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.

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

19 comments:

Kristoffer said...

the agnostic position is obviously always correct.

Magalindan... said...

I think the Orthodox because the particle is in a random spot and can have a different position in relation to different things.

marc2718 said...

Hi Magalindan,

Good to hear from you again. Orthodox is certainly a good pick; enough physicists agreed with you to make it "orthodox" after all. :) But you should know what you're buying: we're not talking about different descriptions of a position: if a ball is on a table and I say it's a meter above the ground or two meters below the ceiling, I'm still talking about the same position. The orthodox view says that the ball is actually at multiple places at the same time! That is, it's not entirely anywhere, but it's "somewhat" everywhere. It's unimaginably weird. Still sold on orthodox?

marc2718 said...

Kris, is that you? Are you back for more punishment now that my fantasy basketball team has whomped yours?

Kristoffer said...

Of course it's me, nobody else spells their name like I do :)

And don't make me lock your precious Jet Terry and Kobe Bryant in a box with a radioactive isotope, a geiger counter, and a cyanide cocktail, cause I'll keep making measurments until he's dead, and even Wigner and all his friend's won't be able to help him.

Rebecca Jensen said...

Ok, boys, no fighting.

And Marc, how could you not know it's Kris(-toffer Kyle)? He's got a corner on the agnostic market.

Magalindan... said...

Oh... that sounds so cool. So sure, why not. I'll stick with Orthodox.

bob said...

you're probably going to say it's orthodox, just because the waveform has to do with probability and therefore requires a "random" trial and it's the most far fetched, but in real life i'm sure every particle has a continuous position and velocity. with the orthodox idea, if you were to try to graph its position over time by taking a bunch of measurements really in a really small amount time, then each measurement would be a random trial. that would be CRAZY! and if you had like REALLY REALLY fast measurements then the average velocity between measurements would almost always exceed c (which i assume is still a magic number in quantum mechanics)
so basically, you're probably going to say it's choice number 2 and I'm probably not going to believe you

-Taylor (not bob)

bob said...

OMG WHY IS MOMENTUM IMAGINARY!?
HOLY CRAP!!
I FEAR WHAT I DON'T UNDERSTAND!!!

Kristoffer said...

bob sounds like a stiff, i like him

marc2718 said...

Hi Taylor (not Bob),

Whew, that's a lot of objections, but I'll try to address them all.

First, I agree: if I'm going to claim that the orthodox position is correct, I'd better be able to back it up with more than poetry about the randomness of the wave function. Telling you that the wave function "sure feels random" wouldn't do it. I'd better have proof (which, as it turns out, I do).

It's worth pointing out, though, that the randomness of the wave function doesn't suggest the orthodox position. It could just as easily be the realist position, in which case the "randomness" would be due to some variable we hadn't found yet.

I have to admit, I'm baffled by your comment about "real life". What have we been talking about all this time?

Your last point, though, is a very good one. If I understand correctly, you're saying that multiple (fast) measurements on the same particle might have it zipping from one side of the graph to the other at faster than the speed of light, violating relativity. That is a problem.

In fact, I intended to address this very problem in the blog in a couple of weeks, but since you asked, I'll give you a sneak peak. The thing is, you're now asking a different question: what happens if we make another measurement on the same particle right after the first? I'll go into more detail later, but the trick is, the first measurement changes the wave function to something else, so there's now no chance of finding the particle somewhere far away with a second measurement.

I'm sorry to hear you won't believe me when I give the answer, though. :)

marc2718 said...

As for the momentum, it's actually not imaginary. It's really just the momentum operator that's imaginary. The distinction is important, because imaginary numbers can never show up in measurements (like a measurement of momentum), but the machinery you use to get the measurement can be (and often is) imaginary.

marc2718 said...

And Kris, play nice with the TAG students. You were young and impetuous once, too.

Kristoffer said...

I still am young an impetuous, thats why I like him :)

Thomas said...

I've been learning about quantum mechanics just recently in my physics class, and I'm pretty frustrated with it. It seems like total bunk to me, not in the sense that it's necessarily incorrect, but rather that my physics professor is selling me all of these things without justifying them. At least when you talk about special relativity you can found everything on a simple thought experiment and a few reasonable assumptions, but quantum physics? Ugh. The "orthodox" view, with its "collapsing of probability" and all seems abhorrent to me. I have no support for this outlandish notion beyond my professor's claims that "it works remarkably well when dealing with elementary particles." He's a theoretical particle physicist, so I guess I should take his word for it, and I certainly don't know enough to counter his statement on scientific grounds, but the idea is so awful that I'll just go with the agnostic answer.

... Ack. Sorry for the rant.

marc2718 said...

Hi Thomas,

Thanks for the comment; it's always exciting to hear from new people. Can I gather from your post that you're in undergrad right now?

In one sense, I very much agree with you; it's not enough to say "trust me, I'm a physicist". One of the things I love about science is that there's no room for appeals to authority, and being an expert is no defense against being wrong. (I've gotten to see a couple of world experts be publicly very wrong.)

So if we're going to claim that one position is right and the other two aren't, we'd better be able to back that up with a pretty convincing experiment. As it turns out there is one, but I'm struggling with whether or not to put it in the blog, because it's fairly involved. It's called Bell's theorem (or sometimes Bell's inequality). If you're skeptical about my claim (and you should be!) I invite you to take a look at it on, say, Wikipedia, and decide for yourself.

Of course, you may decide you're not convinced, which is perfectly fine. But I encourage you not to look at the orthodox view in a negative light. No doubt, it's inconceivably weird, but to me this is a sign that we're really on to something. Nowhere is it written that physical laws will be comprehensible to humans, and it happens that these ones aren't. But that's not awful; it's actually pretty cool.

RW3141 said...

So, I have a question: you say that you have "dozens of operators" to determine a particle's potential energy (probability); why so many? Do they all yield the same result? Are some operators better than others?

marc2718 said...

Hi rw3141,

This is a really good question. The reason we have so many potential energy operators is that we have so many different kinds of potential energy.

For example, in classical mechanics the potential energy of something near the earth is mgh. If you get further out, gets closer to -GmM/r. The energy in a spring is (1/2)kx^2. And there's always the "trivial" potential energy 0 of a particle floating in free space. If you're in Rebecca's EM class next semester, you'll see a bunch more examples from particles in electric and magnetic fields.

Ultimately, there are so many because potential energy in a system can depend on so many different things, and for any one thing, it can depend on it in a lot of different ways. This is very different from kinetic energy, which is always the same old (1/2)mv^2.

bob said...

Also, the Bible SPECIFICALLY says only God can determine a particle's position, and that Adam was kicked out of Eden for describing an apple's position with a waveform.

 
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