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:

∫_a^b 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.

https://mywebspace.wisc.edu/mweinberg/web/waveFunction.pdf