# What is the maximum number of electrons that can occupy an orbital if they have different spins?

Jun 12, 2017

This follows directly from the Pauli Exclusion Principle, which states that no two electrons can share the same quantum state. This means that no two electrons can have the same quantum numbers $n$, $l$, ${m}_{l}$, and ${m}_{s}$.

• $n$ is the principal quantum number, usually indicating what atomic energy level we are in.
• $l$ is the angular momentum quantum number, corresponding to the shape of the orbital. It defines the orbital subshell.
• ${m}_{l}$ is the magnetic quantum number, corresponding to each particular orbital in a given subshell defined by $l$.
• ${m}_{s} = \pm \frac{1}{2}$ is the electron spin quantum number, where $+$ indicates spin-up (and thus of course, $-$ indicates spin-down).

Knowing what $n$ you have fixes you to ${n}^{2}$ orbitals to choose from.

Knowing what $l$ you have fixes you to a particular subshell.

And knowing what ${m}_{l}$ you have fixes you to a particular orbital.

A particular orbital therefore has a single $n$, $l$, and ${m}_{l}$ associated with it. As a result, any electron in a specific orbital already shares three of four quantum numbers.

It follows that when electrons have two different spins in a particular orbital (and there exist only two spins for the electron!), these two electrons have occupied all the allowed quantum states within that orbital.