Pauli exclusion principle explained

[Silviu]

Hello! I am a bit confused about the Pauli exclusion principle. Let's say I have 3 electrons. Due to energy considerations the first 2 go to the ground state, and they can be only 2 electrons there, because the position wavefunction has only one option $\psi_{100}$ (and again due to energy configurations it chooses the symmetrical state) and the spin picks the anti-symmetrical so that it respect the fermions statistics overall. Hence, as there is no free state here, the 3rd electron goes to the next energy level $\psi_{200}$. What confuses me is, how does the 3rd electron know the spins of the first 2? The 2 electrons in the ground state, don't need to have a precise value of the z component of the spin, it can be a linear combination of up and down. Of course, upon measurement, if one of them turns out to be up, the other one is necessary down, but for a random atom, before any measurement the up and down positions of the spin (in the z direction) are not occupied yet, as they are not measured, so how is the 3rd electron prevented to go there? I.e. the state (1,0,0,up) and (1,0,0,down) are not taken, as the 2 electrons are a linear combinations of them.

 

[DrClaude]

The Aufbau principle, where the atom is built up by adding electrons to occupy orbitals of ever increasing energy is a nice heuristic, very helpful to figure out electronic configurations. But Nature doesn't proceed like that.

What you have to consider in the case you describe is that you have three electrons, and you have to figure out what combinations of orbitals are possible, such that the total wave function is anti-symmetric with respect to the exchange of two electrons (Pauli principle). You will find that the lowest energy result corresponds to two electrons in the lowest energy spatial wf, with opposite spin, and one electron in a higher-energy orbital, with arbitrary spin (many solutions are possible here).

Note that I didn't say "the first and second electron" and so on, because that would not result in a wf with a definite exchange parity. The results is one where each electron can be in any of the orbitals, a linear combination of each electron being in each state. You can look up Slater determinant to figure out what these wave functions can look like.

 

[PeterDonis]

how does the 3rd electron know the spins of the first 2?

Electrons don't have labels; you can't pick out individual electrons and say that one is the "first", one is the "second", one is the "third", and match up each label with an orbital. Electrons are indistinguishable; that means that all you can say about the state in your example is that there are two electrons in the lowest energy level and one electron in the next higher energy level. In other words, you don't have three one-electron states where one of the three has to "know" that it can't be in the lowest energy-level; you have one three-electron state that already "knows" all of the energy levels being occupied.