It's a classic thought experiment in quantum mechanics, from before the days when people were conscious about things like cruelty to animals. It goes like so: you put a cat in a sealed box (it's a thought experiment, so it doesn't matter that it can't breathe) with a loaded gun pointed at the cat. The gun is special, designed to fire upon the decay of a certain element that has an exactly 50% chance of decaying at any particular time. So, in essence, it's like roulette; the cat always has a 50% chance of being shot and dying. Now, because the whole schebang is in a sealed box, nobody can observe it. The central idea of quantum mechanics states that, until someone opens the box and looks at the cat, and as long as there's no other way of finding out if the cat is still alive, the cat is actually in a strange state of half-dead, half-alive, and doesn't actually become dead or alive until someone opens the box and looks at it.

Bear in mind that this is really a theoretical exercise, a lot like "if a tree falls in the forest and nobody hears it, does it make a sound?" The point is, if nobody can see the cat, then for all intents and purposes, it exists in a mix of all possible states (dead or alive, dead-alive). When someone observes the cat, it instantly resolves itself into one of those states (this, in quantum lingo, is known as the "collapse of the quantum-mechanical wavefunction). So while our cat had a probability of being either dead or alive, we say that it existed in a mix of states. And once it was observed, it "decided" which state it would end up in.

Quantum theory can be applied to things besides cats. Do you remember any physics? Specifically, the part about interference patterns? I'll sum up: if you shine a light through two slits (or a series of slits), the wave patterns of the light waves from the slits interfere with each other, sometimes canceling each other out, and sometimes reinforcing each other, forming a series of light and dark bands on a screen. That's the classic Young's double-slit experiment.

The other half of this story ties this in to the age-old debate about whether light (and indeed any elementary particle) is a wave or a particle. Interference like Young's experiment illustrates is only possible with waves, of course; little particles bouncing around would never form bands, they'd just scatter diffusely all over the place.

But alas, particle proponents have their own weapon: the photoelectric effect. To refresh your memory, this is demonstrated when you shoot a beam of light at some metal, and the photons hit the metal and knock off some electrons. When you lower the intensity (amplitude) of the light, interesting things happen. Specifically, fewer electrons get knocked off, but they still shoot off at the same speed as always. What we'd expect, if light was a wave, is that the wave would hit the metal, knock off the same number of electrons, but not give them as much energy, since the wave is lower-amplitude. Instead, quite the reverse happens. It would be like a gentle ocean wave rolling up to the harbor, picking up a single boat, and throwing it twenty feet into the air. The photoelectric effect is easy to explain, however, if light is a particle; since the light we're shining is lower-intensity, there are fewer photons, but of course, like little bullets, they still knock off the electrons at the same speed. How is this confusing paradox explained, where intereference says light has to be a wave and the photoelectric effect says it has to be a particle?

Let me mix things up a bit more for you: you probably didn't know this, but electrons display interference patterns, too. Not just the lowly beam of light, but this big honking thing that we seem to absolutely know must be a particle. Shoot some electrons at a pair of slits, and start to map where they hit the screen on the other side, and the little dots on your map slowly start to form a perfect picture of the same bands that light makes. How do the electrons know where to go, to make the banding patterns? They can't tell each other, "Hey, I just hit this spot, now you go hit that spot."

Here's where we finally get to the explanation. While the electron is in flight, between the emitter and its final destination on the screen, nobody can observe it. It's just like the cat; it is in a fuzzy blend of all possible locations and states. This fuzzy blend that is the electron actually has the characteristics of a wave. If you could see it, it would show you the probability of the electron being in a certain place. Where this quantum wave has peaks, the electron has a high probability of existing. Where the wave has troughs, the electron has a very low probability of existing. Our cat has a very simple wave, since it has only two possible states. Its wave looks like a flat line; since all chances are equal (it can be dead or alive), there are no peaks (where it's more likely to be alive) or troughs (where its more likely to be dead). This flat wave is our mixed dead-alive cat. Our electron, on the other hand, is flying through space and obeying all kinds of laws and things, so there are certain places where it's more likely to be, and certain places where you'll never find it, and therefore its quantum probability wave has peaks and troughs to reflect that.

Now it gets funky; while the electron is flying towards the screen, and nobody's observing it, its movement is actually described by the quantum wave. The electron is no longer a point, it is spread out in this fuzzy wave, more in some spots than others. The wave propogates, interferes with itself, and finally reaches the screen. Now, when it hits the screen, the electron gets "observed" and, since everyone knows electrons are not big fuzzy waves, but little tiny particles, it must suddenly change from a wave to a particle and appear somewhere. It picks a spot at which to appear based on the probability defined by the waves. Where the quantum waves have reinforced, it has a high probability of appearing. Where they cancel out, it has an almost-zero chance of appearing. This is the "collapse of the quantum-mechanical wavefunction," when the electron finally decides exactly where it ended up. Since all of the electrons do this, and all of their quantum waves are the same, when you map them out, you get banding patterns that show how the quantum probability waves affected their location. Where the waves reinforce, electrons have a high probability of hitting; these are your "light bands." Where waves cancel out, the electrons have a low probability of appearing; these are your "dark bands."

Ultimately it should be noted that this is all still theoretical. Since the electron only exists as a wave as long as nobody is measuring it, then there's no way to actually "see" the wave. As soon as you look at it, the wave collapses and the electron has a definite point location again.

But, if you have lots of electrons (or photons, or anything else), you can look at the patterns, and see how the probabilities map out a wavefunction that describes that particle. And when you have a whole constant stream of particles, like you get in light, you can actually see the photons hitting together and see the "waves" very easily.

Whew. That's enough QM for today, eh class? We'll learn about wave packets tomorrow, perhaps.