Monday, June 30, 2008

E is for Eigenfunction

Shamelessly aping "Talk like a Pirate Day" (9/19), some science punks are lobbying the blogosphere to recognize "Pi Day" (3/14) as "Talk Like a Physicist Day". There's even a TLAPhizz website from which this Heisenberg Uncertainty tat was lifted.

Heisenberg's Principle has been characterized as "The only certainty is uncertainty." but this is clearly false. One of the most common terms in the physicist's vocabulary is the word "eigenfunction". In a sea of quantum uncertainty the eigenfunction (from the German "eigen" which means " personal", "private", "special") stands as a welcome island of 100% predictibility--something in quantum physics that you can always count on.

Today's physicists represent everything in the world from cats to electrons as "state vectors" dwelling in an abstract realm called Hilbert space. In Dirac bra/ket notation the state vector for a cat is written |cat> and for an electron |electron>. This abstract object contains all knowable information about the results of any conceivable observation. You choose what you want to measure (and because of Heisenberg, each choice of observable demands a renunciation of knowing some complementary observable) and by a simple operation on the abstract state vector you can calculate the outcome of a measurement of the chosen observable in real life. The enormous practical success of quantum theory is dimmed only slightly by the fact that what the state vector actually "means" is still a controversial issue. We don't really know what we're doing here--but it works like a charm! After nearly a century of ruthless testing by Nobel-hungry tinkerers, quantum mechanics has given up millions of right answers, but not one single wrong prediction.

To calculate the results of any observation--position, momentum, live, dead--one simple forms the "bra vector" corresponding to the desired observation , to form the bra/ket expression which is a number that represents the probability that the cat is here or there. (actually you have to square this number to get probability but that's an unessential detail.) The bra/ket expression is a measure of the probability of the cat being alive. Since the world is full of uncertainty, we expect the value of these expressions to lie somewhere between 1 (high probability) and 0 (low probability).

For most observables, the probabilities do indeed lie somewhere in this range. But an eigenfunction (as its name implies) is special. For an eigenfunction of an observable, the probability of observing a particular value of that observable (called its eigenvalue) is precisely 1. No uncertainty here at all.

One of the most impressive intellectual achievements of the 20th century was the working out of the exact mathematical description of nature's simplest atomic system--the hydrogen atom. The determination of the hydrogen wavefunction by a small group of European physicists in the late 1920s was a cultural high-water mark comparable to the construction of the cathedral of Notre Dame in the Middle Ages.

The hydrogen wavefunction is NOT an eigenfunction of position--the electron is spread out in space as an extended probability cloud which means its position is uncertain. But the hydrogen wavefunction IS AN EIGENFUNCTION of the observables Energy (E), Angular Momentum (L) and M, the projection of the angular momentum upon the z-axis. If you choose to make a measurement of one of these observables, either E, or L or M (or all three together) on a hydrogen atom in one of its allowed quantum states, then you will get a precise result with zero uncertainty.

The eigenfunction (zero uncertainty for certain observables) is so important that quantum wavefunctions are customarily labeled by the observables (and their associated eigenvalues) for which the results are single-valued and 100% certain. Thus the state vector for a hydrogen atom is customarily written |N, L, M> where N = 1 corresponds to the lowest energy E, N = 2 corresponds to the next-highest energy state.

Eigenfunction is actually not a quantum concept at all but has been borrowed from the old-fashioned classical physics of vibrating systems where the EF describes the "normal vibratory modes" of violin strings, drum heads, and other oscillating media including the resonant vibrations of this rocky sphere we call home. In the classical case we know exactly what the mathematics means; in the quantum case we are not so sure.

Recently Dean Dauger (formerly of UCLA) has released an elegant shareware program (Macintosh only, and now iPhone) that produces colored, 3D animations of the hydrogen wavefunction for a large number of its E, L, M eigenvalues. Dauger's program also allows you to form simple quantum superpositions (non-eigenfunctions) of some of these states and view the results.

Dauger's program (called Atom in a Box) lets you not only talk (like a physicist) but lets you make your own home movies of quantum wavefunctions, probably the closest humans will get in my lifetime to visualizing "what atoms really look like". Sophisticated enough to satisfy a real physicist yet easy enough for a normal person to use to toy with the structure of the universe at a very basic level. It's a great way to learn about quantum mechanics. Many thanks, Dean, for this sweet little tool.