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?

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

• 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.