Heisenberg's uncertainty principle

Well, it’s getting to be about time to wrap up the semester, and I’ve saved the most important thing for last. Perhaps you’ve gotten the sense in the last few weeks that quantum mechanics is one incredibly bizarre thing after another, and this is certainly true. However, in a very real way, all that unbelievable weirdness actually emanates from a single source: the uncertainty principle. More than that, in fact: everything I’ve told you so far about quantum mechanics can actually be derived from the uncertainty principle.

A friend of mine once referred to uncertainty as “the beating heart of quantum mechanics”. This may be overly poetic, but it is true that the uncertainty principle is one of the two pillars on which all of quantum mechanics is based (the other being Pauli’s exclusion principle). In spite of being so important, it’s usually misquoted, and it’s very often misunderstood.

So what exactly is uncertainty? Actually, when Heisenberg originally wrote his paper in 1925, he didn’t really explain it very well. (I personally think he was hedging his bets. Remember, this was long before the debate about the realist vs. orthodox positions had been resolved.) Basically, what he said was this: the more precisely the momentum of a particle is known, the less precisely its position can be known, and vice versa.

To be a little more mathy about it, let’s talk about the quantities Δx and Δp. Strictly, these are the standard deviations of the position and momentum, x and p, though people typically just call them the uncertainties. (I’d rather not actually define these, just because I haven’t mentioned a couple of things that go into the definition. However, if you don’t remember what a standard deviation is, it’s safe to think of Δx and Δp as the “spread” in the position and momentum. Of course, Rebecca might be upset with you if you don’t remember what standard deviations are…) Anyway, what Heisenberg said was this:

ΔxΔp ≥ h_bar/2

That is, the product of the two uncertainties is greater than some constant. This equation may seem fairly innocuous, but it’s actually an unbelievable result: it doesn’t matter what the actual constant is, the fact that the uncertainties must be greater than zero is the incredible thing. I’m really not sure that there’s anything in human experience which might prepare us for this. What Heisenberg had shown, even though he himself may not have realized it at the time, was the incompatibility of position and momentum.

When a lot of authors (including Heisenberg himself) talked about uncertainty, they made it sound as if it was somehow the experimenter’s fault. For instance, one way to measure a particle’s position is to hit it with a beam of light. If you hit it with low-energy light, you can do your measurement without disturbing the particle too much, so its momentum can be fairly well known. The trade off is that low-energy light isn’t good at resolving the particle, so you don’t know much about where exactly it is. Conversely, you could pummel it with high-energy light, in which case you’d get a great sense of where exactly it is, but the high-energy light would send the particle skittering off to wherever, so we’d have no idea what its momentum is.

This is not only silly, it’s downright misleading. The only conclusion that we would draw from that story is that this particular way of measuring a particle may not be very good. Sheesh, maybe the people doing these experiments just aren’t very smart; it sounds like a clever person would just find a less obtrusive way of measuring the particle’s position. For that matter, maybe the problem is even simpler: perhaps we just need to spend more money to buy a better machine.

But this isn’t it at all! What the uncertainty principle really says is much deeper than this. Recall from the orthodox position that a particle doesn’t always have an exact position or momentum. According to uncertainty, the more definitely a particle has a position, the less definitely it has a momentum, and vice versa. It’s not that an experiment that’s good at finding a particle’s position necessarily has to be bad at finding the particle’s momentum, it’s that when you measure the particle’s position, it doesn’t really have a momentum, so there’s nothing to measure!

At the two extremes, the uncertainty relation is actually even more surprising: if Δx = 0 (that is, the position is known exactly), then Δp → ∞ (that is, the particle has literally every momentum at once). On the other hand, if the particle has an exact momentum (Δp = 0), it exists everywhere simultaneously (Δx → ∞).

Alright, no doubt about it, that’s weird. But what makes uncertainty so important? What makes it the source of all weirdness in quantum mechanics? Actually, uncertainty is more general than I’ve let on: I started off talking about position and momentum, but the wonderful thing about uncertainty is that it applies to any two physical observables, be they position, momentum, energy, spin, or whatever else we can dream up that we might want to measure. Recall that every physical observable has its own operator in quantum mechanics. What the generalized uncertainty principle says is this: if A and B are any two operators, then

(ΔA)²(ΔB)² ≥ [A, B]²/4

This second, more general form of the uncertainty principle is the real engine here: we know that physical observables in quantum mechanics are represented by operators, and we know that sometimes two operators don’t commute. What uncertainty does is take this purely mathematical fact and turn it into something physical: because of uncertainty it is now impossible for some observables to have definite values at the same time; it’s why quantum mechanics has wave functions in the first place, and therefore probabilities, and by extension it’s the cause of the realist/orthodox/agnostic debate. So, no fooling, uncertainty really is the motivating force behind everything we’ve talked about in quantum mechanics.

Anyway, now that we know what commutators are (see last week’s blog), we have all the math we need to evaluate this relation. Well, almost. Actually the “absolute value” brackets mean something slightly different when we’re talking about imaginary numbers, but for the problems we’re doing you’ll be fine if you just remember to make sure the square of the commutator is positive. With this, you should be in good shape for this week’s homework.

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