Monday, January 23, 2012

Guide to the Perplexed: Part II

In my last post, I catalogued all my blog articles for 2010, up to August of that year. What followed was a long period of inactivity, which followed a visit to Winnipeg from my grad school buddy Richard Epp. I remember it was a really hot summer day, and started telling him about Quantum Siphoning. Richard objected to the interpretation of the wave function as charge density. He pointed out something I already knew: if the charge is distributed, then the calculation for electrostatic energy is messed up. How could I answer that objection? I had no answer, and every time I tried to work on physics, that was all I could think about.

Then in February I stumbled across a website called The Foundational Questions Institute which was apparently sponsoring an essay contest on the question "Is Reality Analog or Digital?". Incredibly, the deadline for submissions was that very day, so I immediately wrote up an article which I gave a title that says it all: There are No Pea-Shooters for Photons.  The targets of the essay are the three pillars of the photon theory, as universally recognized in the popular narrative: the Blackbody Spectrum, the Photo-electric effect, and the Compton Effect. In my essay, I show how each of these has an intuitively reasonable wave theory explanation, which could not have been understood by Planck, Einstein, or Compton in their time because the wave theory of the electron was not revealed by Schroedinger until 1926, long after the particle paradigm had taken firm hold. The title of the article refers to the well-known post-modern justification for photons based on experiments where you supposedly fire one photon at a time and track where it goes.  I point out, as I did in an earlier blogpost called The Clicking Detectors, that none of these experiments are really quite what they claim to be, for the plain and simple reason that there really are no pea-shooters for photons. 

When I submitted my essay, I was in a rush to meet the deadline, and I didn't feel that I had dealt as well as I might have with the question of the blackbody spectrum, so I went back to my blog and that became the subject of my next article, which soon mushroomed into a nine-part series. The basic idea was that instead of attacking the ultraviolet catastrophe at the electromagnetic level, you take it on at the mechanical level. All electromagnetic radiation has to have its source in the mechanical vibration of charges, and if there is no mechanism to set those vibrations in motion, you don't have a problem at the electromagnetic level. The one nagging problem with my argument is that thermodynamics doesn't depend on specific mechanism:  there's something called the Equipartition Theorem which supposedly rules no matter what, and I had to somehow explain it away. I struggled with this through my next four blogposts until I made a huge breakthrough.

There's something people do in physics which drives me crazy and that is putting all their arguments in the most abstract, mathematical form. I need to see real examples and real mechanisms, and in particular I wanted to figure out just how an equilibrium is established between a mechanical oscillator and the electromagnetic field. This doesn't look like it should be an insurmountalbe problem: we know how to do the driven harmonic oscillator with damping: just apply that to an atomic oscillator with the electromagnetic field as the driving force. How hard could it be?

The problem is that we don't just have a driving field of known intensity: we have a random, distributed field. There is no simple number we can pick out of the blue and say "we are driving the atom with an oscillating field of so-and-so-many volts-per-meter: the field strength is expressed in volts-per-meter-per-hertz, and how the hell do you interpret that. What happened is after years and years of not knowning how to handle this question, all of a sudden I figured it out! It's a beautiful, very pictorial explanation that starts off by considering that familiar old chestnut of statistical theory, The Drunkard's Walk . Analyzing the driving force on the oscillator as a special case of the Drunkard's Walk, I show that you are allowed to truncate your frequency distribution at any arbitrary limits and you still get the same oscillation regardless. I then carefully count up the cavity modes of the electromagnetic field, and then run a numerical example of a special case to come up with the amazing conclusion, which is the basis of the Rayleigh-Jeans derivation of the ultraviolet catastrophe: the energy per mode of the electromagnetic field is equal to the energy per mode of the mechanical oscillators!

What this means is if you can show that the high-frequency oscillations are suppressed at the mechanical level, then they are automatically suppressed at the electrical level. You don't need to throw out Maxwell's equations to avoid the ultraviolet catastrophe.

My next post was four weeks later. A year previously, I had been very excited to work out the solution for the problem of two electrons sharing the same potential well. It came as a huge surprise to learn that in contrast to the well-known single-electron case, with two electrons the shape of the solution depended on the size of the box. I had a huge argument with a guy named SpectraCat in over this: he said that the shape of the solution depended on the strength of the interaction, and I said that was exactly the same as saying it depended on the size of the box. Of course I was right: the very suprising thing about it was that the case of the very small box corresponded the case of independent particles, and vice versa: the strong interaction corresponded to the case of the very large box! It may seem counterintutitive, but in my analysis of the iso-electronic series of helium, I show that's exactly how it works. A "helium-like atom" is any atom stripped down to its last two electrons. The series actually begins with hydrogen: it turns out that the negative hydrogen ion, consisting of a proton and two electrons, is marginally stable. As you add more protons to the nucleus, the electrons get more tightly bound: helium, lithium, beryllium, etc: I actually found binding energies for all those atoms and you can clearly see from the values for the energy, that as you go to higher atomic numbers, the electron configuration approaches the simple product state of hydrogen-like orbitals. In other words, as the box gets smaller, the interaction of the electrons becomes insignificant.

What brought me back to this question was I realized I had made a big mistake: I had botched the symmetrization! The wave function was actually quite a bit more complicated than I had drawn it, because my answer did not preserve the correct symmetry for fermions: the function must reverse polarity when you switch particles. You can always do this by taking appropriate sums and differences of whatever function you already had, and that's what I do in this article.  It turns out this is the exact same symmetrization method that needs to be applied to the case of two isolated hydrogen atoms. You are not allowed to simply say that the electron here is spin-up, and the electron there is spin-down: try writing it down that way, and then interchange electron A with electron B. You know that according to theory, you must get back the same function with a negative sign, and you'll see that you don't. The wave function isn't right until you symmetrize it the way I've shown in my article, and it turns out that this leads to some very distrubing and surprising consequences.

In the meantime, I wrote a couple of other articles in April about Fourier Transforms, Ladder Operators, and Pertubation theory. My son's friend was taking a course in Mathematical Physics, and I had been helping him with assignments. It drives me crazy the way they suck all the physics out of these things and just give you math question that amount to manipulation of symbols according to a set of rules. That's not physics to me, and it's not even math. For me, it's all about the interpretation, and I got into some very cool stuff in this article about solving differential equations with Fourier transforms. The very last problem on the homework assignment was a weird-looking differential equation that I vaguely recognized as having something to do with the quantum harmonic oscillator: I couldn't quite put my finger on it but finally figured out that it had to do with ladder operators. One thing led to another and I started writing some very cool stuff about Perturbation Theory. This is something that's taught, as usual, as a set of rules for symbolic manipulation of functions, but here I make it into something pictorial, relating it to ladder operators and Taylor expansions. It's a bit half-baked, but it's still good.

And then it stops. I don't know exactly what happened, except I remember I was working all summer on a construction survey crew, and we had a lot of fifty-hour weeks. My next article wasn't until almost six months later, and that's where we'll continue when I come back again.

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