Question: explain non-commuting variables and what they mean for the uncertainty principle.

Sam Geen answered on 20 Jun 2013:

OK, so Heisenberg is driving along, and he gets pulled over by the police for speeding. The police ask him “Sir, do you know how fast you were going?” and Heisenberg says “No, but I know exactly where I am!” and they arrest him because he was commuting at the time.

Let’s explain the joke! Because jokes are funnier when you explain them. Heisenberg’s uncertainty principle says: take the accuracy you know your position. And take the accuracy that you know your speed. Multiply them together. The result is greater than zero – always. What this means is that you can’t know your position and your speed at the same time precisely – there’s always a small uncertainty. However! There are some things you can measure together that you can know at the same time. Why? Well, in quantum mechanics, your speed and your position aren’t just numbers – they’re “operators”, things that you can do to numbers to change them. And if you multiply position by speed, you get a different result than if you multiply speed by position. This is known as “non-commutativity” – things commute when doing them one way around is the same as doing them the other way around. For example, addition and multiplication are commutative – 5 + 2 is the same as 2 + 5, and 3 x 4 is the same as 4 x 3. But division and subtraction aren’t! 2 – 5 isn’t 5 – 2, and 3/4 isn’t 4/3.

If you want to know in detail what happens, well, do a university course! But this is basically it. Quantum mechanics says that everything is a “probability wave”, i.e. it’s like a wave on the ocean, but the wave is the probability that a particle (like an electron) is at any given point along the wave – the higher the wave, the more likely that the electron is there. Position and speed are just “operators” on the wave, things that change the wave into a position or speed. And doing them both one after the other gives a different answer depending on what order you do them in, which is why you can’t know for certain your exact position and speed at once.

Which is why Heisenberg had his driving license taken away and has to get the bus to work from now on.