Polar Beer

And now: the effects of blogging on Ultracold Polar Molecules

23 February 2006

The Canonical Link List

A list of incredibly useful physics websites. All my (zero) readers out there: send me others which you think are equally brilliant, and I'll post them up here. This will keep getting updated of course.
  1. Silicon Sam's Electrical/Laser FAQ ... don't enter your lab without it !
  2. Globalspec.com ... if you want to know what a D-sub 9 pin connector is, and who will sell it to you. An ultrasuper technical reference is at Gino De Beer's Educypedia: links to an astounding amount of information. Definitely look at it before wiring up a PIN diode to detect a Gunn diode.
  3. Abramowitz & Stegun ... all the juicy details about expanding Coulomb wavefunctions in Bessel functions (and rewriting those in terms of Chebyshev polynomials, if you're feeling particuarly masochistic). Of course, all of these should eventually get reworked into NIST's Digital Library of Mathematical Functions.
  4. The Particle Data Book, for those times when you wonder: "What exactly is the uncertainty on the electron's g-2 ?". Equally radical questions get answered on the NIST Physical Reference Data pages & the CRC Handbook of Chemistry and Physics (note the ordering here .. $%*#@), although the latter will not tell you the ground state term for an OH molecule :(. 5 points if you get the bad joke in the preceding line.
  5. Atomic spectroscopy basics, if you have somehow forgotten the hierarchy of term symbols in a Ca supermultiplet. For the diatom-inflicted folks: Oliver Dulieu's slides on ultracold molecules from the 2004 Les Houches school and NIST's HOWTO on calculating rotational line intensities - perfect for some relief from Hund's case blues. And before building an accelerator in your basement, learn all about it from the boys who know it best: CERN's accelerator school lectures on everything from RF cavity design to vacuum bake-out tips.
Thus ends Version 1.0 of the Canonical Link List. More like a Microcanonical list, methinks.

18 February 2006

Can one hear the shape of a molecule ?

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 ?

17 February 2006

Ground work

In times like these, there are very few reasons to stop and think before inflicting another blog on the net. My excuse must be a recent infatuation with molecules and their wonderfully complicated physics. Hopefully this space will get filled up with lots of ultracool quantum mechanics, horrifically bad puns and insights into the private lives of molecules and their innards. For those who prefer a little more entertainment, there are also vague plans to finally broadcast The Curse of the Vibrating Polar Bear - chronicling the adventures of a daring/dashing/very cool pirate crew as they attempt to win back Diatomic Molecules (and save the world) from the Dastardly Chemists.

For starters, a chain of thought: Is there an analogue of Hund's rules that one can mindlessly crank to figure out the ground state term of a molecule ? After a windy afternoon trying to cook up Hundish rules, here is a summary of some rather doggerel thoughts:

  • Too many molecules have 1Σ ground states. Is this some laziness-induced selection at work, or as Landau claims, is the ground state often the maximally symmetric state ?
  • Bonding/anti-bonding molecular orbitals are exactly the same as exchange-coupled spin anti-symmetric/symmetric electron pairs ! Molecules with anti-bonding HOMOs are in spin triplets, while the bonding HOMOs lead to spin singlets.
  • Details of the tensor-Stark shift determine whether a σ bonding orbital is lower in energy than a π orbital - which is another way of saying: sometimes the σ is lower than the π± and sometimes not ... so blame the Stark shift.
  • Hund-like filling rules for electrons into the molecular orbitals determine the Λ for the ground state term. The π± are degenerate and electrons fill them repulsively.
After doodling for a while, the ground state terms for a lot of Herzberg's list seem to make sense. The preponderance of 1Σ terms is likely because of the preponderance of filled bonding orbitals whose Λ 's (σ: Λ = 0, π±: Λ = ±1) add up to zero. O2 and He2 both have 3Σ ground terms, but the singly ionized molecular ground states show that the antibonding σ is lower than the antibonding π in He2, and so on ...