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In describing electron states in hydrogen, we have a very "simple" picture, at least in intro-quantum. But this only has one electron! As we permit more electrons, we also have things like the exclusion principle, which comes from the spin-statistics theorem, but that relies on the electrons having some kind of coupled wave functions, right? Why do electrons always come in pairs?

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A potential well in the Schrodinger equation will produce energy levels similar to the energy levels of the hydrogen atom

A nucleus with two protons to be neutral will need two electrons. These will be accommodated in the lowest two energy levels: because of the spin statistics of electrons they cannot occupy the exact same energy level. A nucleus with three protons (Z=3) will need three electrons occupying the lowest three energy levels etc. The electrons may thermally or by photon interactions be kicked up to higher energy levels giving the characteristic spectral lines of the specific atom when returning to the ground state.

Electrons do not come in pairs. The fill up the levels so that the atoms are neutral.

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