Thursday, November 29, 2007

Bell's theorem

Hey folks,

For those of you who are interested, I went ahead and wrote up the proof for the "orthodox" position. Let me know what you think.

Monday, November 26, 2007

The Schrodinger equation

I just know you’ve been in suspense for two weeks now, so I think it’s time I told you which of the three viewpoints (realist, orthodox, or agnostic) is correct. It’s worth mentioning that only recently did physicists discover there even was a right answer to this question. (Well, not that recently; the news is over forty years old by now.) Before the 60s you could basically pick whichever viewpoint appealed most to you, and each one had a large group of fans.

In 1964 though, an Irish physicist named John Bell demonstrated that it makes a real, observable difference whether the particle had a definite (though unknown) position before the measurement. As Griffiths puts it: “Bell’s discovery effectively eliminated agnosticism as a viable option, and made it an experimental question whether the realist or orthodox position is the correct choice.” So if any of you picked agnostic thinking it was the safest route, you might be surprised to learn it was the first one to bite the dust.

I’ve struggled a lot with the question of whether to show you Bell’s proof, but I finally decided against it. I realize this probably means that Taylor will never believe me, but unfortunately the proof requires a fair bit of background information, as well as being somewhat complicated in its own right.

But let’s say for the sake of argument that you believe me, and agree that there is a right answer. So what is it? Alright, fine, I’ll tell you: it turns out that the correct position is… the orthodox one. Believe it or not, the particle did not have a position before you measured it. This is why quantum mechanics is so insanely weird. Before the measurement, that particle wasn’t entirely anywhere; however, it was somewhat everywhere. It’s a safe bet that no human on earth can really “comprehend” this, and it’s difficult to even communicate this fact through language, but there it is anyway: particles just don’t have positions.

And this isn’t just true of positions! Get down small enough, and you’ll find that particles don’t have definite momenta either, or angular momenta, or energy, or anything else that can be measured. The act of measurement gives these things definite values. If you thought relativity was weird, get a load of that.

In the comments last week, Taylor made an extremely good point. The complaint was this: suppose you measure a particle’s position and find it was at point P (from the graph in the “wave function” blog post). Then, immediately after that, you make a second measurement. Who’s to say you won’t find it far away in, say, Atlanta? If you make the measurement fast enough, wouldn’t the particle have had to travel faster than the speed of light?

That’s a legitimate objection, and if quantum mechanics allowed things like that to happen, we’d have a real problem. But wait! We’re now asking a very different question! We’re now asking what happens when we perform a second measurement on the same particle. The trick is, the first measurement changes the particle’s wave function, so now it looks like this:

As you can see, there’s now zero chance of finding the particle anywhere except at point P. So if you make a second measurement (and third, forth, etc.) you will continue to get the same value back. That is, before the first measurement there was no telling where it would be, but once you’ve made that measurement, any subsequent measurement will give the same value. (To throw around a bit of jargon, the first measurement is often said to “collapse” the wave function into the function above, which is called a “Dirac delta function”.)

Alright, time to get back to work. In the “wave function” blog post I did a sort of comparison between classical and quantum mechanics. In both cases, we consider a particle of mass m, and for simplicity, we only let it move in the x-direction. Then we subject it to some kind of force, F(x, t), and we’re ready to roll. In classical mechanics, the goal is to figure out the particle’s position at any particular time, x(t), and in quantum mechanics (where particles often don’t have positions), the goal is to figure out the particle’s wave function at any particular time and place, Ψ(x, t).

But how do you get these in the first place? In classical mechanics, the answer is easy: that’s what Newton’s second law is for! Once you know that F = ma, your worries are pretty much over. Sometimes it’s a teeny bit more complicated, because you’ll be given a potential energy instead of a force. But that’s no big deal: you know from the definition of work that F = -dU/dx (in this case, I’m using U to mean potential energy). If we put this in, Newton’s second law reads: md2x/dt2 = -dU/dx. Plug stuff in with the right initial conditions and you’re done.

So that’s how you get x(t) in classical mechanics, but how do you get Ψ(x, t) in quantum mechanics? As it turns out, we get it by solving something called the Schrodinger equation. So the Schrodinger equation does the exact same thing in quantum mechanics that Newton’s second law does in classical mechanics. This may not seem like a big point, but I went through an entire year of quantum mechanics before anyone bothered to tell me this. So in case you ever take a course in QM, maybe this will save you some grief.

Anyway, maybe you’re wondering why I haven’t actually shown you what the Schrodinger equation looks like yet. That’s because it’s your job to tell me, which brings us to this week’s homework.

Thursday, November 8, 2007

The wave function

Okay, now I’m going to sum up in one paragraph almost every problem you can get in your mechanics class. Let’s say we find a particle of mass m and we let it move back and forth in the x-direction. While we’re at it, I’m going to apply a force to the particle, F(x, t). Basically the whole idea of classical mechanics is to figure out, given that force, what the particle’s position will be at any particular time, x(t). Pretty much everything else follows from that. Yours for the asking are the velocity, dx/dt; the momentum, mv; the kinetic energy, (1/2)mv2; or whatever else Rebecca decides to ask you for.

The point of this is simply to give you something to compare quantum mechanics to, to help you get oriented. So let’s compare: suppose we have the exact same setup (a particle of mass m moving in the x-direction) in quantum mechanics. The problem is that at the microscopic level, particles don’t necessarily have positions, so it doesn’t make sense to ask about x(t). Instead, what we’d like to find out is something called the wave function of the particle: Ψ(x, t). This is the interesting thing, and once we’ve got it we can figure out anything else we want.

So the wave function, Ψ, does much the same thing for you in quantum mechanics that the position, x, does for you in classical mechanics. It’s the big thing, from which pretty much anything else can be figured out. You say you want to know about the particle’s momentum? Simply apply the momentum operator to Ψ and see what happens. The kinetic energy? We’ve got an operator for that, too. Potential energy? Oh, jeez, we’ve got dozens of operators for that. (Of course, you can’t actually apply any of these operators yet, because I haven’t actually told you what they are.)

Alright, so that’s what the wave function does, but what exactly is it? The best answer was provided by a German dude named Max Born, who provided the “statistical interpretation”:

If Ψ(x, t) is the wave function of a particle, then Ψ2 is the probability of finding the particle at point x, at time t.

To be a teeny bit more mathy about it:

ab Ψ(x, t)2 dx = {probability of finding the particle between a and b, at time t}

That is, if you’ve got the wave function of a particle Ψ, and you’d like to know the probability of finding it between two particular points, just take the area under the curve of Ψ2 from a to b. For example, the picture is a graph of Ψ2, so we would be much more likely to find the particle near the middle than out at the edges.

If you’re thinking all of this is a little bit hokey, I can hardly blame you. What kind of a theory is this? We get the wave function, the most important thing we can learn about this particle, and the best we can do with it is find the probability that a particle will be in a certain place? Why can’t we say beforehand exactly where it’s going to be when we measure it? Is there just something wrong with quantum mechanics?

Actually, all this can be summed up in one question. Let’s say a particle has the wave function in the picture, and I actually do measure the particle’s position. It so happens that I discover the particle was somewhere right around point P. The question is: where was the particle the instant before we measured it?

There are three general opinions you might have on this (courtesy of David Griffiths):

1. The realist position: “Well, obviously, the particle had to have been at point P. That’s where we found it after all. If you find your socks under your bed, clearly they were under you bed right before you found them, you just didn’t know it.” It certainly seems hard to argue with this logic, and Einstein himself took this position. If it’s true, the theory of quantum mechanics is a bit of a letdown, because it must be incomplete. That is, the particle had a position before we measured it, and quantum mechanics just couldn’t tell us what it was.

2. The orthodox position: “Believe it or not, the particle wasn’t really anywhere until you measured it. The measurement itself caused the particle to have a position.” So how about that? In addition to being really, really weird, this claim requires us to believe some pretty farfetched things. In particular, it means there really is such a thing as a “random” event: it was not possible, even in principle, to know where the particle was going to be before you measured it. Even if you knew every true thing in the entire universe, you still could not have said where that particle would be.

If this surprises you, maybe you can relate to this: when I was a kid, I thought everything in the universe was completely determined. A coin flip, for instance, only seems random; if you knew exactly how it was flipped, and exactly how the air moved around it, you could figure out whether it would come up heads or tails every time. A computer claims to generate “random” numbers, but it really just takes a seed number from its internal clock and performs calculations on it. Even humans would just be machines, following deterministic paths (though they might think they had free will). In short, I figured if you knew all the starting conditions of the universe, you could figure out everything that would happen, from a coin flip to what a person would say at a particular moment. However, if the orthodox position is right, even with a perfect knowledge of the universe, there are some things that can’t possibly be known beforehand.

3. The agnostic position: “I refuse to answer the question. I don’t want to, and you can’t make me.” Actually, this isn’t as ridiculous as it sounds. The point is, it makes no sense to ask about where the particle was before a measurement: the only way to know that would be to make a measurement! In that case, it wouldn’t be before the measurement anymore, would it? It’s just goofy philosophy to ask about something that can’t, by definition, be known. It’s like asking “what if time suddenly started to run at half-speed?” There’d be no way to check. It may sound like you’re describing two different things, but you’re actually just describing the same thing in two different ways.

This question bugged me for years, but I always thought it was basically a matter of opinion. Believe it or not, though, there actually is a correct answer! Incredibly, it makes an observable, testable difference which position is right. I’ll give it away next week, but I’d like to know: what do you think? Position 1, 2, or 3? Or do you have a different position altogether? Write a comment and let me know.

In the meantime, here's the problem for this week. Feel free to write me if you get stuck, too.
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