Tuesday, November 25, 2008

Dancing Droplets and Spherical Harmonics

My first encounter with Spherical Harmonics must have been in the course on electrodynamics, when these intimidatingly looking functions showed up in series expansion of the Coulomb potential and in the multipole expansions of charge distributions. Half a year later, they appeared again, in the solution of the hydrogen problem in the quantum mechanics class. Here, at least, they were used to produce nice figures of electron orbitals.

But I did miss out on one of the most elementary occurrences of Spherical Harmonics.

Spherical Harmonics Ym(θ, φ) describe the angular part of the solution to Laplace's equation in spherical coordinates. As such, they are ubiquitous in physical problems with a spherical symmetry. Thus, they describe not only the behaviour of the electron in the hydrogen atom, but also the wobbling deformations of an oscillating, elastic sphere. What sine and cosine are for a one-dimensional, linear string, the Spherical Harmonics are for the surface of sphere.



Deformation modes of a bouncing oil droplet (radius R = 0.765 mm) described by the Spherical Harmonics Y20, Y30 and Y40, as observed with a high-speed camera. From S. Dorbolo et al.: Resonant and rolling droplet, New J. Phys. 10 (2008) 113021.


This has been demonstrated very nicely in a paper by S. Dorbolo, D. Terwagne, N. Vandewalle and T. Gilet just published in the New Journal of Physics, Resonant and rolling droplet. In the experiment, a tiny oil droplet is placed on an oil bath which is set into vertical vibrations to prevent coalescence of the droplet with the bath. The droplet, which at rest would have a spherical form due to surface tension, bounces periodically on the bath. At the right frequencies of the vibrating surface, the droplet oscillates in resonance – and deforms according to spherical harmonics!

A movie (Quicktime, 11.0 MB – someone should explain to the NJP how to upload these movies to YouTube ... ) shows the oscillations of the drop and the corresponding calculations using Spherical Harmonics Ym with ℓ = 2, 3, 4 and m = 0. "Magnetic quantum number" m = 0 means rotational symmetry of the wobbling around the vertical axis. For m ≠ 0, deformations are not symmetric with respect to the vertical, and in this case, the droplet starts to move around on the oil bath. This can be seen in a second movie (QuickTime, 5.6 MB).

That's a beautiful example of Spherical Harmonics in action I would like to have known when I was struggling for the first time with multipole expansions!



The first application of Spherical Harmonics to describe the wobbling of a droplet was by Lord Rayleigh, in the appendix of a paper On the Capillary Phenomena of Jets, Proceedings of the Royal Society of London 29 (1879) 71-97. You can check out his calculation - the PDF (2.5 MB) is available for free.




7 comments:

Plato said...

This to me prepares one to accept a much larger global perspective and the geometrics behind such expression, if seen in context of "backreaction." A recursive lifestyle.

First, it's defined. Then, a resulting singularity, which motives "original design of energy in expression?" Reaching, the source of definition?

You may not understand it now, but you will:)

Best,

UIncle Al said...

Legendre polynomials and the LCAO model of chemical bonding are, of course, faery dust. MO theory is the truth! Any undergrad using MO theory in first term organic will be Blutwurst by the midterm. LCAO is the trivial answer to everything until the Woodward Hoffmann rules.

A good approximation is better than an exact solution plus a supercomputer to approximate it. Advances in theory, however, require the exact solution or all one obtains is pericycles and heteroskedasticity (excuses). A vibrating ball is a wonderful analog computer (3D Chladni figures).

Neil' said...

Don't spherical harmonics also describe the radiating atom? A common, "naive" view is that a radiating atom spreads a uniform shell of a wave function into space, but that isn't so. IIUC, most modes of oscillation send more "likelihood of being there" (I almost said "energy" but this is about a single quantum) out around a preferred plane and less along an axis (like a macro antenna) even though the atom per se is "spherical" and not a little solar system. The most likely polarization is also non-isotropic, again like a real antenna.

This WF is actually locally defined as "real" (in principle), right, since it is not entangled with another one? But we can't examine the atom to find out what it should have sent out, like we can our own photon guns - ?

Plato said...

Elements and Atoms: Chapter 11
An Unsystematic Foreshadowing: J. A. R. Newlands

Having revealed where I stand, I invite you to examine and assess Newlands' work on the subject. This selection includes four short papers of Newlands and a report of another paper which show him struggling toward and eventually formulating the system he dubbed the "law of octaves

Of course, this philosophically has meaning for me.:)

Maybe the reference to the orbital action is specific in regards to spintronics for the geometrical inclination of the position in the wave and element, as it is interpreted from that global perspective?

Best,

Phil Warnell said...

Hi Stefan,

A nice piece, although it’s too bad that the Quick times won’t work for me beyond just loading and sitting frozen. It’s perhaps interesting to note that it was Johannes Kepler who first imagined that harmonics played some role in spherical systems which in this case was the solar system with his third law of planetary motion . He didn’t of course have it all right yet the intuition was there.

Best,

Phil

stefan said...

Hi Neil,

higher multipole radiation indeed is described by the corresponding spherical harmonics. But if I remember correctly from atomic physics, electronic transitions in the atom are mostly dipole, which is responsible for the standard angular momentum selection rules - the photon has spin 1, and the radiation doesn't carry additional orbital angular momentum.

Best, Stefan

Plato said...

I mean most certainly we push perspective. So our views of the cosmos become very different.

So your watching events at locations in the universe.

Now of course we like to think our world is very round but under closer examination with Grace, it's not really that way. While some might focus on the rubber sheet analogy I would like to be able to see "the value of the element" in the way we see the world too.:)

So you get this sense of seeing in other ways given measure its value and concepts that have to this day have been changed?

Helioseismology and Amara knows all about that. We want to know what the sun is doing in advance.

Best,