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I'd like to prefix this with an apology: I have no formal training in QP, and most of what I know has been obtained by reading Wikipedia. As such, it'd be really helpful if any answers took my lack of real knowledge into account!

I'm a bit confused about how the Pauli exclusion principle works in the case of electron shells where there are more than two electrons. My understanding of the Pauli exclusion principle is that two electrons cannot have the same quantum state at the same time.

On the Wikipedia page for the Pauli exclusion principle, it says:

For two electrons residing in the same orbital, $n$, $ℓ$, and $m_ℓ$ are the same, so $m_s$ must be different and the electrons have opposite spins.

This makes sense to me, because I know that in the lowest orbital there are a pair of electrons with opposite spins - positive and negative one half.

Then, on the Quantum number page, it says (emphasis mine):

An electron has spin $s = \frac12$, consequently $m_s$ will be $\pm\frac12$, corresponding with "spin" and "opposite spin." Each electron in any individual orbital must have different spins because of the Pauli exclusion principle, therefore an orbital never contains more than two electrons.

This is where I get confused. My understanding was that electrons were grouped into shells of total counts $\{2, 8, 18, 32\}$, based on cumulative subshell sets containing $\{2, 6, 10, 14\}$ electrons each. But if only two electrons can possibly be in the same shell, how is it possible to have a shell that has 6, 10, or even 14 electrons in it?

The four assertions I would make are:

  • The principal, azimuthal and magnetic numbers of electrons in a single state are always equal.
  • The only variable quantum number is spin projection.
  • Electrons can only be at spin $\pm\frac12$, so $m_s$ can only have one of two values.
  • Subshells can contain 2, 6, 10, or 14 electrons each.

One of these must be false, or I'm missing something. Where did I go wrong? How does the electron shell model reconcile with the Pauli exclusion principle?

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I think your confusion is cleared up quite simply: you're confusing the terms "orbital" and "electron shell". An orbital is characterized by the three quantum numbers $n,\ell,m_\ell$. This terminology makes sense, because these three numbers together completely determine the spatial component of the wave function.

However, this leaves a freedom in the spin component: $m_s=\pm\frac{1}{2}$. Therefore, it follows straightforwardly that every orbital (i.e. fixed spatial wave function) corresponds to at most $2$ electrons.

You are completely correct that the shells, which are labeled only by the principal quantum number $n$ and have a fixed energy (at least in a first approximation), can contain $2n^2$ electrons. If you read carefully, you'll see that no one ever claims that an electron shell can only contain $2$ electrons (unless it's the $n=1$ shell). A recent, related answer of mine can be found here.

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