Quantum mechanics

Offhand, I can’t think of any physics theory that was discovered just by one person. Often, though, there’s one go-to guy who wrapped the whole thing up in a paper somewhere, and whenever we think of that theory we think of them. Newton’s mechanics, for example, or Maxwell’s electrodynamics, or even Einstein’s relativity. But quantum mechanics isn’t like that; it was put together by a collection of extremely clever folks over the span of maybe twenty years or so. Many of them didn’t even like each other. Heisenberg, for example, couldn’t stand Schrodinger’s work, even though it turned out they were doing the same thing.

As a result, even today no two physicists completely agree on what quantum mechanics is. No one knows what its fundamental principles are, or how it should be taught. (Although a friend of mine likes to say that “quantum mechanics is just Heisenberg’s uncertainty plus Pauli’s exclusion.” I’m not sure what that means, but I like it because it’s short and snappy.) I heard a story once about a kid who misses the very first day of kindergarten, when the teacher explains what “numbers” are. He gets there the second day and everyone already seems to know, so he’s too embarrassed to ask. He doesn’t want anyone to catch on that he’s got no idea what a number is, so he works incredibly hard. He learns arithmetic and algebra, and later trigonometry and calculus. Eventually he becomes a famous mathematician and wins the Fields Medal. But inside he always feels like a fraud, because the truth is he still doesn’t know what a number is, just how it works. That’s sort of how I feel when I “do” quantum mechanics.

Over the next few weeks, I’d like to crack the QM door and let you peek in. Unfortunately, a peek is really all that’s possible, because there’s a fair amount of math that goes along with the subject. Things like spherical harmonics, Fourier transforms, and Hilbert spaces, along with a host of others. Probably you’ve never heard of any of that stuff before, but you’ll run into some of it if you’re interested in technical careers like engineering or the sciences.

Okay, so in my opinion, the very first thing to know about quantum mechanics is the idea of operators. In physics there are all kinds of observable quantities that might interest us; things like position, velocity, momentum, angular momentum, potential energy, kinetic energy, and so forth. In classical mechanics (what you’ve been studying) these quantities are just variables, which hopefully you’re familiar with by now. In quantum mechanics, each of these quantities gets an upgrade—they’re turned into operators.

So what is an operator? Well, I suppose it doesn’t help to say they operate on something, so maybe an example is the way to go here. Let’s say I invent an operator called D. What D does is take the derivative of something with respect to x. That is,

D = d/dx

Sitting out there, all by itself, D doesn’t mean very much. It’s a derivative, sure, but of what? To be useful, D has to operate on something, say a function f(x). Suppose f(x) is something simple, like, oh, 3x⁴. If we let D operate on it, we get

Df(x) = 12x³

I just made up the operator D, but I should actually mention that it’s not the first operator you’ve run into. In trigonometry, things like sine and cosine are operators. Obviously a “cos” just sitting around doesn’t mean anything, but everyone’s seen cos θ. It doesn’t mean you should multiply θ by cos, rather the cosine operates on θ.

Groovy? Hopefully you’re happy with this, but if not you can always write me with questions. While we’re at it, I should also mention a couple of pitfalls so you’ll know to avoid them. Let’s go back to my made up operator D again. What if this time it operates on the function g(x) = e^2x? In that case, we get

Dg = 2e^2x = 2g

Does this mean we can cancel out the g’s and conclude that D = 2? Of course not! Remember, we’re not multiplying by g here, we’re operating on g. In this case, g was an exponential, so we got it back when we used D on it. If you’re still not convinced, consider this equation:

cos θ = sin θ

Is it okay to cancel out the θ’s and say cos = sin? Nope.

Here’s another thing you should be aware of. What does it mean if I have D²? Should you take the derivative of something and then square it? You might think so, but that’s not it. When I say D², I’m really saying you should apply the operator D twice. For example, going back to the f(x) from earlier,

D²f = DDf = D(Df) = D(12x³) = 36x²

And there you go. To be a bit more explicit, since D is a derivative, D² is a second derivative. (If you applied them to a particle’s position, D would give the velocity, and D² would give the acceleration.) Armed with this knowledge, you should be more than a match for this week’s homework.

https://mywebspace.wisc.edu/mweinberg/web/Operators.pdf?uniq=-9cdw1b

G’bye to special relativity

So that about wraps up the special relativity I wanted to cover with you guys. I suppose it’s time we move on to other things. If you survived (and possibly even enjoyed) the problems from the last few weeks, you should be fairly impressed with yourselves. At many universities, the material we covered would be part of a sophomore-level course, so you’re seeing it some two years early. Perhaps you breezed through them, but I’ve found from my own experience that the biggest challenge to working problems like this is just learning not to be intimidated by them. They use new language and deal with very new ideas, and it can be very tempting to give up without actually trying them. Personally, it took me years to learn that lesson, and it wasn’t until midway through my undergrad that I stopped freaking out every time I saw something new in physics. Well, mostly. I still freak out every now and again.

If you really liked special relativity, you might be happy to hear that we really only scratched the surface. There are deeper and more beautiful ideas in relativity, like spacetime structure, covariant and contravariant vectors, relativistic energy and momentum, field transformations, and tensor potentials. So if this stuff really grabbed you—and you’re willing to learn a little more math—you might consider taking a course on relativity in college sometime.

Length contraction problems

Well, since no one left a comment, I’m assuming that no one had any thoughts the length problem (or, surprisingly, the Veyron). But as you might anticipate, the explanation is similar to the one for time dilation. So, if A says that B’s measuring stick is short, and B says that A’s measuring stick is short, who is right? Answer: they both are. It makes more sense if we think about how length is actually measured.

Let’s say, once again, that A is driving the Maserati, and we’d like to measure its length on the ground. If A happens to be parked, this is easy: we just lay our ruler on the ground, check the positions of the front and back bumpers, and subtract the two. No problem.

For that matter, it’s not hard to check the length of the car if it happens to be moving, either. Same procedure, but this time obviously we have to be sure to check the two positions at the same time. If you don’t, then clearly the car will move while you’re measuring and you’ll get the wrong answer.

Hmmm. Just maybe, this could be starting to sound a little familiar to you. The thing that bails us out of the paradox, once again, is the first consequence of relativity: two events that are simultaneous in one frame are not going to be simultaneous in another. We say we were careful to measure both points at the same time, but no matter how careful we are, the dude in the car is never going to agree. He’s going to complain that we read the front end of the Maserati first, and then read the back end after the car had already moved forward, so of course we got a number that was too small.

Well, A has always been something of a complainer, and we’re thinking he’s really just full of it, but it’s important to remember that neither of us is “actually” right. In his reference frame, it really is true that we measured the two points at different times. In our reference frame, we really did measure at the same time, and his car really is shorter. For that matter, someone moving in a third frame would tell us we were all wrong: they would claim that we mismeasured, and his car is shorter (exactly how short would depend on how fast they were going, but they’d get a different length than us).

Is there really a paradox hiding somewhere in length contraction? Once again, it sure feels like it, but I’ve been looking for years and I’ve never actually found one. Still, no one’s ever accused me of being too clever, so maybe you’ll have better luck. I encourage you to roll this one around in your heads for a while, see if you can come up with something that looks like a contradiction. To help get you started, this week’s problems involve a couple of entertaining “paradoxes” about length contraction. No worries; you won’t need to figure them out to answer the questions, but if you have any guesses on how to fix the paradox, leave me a comment. (Seriously. I get lonely out here in these French villages.)

Here are this week's problems:

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

and here are a couple of pictures to help you visualize the ladder paradox:

https://mywebspace.wisc.edu/mweinberg/web/The%20ladder%20paradox.pdf

Length contraction

Alright, maybe you found the discussions about simultaneity and time dilation a little difficult to picture. In that case, you’ll probably appreciate the third consequence of relativity a little more. Length contraction simply claims:

Moving objects are shortened.

Creepy, I know. To be a little more specific, moving objects are shortened in the direction of motion. So if a Maserati MC12 goes zipping by you at near light speed, its front bumper will be closer to its back bumper, but its height and width will stay the same. Just as with simultaneity and time dilation, I want to remind you that this is not some trick of observation: it’s not just that the car looks shorter, it’s that it actually is shorter (in the reference frame of the ground). While we’re on the subject, take a guess: by what factor do you predict the length changes?

Okay, well, I hate to keep you in suspense. If you guessed that the factor was γ, just like for time dilation, then you have a devious mind. But you’re also correct. If the length of an object at rest is L’, then the length that someone on the ground measures is L = L’/γ. If you don’t believe it, you can check out my proof:

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

So, huh. That’s weird. Why γ? Why not, I dunno, 3γ? Or some entirely different function of velocity? If you read the proof, you’ll see that the way γ crops up is quite different from the way it came about in time dilation, so what gives? Actually, it’s no coincidence that the factor by which time is increased is exactly the same as the factor by which length is decreased.

You’ve probably just started studying vectors in physics, but of course you know that they’re basically just a magnitude and a direction. If I want to, I’m free to rotate a vector however I like (by multiplying by something called a “rotation matrix”). Of course, the rotation can’t possibly change the magnitude of the vector, just which way it’s pointing. The secret is, special relativity is just a very abstract kind of rotation: instead of rotating a normal three-dimensional vector, say, out of the x-direction and into the y-direction, it rotates a four-dimensional vector out of space entirely, and into time. The magnitude of the “spacetime” vector doesn’t change, which is why the amount that length decreases has to exactly match the amount that time increases.

Of course, the guy driving the Maserati doesn’t think his car is shortened. To measure it, he’d have to use some kind of measuring stick, but all his measuring sticks are also shortened by the same factor! As a result, no matter how carefully he measures, he’s never going to agree that it’s not its normal length.

In fact, just as with time dilation, remember that as far as he’s concerned, he’s stationary, and it’s the road that’s moving underneath him. Actually, if he sees anyone on the ground whip out a measuring stick, he’s going to think they’re the ones that are shortened. So this raises a similar problem to the one in the time dilation section: if Al is the guy in the car and Bob is a scientist on the ground, Al says Bob’s measuring sticks are short, and Bob says Al’s measuring sticks are short. Which one is right? Write me a comment and let me know what you think. In the meantime, here are a couple of questions about length contraction to entertain you. (Remember, if you get stuck, ask me questions; I’m always happy to help out.)

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