Why is the azimuthal quantum number so named?

The name "azimuthal quantum number" is often used for the total orbital angular momentum quantum number $\ell$ in an atom.

What is the origin of this name? It makes no sense to me, since the usual meaning of "azimuthal" is apparently "of or pertaining to the azimuth; in a horizontal circle", but $\ell$ has no information about orientation.

2 Answers

I believe this is what happened. Sommerfeld (1915, p. 430) $=$ (1916a, p. 12) originally introduced an “azimutal” quantum condition and number $n$ such that $\smash{\oint p_\varphi\,d\varphi=nh}$. This $n$ is what we now call the magnetic quantum number $m$.

Then in (1916b) he implicitly switched his azimuth from being $\alpha$ to $\gamma$ in this figure, where the tilted plane would be his “Bahnebene”. As he writes, the resulting “azimutal quantum $n$ splits into quanta $n_1$ and $n_2$ belonging to the coordinates $\varphi$ and $\vartheta$”: $\smash{\oint p_\varphi\,d\varphi=n_1h}$ and $\smash{\oint p_\vartheta\,d\vartheta=n_2h}$. That new $n=n_1+n_2$ is what we now call $\ell$. It measures angular momentum in the direction normal to a putative “orbit’s plane” rather than in a fixed vertical direction.

When the Schrödinger equation is solved in spherical coordinates by separation of variables it is split into three equations whose spectra produce the first three quantum numbers. One of them, ℓ, corresponds to the selection of spherical harmonics and determines the number of planar nodes going through the nucleus (planar node is the midpoint between crest and trough with zero magnitude).

The name "azimuthal quantum number" for ℓ was originally introduced by Sommerfeld, who refined Bohr's semi-classical model by replacing circular orbits with elliptic ones. The spherical orbitals were similar (in the lowest-energy state) to a rope oscillating in a large "horizontal" circle.