Sean Carroll

That's not the only thing that I could measure.

For example, even if I forget about the fact that electrons have spin, I could measure the velocity or the momentum equivalently of the electron.

So naively, or at least taking my words overly literally, you might think, okay, so the wave function is a superposition of every possible measurement outcome, so that would include both positions and momenta of the electrons, okay?

That's not right.

I think actually if you carefully parse the words I've said so far, it's compatible with that, but I didn't make it explicit, so let's make it explicit right now.

If you give me, as it turns out, as Schrodinger himself, you know, noted, if you give me the wave function as a function of just position, which makes sense if you were thinking about it incorrectly as a field, this is why people get mixed up right from the start.

So you think of psi of x, okay, the wave function as a function of the possible positions you could see the electron in, then you're done.

You don't separately give me psi of x and p, x being the letter we use to denote positions and p being the letter we use to denote momenta.

You don't need to do that.

You can actually calculate the probability of getting a momentum measurement outcome from psi of x, from just you give me the wave function as a function of position measurement outcomes.

you can figure out what the probability is of different momentum measurements.

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.