# 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?

Aug 27, 2017

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.

$l = 0 , 1 , 2 , . . . , {l}_{\text{max}}$, $\text{ "" "" "" "" "" "l_"max} = n - 1$
$\left(0 , 1 , 2 , 3 , . . .\right) \leftrightarrow \left(s , p , d , f , . . .\right)$

• The magnetic quantum number ${m}_{l}$ describes the exact orbital the electron is in.

${m}_{l} = \left\{- l , - l + 1 , . . . , 0 , 1 , . . . , l - 1 , l\right\}$

• The spin quantum number ${m}_{s}$ describes the electron's spin.

${m}_{s} = \pm \frac{1}{2}$ 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, ${m}_{s} = + \frac{1}{2}$ or $- \frac{1}{2}$, 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 $1 s$ 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.