quantum mechanics - Does the Pauli exclusion principle instantaneously affect distant electrons?

According to Brian Cox in his A night with the Stars lecture$^1$, the Pauli exclusion principle means that no electron in the universe can have the same energy state as any other electron in the universe, and that if he does something to change the energy state of one group of electrons (rubbing a diamond to heat it up in his demo) then that must cause other electrons somewhere in the universe to change their energy states as the states of the electrons in the diamond change.

But when does this change occur? Surely if the electrons are separated by a significant gap then the change cannot be instant because information can only travel at the speed of light. Wouldn't that mean that if you changed the energy state of one electron to be the same as another electron that was some distance away, then surely the two electrons would be in the same state until the information that one other electron is in the same state reaches the other electron.

Or can information be transferred instantly from one place to another? If it can, then doesn't that mean it's not bound by the same laws as the rest of the universe?

--

$^1$: The Youtube link keeps breaking, so here is a search on Youtube for Brian Cox' A Night with the Stars lecture.

Answer

The Pauli exclusion principle can be stated as "two electrons cannot occupy the same energy state", but this is really only a rough way of stating it. It's more precise to say that the wavefunction of a system is anti-symmetric with respect to exchange of two electrons. The trouble is that now I have to explain to a non-physicist what "anti-symmetric" means and that's hard without going into the maths. I'll have a go at doing this below.

Anyhow, Brian Cox is being a bit liberal with the truth because I'm not sure it makes sense to say the electrons in his bit of diamond and electrons in far away bits of the universe can be described by a single wavefunction. If this isn't a good description then the Pauli exclusion principle doesn't have any meaning for the system.

Suppose you have two electrons in an atom or some other small system. Then that system is described by some wavefunction $\Psi(e_1, e_2)$ where I've used $e_1$ and $e_2$ to denote the two electrons. The Pauli exclusion principle states:

$$\Psi(e_1, e_2) = -\Psi(e_2, e_1)$$

that is if you swap the two electrons $\Psi$ changes to $-\Psi$. But suppose the two electrons were exactly the same. In that case swapping the electrons cannot change $\Psi$ because they're identical. So we'd have:

$$\Psi(e_1, e_2) = \Psi(e_2, e_1)$$

but the exclusion principle states:

$$\Psi(e_1, e_2) = -\Psi(e_2, e_1)$$

therefore if both are true:

$$\Psi(e_2, e_1) = -\Psi(e_2, e_1)$$ ie $$\Psi = -\Psi$$

The only way you can have $\Psi = -\Psi$ is if $\Psi$ is zero, which means $\Psi$ doesn't exist. This is why if the Pauli exclusion is true, two electrons can't be identical i.e. they can't be in the same energy state.

But this only applies because I could write down a wavefunction $\Psi$ to describe the system. When systems become large, e.g. two footballs in a swimming pool instead of two electrons in an atom, it isn't useful to try and write a wavefunction to describe the system and the exclusion principle doesn't apply. NB this doesn't mean the exclusion principle is wrong, it just means it doesn't apply to that system.