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