Pauli Exclusion Principle

Wolfgang Pauli's exclusion principle places a hard limit on how many electrons can share a single orbital: no two electrons in one atom can have the same set of values for all four quantum numbers. Put differently, every single electron in an atom must have its own unique combination of $n$, $l$, $m$ and $s$ - no two electrons are ever allowed to be identical in all four respects at once.

Hydrogen and helium as the simplest cases. Hydrogen's single electron has quantum numbers $n=1,\ l=0,\ m=0,\ s=+\tfrac{1}{2}$. Helium has two electrons occupying the same 1s orbital; the first shares hydrogen's exact values $n=1,\ l=0,\ m=0,\ s=+\tfrac{1}{2}$, but the second must differ in at least one quantum number to satisfy Pauli's rule - and since $n$, $l$ and $m$ are already fixed by "being in the 1s orbital," the only quantum number left free to differ is spin: the second electron has $n=1,\ l=0,\ m=0,\ s=-\tfrac{1}{2}$. …