Sean Carroll

And the answer is, what happens when we have two electrons?

Because your guess would be there'd be a wave function, a field for electron number one, and a different field or a different wave function for electron number two, but that is not what quantum mechanics says.

It says that there is a single wave function for the combination of the two electrons.

And if you're worried about subtleties, about electrons being indistinguishable, just think of an electron and a proton, or an electron and a positron, whatever, two different particles, two distinguishable particles.

So what does it mean to say that there is only one wave function for two particles?

Well, if you think of the wave function not as a field living in space, but rather as the superposition of every possible measurement outcome, then when you have two electrons, these ideas, these concepts become distinguishable, right?

Because what are the possible measurement outcomes of a system of two particles?

It's every possible location of particle 1 and every possible location of particle 2 considered separately.

So for those of you who are a little bit mathematically inclined, the correct Greek letter to attach to the wave function is psi, P-S-I.

And what we're saying here is instead of psi of electron 1 and psi for electron 2, there's only one psi.

There's only one wave function.

It's the wave function of the universe, ultimately.

And it's a function of the position of electron one and the position of electron two.

Okay?

So that's very different, radically different than classical mechanics where you would just have whatever particle one is doing and whatever particle two is doing.

quantum mechanics is telling you that this superposition of both electrons at once is the state of the universe.

And the reason why, or the state of the two-electron system anyway, the reason why that implies entanglement is because there might be relationships between the probability of seeing one electron in one location and another electron in another location.

Like, let's actually make our lives easy and switch to an electron and a proton.

So they're two obviously different particles, okay?

Maybe the wave function of your two-particle system could be the electron is probably here or there.