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.

2 comments:

bob said...

Thank you Marc
so what I meant by "the real world" was that I had my doubts about whether or not the realist view, (which when taking into account things like technological and experimental limitations etc has to be expressed with probability, just like any pseudo-random event) and the orthodox view (which is that each particle can ONLY be described with probability) had any mathematical difference and that, along with other assertions, would lead many physicists to follow the orthodox religion, but apparently John Bell proved the prthodox view. I should like to see this proof.

More importantly, I prepared myself for the possible explanations for the orthodox view. so the waveform changes when the particle is located and is actually determined by locating it. WELL HERE'S A THOUGHT! what if someone ELSE measured the particle's location BEFORE you!? and you didn't know about it!? and then you go and describe its position with this probability based orthodox waveform. That waveform is WRONG, it's position is not random, it has been determined by that other guy's measurements, but YOU DON'T KNOW THAT! how about you go and measure its position and you think YOU determined it's position, you'd be wrong and not even know it. so let's go even further, some crazy advanced observer, let's call him steven, is able to measure the position of every particle everywhere, would that screw up the orthodox theory for some developing civilization that is just like us forty years ago, or would they come up with Bell's theorem on their own, thinking that they are determining some particles' position with measurements even though steven beat them to it and every particle already has a position
i guess what i'm saying is, would it have affected the mathematics of our observations at all if someone, say beyond the realms of our understanding, predetermined a particle's position by measuring it and not telling us? if there's no mathematical difference, then we really wouldn't know if these particles had positions before measured them at all, in fact, not even steven would be able to tell if particles had positions beforehand. if there IS a mathematical difference, then not only would the developing civilization which had the misfortune of cohabiting a universe with steven be misled into believing the realist view with no math to support or even suggest orthodoxism, but a physicist in our universe might come across a particle that was measured beforehand and be like "oh snap, according to this math, someone beat me to this. what are the odds!?".
my point is if we can labor under the delusion that we just determined a particle's position completely and mathematically unaware of the fact that some god named steven already did all that, who's to say we, in the real world, aren't just mathematically unaware of the fact that every particle really does always have a position that we just don't know until we measure it.

and another thing, since everything is like the sum of a bunch of particles which don't have position , does a regular object even have a position? If a tree falls in the forest, and no one has ever measured it's position, does it even HAVE a position? could there eventually be a particle-green party that makes particle parks to preserve the natural beauty of their lack of position, and fights mankind's interfering with waveforms?

I think i digressed a little bit there, and I have no intention of proofreading or even rereading this so sorry if it lacks focus or fails to make sense. I'm especially sorry if this ends up being damn long.

I appreciate you doing this for us, Marc. It feels good asking these questions. Now I may be slow to accept certain concepts, but I'm never one to argue with mathematical proofs, and I'm sure i'd open up to orthodoxolicism if you explained Bell's theorem, but if it's hard to explain or anything i'll be fine without it.

-Taylor (not bob)

marc2718 said...

Hi Taylor,

Holy smokes. This is quite a comment to wake up to. Well, here goes...

So what Bell showed was that there is indeed a mathematical difference between the realist view and the orthodox view, but I think you're misunderstanding the realist view. It's not because our machines or our experiments aren't good enough: quantum mechanics itself says the particle's position is random, so if it's not, something is very wrong with the theory. But since you asked, I'll try to write up Bell's proof in the next couple of days and post it. Also, I disagree with calling it a religion, just because people believed it without much evidence; I believe the Dallas Mavericks are going to win the NBA Finals this year, but that's not a religion either.

On to your second paragraph. Let's say someone else measured the particle's position before you. Would you be able to tell? The answer, according to Bell, is yes! Your theory that Steven measured all the particles in the world is just a hidden variable theory (that is, it claims that the particles are at particular points, but we don't know them because the variable that determines them is hidden from quantum mechanics) In this case the hidden variable is just a list that Steven made of the position of every particle. So this isn't a new objection, it's just a rewording of the old one! Your question (midway down the second paragraph) is: does it make a mathematical difference whether the particles had a position before you measured them. And the answer again is yes, it does. I can prove that the particles did not have positions before they were measured. Or, if they did (because Steven really did measure them forty years ago), I can prove that quantum mechanics will give the wrong answers, so we can tell the positions were already measured.

Now for your third paragraph. You're absolutely right: since regular objects are composed of particles without exact positions, regular objects also don't have exact positions! In fact, if you could look closely enough at an object (like, say, a pencil) you'd discover that it was actually fuzzy at the edges. Actually, part of your pencil, or your computer, or you yourself, exists everywhere in the universe. Now, Planck's constant is extremely small, so objects like you or your pencil are extremely well localized, but not completely.

Thanks for the comments; I'm really glad you're getting something out of this, and I'm very impressed that you've thought about it so much. That skepticism will serve you well. As for the proof, I'll write it up in the next couple of days and post it for anyone who wants to take a look.

 
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