Question

How does the Pauli Exclusion Principle lead to the restriction that electron with `n = 1` and `l = 0` can only take on (a linear combination of) two possible spins?

Answer 1

It has to do with the quantum numbers that belong to those electrons.

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• The principal quantum number `n` describes the energy level, or the "shell", that the electron is in.

`n = 1, 2, 3, . . .`

• The angular momentum quantum number `l` describes the energy sublevel, or "subshell", that the electron is in.

`l = 0, 1, 2, . . . , l_"max"`, `" "" "" "" "" "" "l_"max" = n-1`
`(0, 1, 2, 3, . . . ) harr (s, p, d, f, . . . )`

• The magnetic quantum number `m_l` describes the exact orbital the electron is in.

`m_l = {-l, -l+1, . . . , 0, 1, . . . , l-1, l}`

• The spin quantum number `m_s` describes the electron's spin.

`m_s = pm1/2` for electrons.

And thus all four collectively describe a given quantum state. Pauli's Exclusion Principle states that no two electrons can share the same two quantum states.

Since `n`, `l`, and `m_l` exactly specify the orbital, while giving the `m_s` fully specifies the quantum state, it follows that the `m_s`, i.e. the electron spin must differ, for a given electron in a single orbital.

Electrons can only be spin-up or down, `m_s = +1/2` or `-1/2`, so only two electrons can be in one orbital.

At `n = 1`, there only exists `l = 0` (as the maximum `l` is `n - 1`), and we specify the `1s` orbital.

With only one orbital on `n = 1`, there can only be two electrons within `n = 1`, the first "shell". If any more electrons were shoved in, they would all vanish.