) ground-state wave function. The surface-area element on each contour gets weighted by a measure factor, which is the absolute value of the gradient of the wavefunction. On a shell of fixed volume, the contribution to the energy eigenvalue turns out to be minimized when this measure factor is translation-independent on
the iso-probability contour i.e. when it is a constant for a contour. The eigenstate energy is a weighted sum of the surface-areas of these contours, and the probability enclosed by each one is fixed. Hence the least energy eigenvalue is one whose iso-probability contours are spherical - in particular the wall of the well (a contour with value zero) is spherical.
People out there continue to worry about various versions of this problem - this one (math/0312287
) asks if the lowest eigenvalues for the discrete Laplacian on a graph happen for "spherical" graphs !Hearing shapes
Having seen how the geometry of boundary conditions can affect Laplacian eigenmodes, an interesting question is whether a free-particle's energy states completely determine the shape of its container. This is precisely the question that stars in this superb lecture:
Mark Kac, Can one hear the shape of a drum ?, Am. Math. Monthly 73:4 (1966)
Kac's guess that this wouldn't always be possible for a planar drum was verified in 1992, and Wikipedia's page
has an example of two shapes which have the same tones. I won't ruin the beauty of Kac's lecture by trying to rephrase any of its contents. The interesting thing is that Kac tries to glean as much as possible from just the asymptotic properties of 'large' eigenvalues. One of the related conjectures that had been conclusively demonstrated by Hermann Weyl around 1915 was this poser by James Jeans, via H A Lorentz: at sufficiently high frequencies, the number of electromagnetic cavity modes in a frequency interval is independent of the shape of the container, and proportional only to its volume
- something over which every cond-mat artist these days glibly waves his hands. Anyone pointing me to a readable version of Weyl's proof (and/or the video of Kac's talk) will have my eternal gratitude.
Kac proves this result using the diffusion (or heat) equation, and also goes on to show that one can extract the circumference of the boundary as well ... which then leads to our old isoperimetric friend and the result that a sufficiently high-bandwidth listener can tell if a drum is circular by listening to it ! Given that the heat equation snuck in here, it shouldn't be too implausible to believe what Michael Atiyah tells us in this exemplary biography of Hermann Weyl
, that Weyl's work eventually led to high-brow magic like the index theorem, topological anomalies and that sort of thing.
So why is a wannabe spectroscopist like me mucking around with spectral analysis ? Because even for supremely simple things like diatomic ions, figuring out molecular potential curves
seems to be mostly a matter of performing a bazillion parameter match of some favoured interaction potential to measured spectral lines. Because I am reasonably certain that even this sort of thing is impossible for estimating inter-hadronic potentials from nuclear spectra. Ergo, some randomly generated challenges for any lurking theorists or a boring day:
- Are there similar isoperimetric-oid eigenvalue bounds for finitely deep potential wells ? The finite depth means, of course, that the problem is not very Dirichlety any more.
- Can one 'hear' the shape of a molecule ? Can the 'large' eigenvalues (highest vibrational states, which would seem to sample a large part of the potential well) say anything interesting about the size of a molecular state ?
- Why the heck do I sense a recurring conspiracy to make ground states extremely symmetric ?