Fixed quantum angular momentum in a given direction.

Can range between $±l$.

E.g. consider gallium which is 1s2 2s2 2p6 3s2 3p6 4s2 3d10 4p1:

- the electrons in s-orbitals such as 1s, 2d, and 3d are $l=0$, and so the only value for $m_{l}$ is 0
- the electrons in p-orbitals such as 2p, 3p and 4p are $l=1$, and so the possible values for $m_{l}$ are -1, 0 and 1
- the electrons in d-orbitals such as 2d are $l=2$, and so the possible values for $m_{l}$ are -2, -1, 0 and 1 and 2

The z component of the quantum angular momentum is simply:
so e.g. again for gallium:

$L_{z}=m_{l}ℏ$

- s-orbitals: necessarily have 0 z angular momentum
- p-orbitals: have either 0, $−ℏ$ or $+ℏ$ z angular momentum

Note that this direction is arbitrary, since for a fixed azimuthal quantum number (and therefore fixed total angular momentum), we can only know one direction for sure. $z$ is normally used by convention.

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