Sean Carroll

In fact, this is deeply, closely, intimately related to the uncertainty principle in quantum mechanics.

Heisenberg's uncertainty principle says that you don't have any quantum states that are simultaneously definite in both position and momentum.

And that's because position and momentum, as it turns out, are two different ways, two different angles you can take

looking at the same thing looking at the quantum state looking at the wave function okay and once you know the wave function as a function of position you can do a thing called the fourier transform you can turn it into you can transform it into a wave function as a function of momentum if you just gave me the wave function as a probability of every possible measurement outcome of momentum

you could also go back and figure out the probability of every possible measurement outcome for position so you just need psi of x or psi of p you don't need psi of x and p you don't need to know both at once that's the origin of the uncertainty principle in quantum mechanics and just to get a little tiny bit technical if you if you envision in your mind

a two-dimensional vector space, okay, with x and y axes, I could imagine rotating the axes to x plus y and x minus y, the two diagonals that go through the origin.

And different versions of momentum are kind of like those diagonal elements, and different versions of position are kind of like the original horizontal and vertical axes that you drew, x and y. And so if I have a vector

in x-y coordinates, and I specify it by a point in x and y, I don't need to give you extra information to specify it in the rotated coordinate axes.

It's already implicit there.

There is a formula.

It's easy to find the components of the same point in the new axes.

And that's exactly what position and momentum

how they're related in quantum mechanics.

So the rule, again, just getting a little bit technical, the jargon, if you want to throw around at cocktail parties, is that an entire quantum wave function is expressed as a function of some complete set of commuting observables.

So when I say commuting observables, I mean there's different observables I can do that I can measure one and then measure the other just fine.

One does not interfere with the other.

They do commute with each other.

So if I measure the position of an electron and the spin of the same electron...

or I measure the spin of an electron and the spin of a different particle, a proton or whatever, those are commuting observables, they don't bother each other.

But the position and the momentum of the same electron, those don't commute with each other, so you don't need to include one of them.