Sunday, February 14, 2010

The clicking detectors

The detection of clicks or spots on a photographic plate has long been held to be clear evidence for the particle nature of light. But attempts to make a mathematical case have always fallen just short. The reason is that the statistics are ambiguous. It turns out the classical theory supports the facts just as well as the quantum theory.

By "classical" I'm talking about the theory where the waves are classical but the atoms are quantum. You have to make one assumption: that the detection probability, say of a silver halide crystal changing state, is proportional to the intensity of the incident wave. Why should a crystal behave this way? That's another question, but at least it's a reasonable working theory. It is of course the only theory which makes the classical wave agree with the photon picture, and it happens to be a reasonably plausible theory.

It works fine when you have lots of photons, but gets tricky when you go down to "single photons". You do this by attenuating your source so the photons only strike once every ten minutes. Wouldn't the waves be too weak to concentrate their power into a detection event? Not according to our working assumption. Remember, by the way, the appearance of a dot on a photographic plate is what we call thermodynamically spontaneous - you don't need a net energy input to drive the transition. The wave theory still works even down to the very lowest levels.
But now let's add a twist. We're going to split the beam with a half-silvered mirror. Now the wave and particle theories make different predictions. The wave procedes at half strength to each detector, so there's some chance that both will click. But the particle goes one way or the other, so both detectors can never click.

Or can they? The problem is you're never quite sure that you have only one photon, no matter how weak the beam. Let's do the statistics. Take a beam with one photon every ten seconds. Use a one-second "coincidence" window. The beam splits in two so each detector fires every twenty seconds (100% efficiency). The wave theory says that there is a certain, constant wave intensity at each detector, that they fire on average every twenty seconds, and it doesn't matter if one or the other fires. In any given second, there is a one-in-twenty chance of either detector firing, and a one-in-four-hundred chance of both.

What do photons do? Photons are different, because if Detector A fires, it "uses up" the energy that "might have" also fired Detector B. That's the theory anyways. What do the statistics say?
Let the photons obey Poisson statistics (typical for laser). You can see where this is going. In one second there is a one-in-ten chance of a photon. What are the chances of two photons within the same "window"? One in...get this...two hundred. (Write the formula for Poisson if you don't believe me). But wait...just because there are two photons doesn't mean that you get two clicks. They might both go to the same detector. Four ways...AA, AB, BA, and BB. Only half the options give you coincidence counts. The odds? One in four hundred.

You can see that we can make the beam weaker and the window tighter, and it still doesn't hlep. The photon theory and the wave theory will always predict exactly the same number of clicks.
Now...IF you could fire one photon at a time, there is a very simple experiment which would totally go against the wave theory. You just put up a beam splitter and wait. With single photons, you can NEVER get two clicks. And the wave theory predicts that whatever your detection efficiency, call it n, you should get two clicks with a probability of n-squared/4. (25% of the time for 100% efficiency). It's a very simple experiment.

Interestingly enough, the much-referenced paper by Thorne et al
proposes an experiment which is generally similar to what I've described except it is decidedly more complicated. There is a third beam used for gating, and there are electronic coincidence detectors. Then you have to calculate something called "second-order coherence". It's just overly complicated enough that I might plausibly claim to not be convinced by the results. But of course that would be unreasonable of me.

Yet I have to wonder. The authors of this experiment make it a major selling point that it clearly demonstrates the particle nature of light at a level suitable for an undergraduate lab. Yet their version is far more complicated than my bare-bones version. If the case is so clear-cut, why the complications?

Wednesday, February 10, 2010

Something I just figured out yesterday

People go on and on about the deep mysteries of quantum mechanics. It seems like the double slit experiment is a huge mind-bender for people. Not for me. I can do the double-slit calculations and they make sense. Basically I can handle stuff in quantum mechanics when there is one electron at a time. It's when you add the second electron that it gets messed up.

I finally decided that it should be possible to work some things out by taking the very simplest two-electron case. No, it's not the Helium atom, although that's also an important one. No, it's not the Hydrogen molecule either. No, it's not the scattering of one electron by another. And it's not even the infinite potential well with two electrons in it. I can't do any of these problems.

What I've tried to do is the very simplest possible case. It's the case of two isolated hydrogen atoms, treated as a single system with two electrons. Now why would you do it that way? You can solve for a single hydrogen atom, and then if you have a second one, you've already solved it. It's not exactly easy to solve for one electron in the sense that you've got to differentiate stuff in polar coordinates, but it's at least straighforward.

But that's why I chose to do it as a two-electron problem. You just get the familiar solution centered around each atom, and then combine them into symmetric and antisymmetric combinations. The symmetric state is the lowest energy.

Now bring in the electrons. You can put the first electron in that ground state and it is then shared between the two atoms. If you want to localize the electron at one atom, you have to use the combination of symmetric plus antisymmetric. But since the energies of these two states are slightly different, they gradually go out of phase with each other and eventually you find the electron at the other atom, even if that other atom is quite far away. That's what they call tunneling.

What gets interesting is the second electron. Forget everything you heard about two electrons being able to occupy the same state if they have the opposite spin: technically that's correct, but it fools people into thinking they can just drop two electrons into the states they've already calculated. While that would give you two nice hydrogen atoms at first glance, it's not really correct. It ignores the interaction energy between the two electrons.

If you've ever glanced at the solution for the helium atom it starts by writing the six-dimensional Schroedinger equation which basically has terms for the potential and kinetic energy similar to the hydrogen atom, plus a new term in 1/(r1-r2) which is the interaction potential. Now, what occurs to me in the case of two hydrogen atoms, is that it's exactly the same equation. The way nature solves this equation in practise is that it anti-correlates the two electrons by putting one here whenever the other one is there, so the interaction term basically gives you zero and you get the familiar solutions. But it occurs to me: why can't you let the hydrogen atoms solve this problem exactly the same way that helium does? In other words, take the "known" solution for helium - well, you can't easily write it down but somehow nature "knows" a solution" - and scale it by a factor of 2 on length so it fits the hydrogen atom. How can this not possibly be a correct solution of the Schroedinger equation? True, it's not the lowest energy solution, but it's still a solution.

So each hydrogen atom becomes a mini-helium, with two half-electrons vying for the same space. The energy is exaclty one-quarter of the helium ground state energy of -76 eV, which comes to something like -9.5 eV per electron. Of course, you can't ionize it with 9.5 eV, you need to use 19 eV because its a coupled system and you have to ionize both at the same time. But regardless, you've got a mathematical state which it seems to me should have some physical consequences.

Friday, February 5, 2010

Let's get right to the point.

I hate physics because I can't do it anymore. Once I could, or thought I could. Now I can't.

I can still do this and that, but when it comes down to it I don't really know what I'm doing. And that bothers me. Because I think I ought to know what I'm doing.

When I used to be able to do physics, it was because I made up my own rules. All you had to do was point me in the direction of the problem and I'd sit down figure out what was going on. It had to make sense, and in the end it always did. It seemed like I could do anything.

I remember in first year physics we had a lab on the Helmholtz resonator. You blow across the top of a beer bottle and you get a sound out of it. We had to do some measurements and plot the frequency against the volume of air in the bottle. They had a formula for us and it was no big deal. But I didn't want to look at the formula. I remember I went for a walk one night and just thought about it. I thought about it until I figured out what was going on, and then I came up with my own way to calculate the frequency. Oh, I might have been out by a factor of 2 or so, but I had an angle on the problem and I could work it out. I didn't need anyone to rely on a formula from the textbook. And if my calculation was partly wrong, there was at least a reason why it was wrong; and I could understand what the reason was.

Of course there were things I couldn't do. There is a calculation done by Maxwell in the 1870's where he took molecules of air in an enclosure, gave them a certain amount of total energy, and then calculated what percentage of them would have a given velocity: in other words, the whole statistical distribution. And I couldn't do it. But it didn't really matter, because I could still see what was going on. You have two molecules and when they glance off each other, there is an exchange which conserves energy and momentum, but the individual energies get re-arranged. And after a long enough time there would be a statistical equilibrium where the velocities and energies, no matter how they had started out, would be mixed up into a stable unchanging state. And that made sense, even if I didn't have the mathematical tricks to actually calculate the distribution. So I felt alright about it.

I don't feel alright anymore, and it's quantum mechanics that's to blame.