JOHN MEANEY

10.3.06



THAT FELINE PARADOX...

...of Erwin Schroedinger. Can I just muse on this for a while? You'll know the setup for the famous thought experiment. I'm just wondering whether everyone knows where the paradox is. Actually, I'm wondering if I know exactly where it is.

You trap a cat in a box. Inside the box is a nefarious device set to activate at random (e.g. by radioactive decay) which will release poison gas. When a biologist (since no physicist would be cruel enough to actually perform the experiment) opens the box, they'll either find a dead cat or a live cat. Before they open the box, the cat is neither alive nor dead but somehow both. Mathematically, the feline wave function is a superposition of two possible states (or eigenvalues).

In fact, I think I read that much in a lit'ry novel last year. Um, Saturday, by Ian McEwan? That's the one where it take twenty pages for the protagonist to walk from his bed to a window, stare out, and then return to his bedside. Hmm.

Anyhow, where's the paradox? And who was this Schroedinger guy, anyway?

Well, when you're trying to work out how something behaves in a quantum way, you typically attempt to write down the particular variant of the Schroedinger wave equation that describes that particular situation, and then solve it. That's easier said than done. (And it wasn't that easy to say...)

For a start, Schroedinger was a clever dude. Did he really think a cat could be simultaneously alive and dead? In fact the cat's in a state of kiss-your-skin-goodbye-as-soon-as-I-get-my-claws-on-you, or similar. (Either Terry Pratchett or Ian Stewart or Jack Cohen derived the correct solution, in one of their books.)

We may be zeroing in on a paradox here. A paradox of some kind.

Why do sodium-vapour streetlamps always shine a particular shade of orange? Because an outer electron can be in one of two particular states (there are other states of course, but it's still a restricted range of allowed values), and if it's in the more energetic state and releases its energy, it emits one photon and drops to the lower level. The energy difference is always exactly the same, the one photon (carrying away the energy) therefore has exactly the same energy, therefore the same frequency... and therefore the same colour, as perceived by us.

If you narrowed your focus to these electrons, you would not know whether a particular electron was in the higher or lower state unless you directly measured it. Before measuring, you could say that the electron's wave function (related to the probability of finding it in a particular state and place) is a mixture of both possibilities.

In general, the wave function could be a mix of umpteen possibilities, some of which might be more likely than others (different probabilities).

So far, this just seems to state that you don't know what's really there until you look. Right. Where this departs from everday logic is when you look at scenarios like diffraction. If you squint and look at one of those streetlamps through your eyelashes, you see weird patterns. (Even if you haven't eaten strange mushrooms recently.) That's interference, where light waves reinforce and destroy each other at various angle to produce thos patterns.

Remember, this is a real effect. Go look at a streetlamp tonight.

The simplest form of interference pattern is when you slice two razor-thin vertical slits in a piece of card, and shine light through. Beyond the card, if lay a solid screen across, you'll see a neat striped diffraction pattern. If you've studied this, you'll know what I'm leading up to...

...which is, if light is made of particles, how do you get interfering waves?

Each photon goes through one or other of the two slits, but behaves as if it knows where the other slit is. How do I know this? Because the accumulated photons, once through the slits, building up a striped pattern on the screen they hit. There is ZERO probability that a single photon would hit one of the dark areas -- where destructive interference occurs.
But if the photon was ignorant of the second slit, it would in fact occasionally strike one of those areas, as it simply goes more-or-less straight through one slit.

So the photon is said to be in a superposition of two states AS IT IS GOING THROUGH THE SLITS until it strikes the screen, where it turns out that it did IN THE PAST travel through one slit only.

Uh-huh.

Cats don't behave like this.

Actually, maybe they do. Cats are mystic creatures. Can we start all over again with the Schroedinger's Dog experiment? Okay, the dog is either alive or dead, it's not both at the same time.

A dog is not a photon.

But a dog consists of many, many, many subatomic particles built up into a vastly complicated structure that goes woof and wags its tail. Electrons and protons and photons and all the rest DO behave in that weird quantum way, but entire dogs do not. And neither do cats, except when they're engaged in private mystical experiments that are none of our business.

I believe this is the paradox that Herr Doktor Schroedinger had in mind.

One particle obeys his wave equation (which is fiendishly difficult to solve algebraically for all but the simplest situations). Two particles... yep, you can write down the equation, it's just harder to solve. Three particles... okay. Not sure I can solve the equation, but I can still write it down and it looks well behaved. Four particles... Sure, still going strong.
Somehow, when we get to dog-number of particles, the rules completely change.

So, is this a paradox?

Yeah, I think so. It's in the same class as Zeno's Paradox, in any of its variants. You know, Achilles runs ten times faster than the tortoise he's chasing. Say he starts ten yards behind the tortoise. When he reaches the place the tortoise was (at the time Achilles broke into a sprint) the tortoise has moved on by one yard. When Achilles reaches that point, the tortoise has move on by one-tenth of a yard. And when Achilles reaches THAT point, the tortoise has moved on by one-hundredth of a yard...

Now the modern accepted view of Zeno's Paradox is that it's based on a misunderstanding. Achilles can't overtake the tortoise until he's passed through an infinite number of ever-smaller distances in ever-shorter time periods. Those old Greeks assumed that it took forever to add up an infinite number of numbers... but nowadays, that's seen as a mistaken assumption. An infinite series can add up to a finite value.

I believe that not every mathematician accepts that we've made the problem go away. Zeno wasn't stupid, any more than Schroedinger.

So what about the cat/dog? Seems to me it's about adding up (for a certain complicated meaning of the word 'add') lots of little numbers, both for Zeno's and for Schroedinger's paradoxes. Somehow, the addition of lots of terms (and it's definitely a FINITE, not infinite number, in the Schroedinger case) qualitatively changes the rules. Jeez... can anybody spell 'emergent properties'?

I hope you're not expecting me to solve the paradox here, although the preceding sentence may hold a clue!

At least, we now know that Schroedinger wasn't bonkers, he was brilliant.

Since we've established that, we might go on to explore the heart of the large-number-of-particles problem, aka the decoherence problem. In doing so we'll come across a third paradox, the one identified by Eugene Wigner... but perhaps that deserves a blog entry in its own right.

Time to rest those superheated brains, Pilots. Fly straight.

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