Direct experimental observation of magnetic orbital quantum number $m_l$ Is there an experimental way to observe magnetic quantum number $m_l$ values directly, the way electron spin was detected by Stern Gerlach experiment or proton's spin by nuclear magnetic resonance experiments? The Zeeman effect comes to mind, but in the Zeeman effect one cannot ignore the electron spin. In short, how can one experimentally or spectroscopically see that if $l=2$,, $m_l$ will be -2, -1, 0, +1, and +2?
 A: A hint to the literature of fine-structure spectroscopists: an $ℓ=2$ state is sometimes known as a “quintet” because of its splitting into five sublevels. See also singlet, doublet, triplet, etc.
You suggest in a comment that we consider the famous sodium doublet, which is visible without any magnetic field. That doublet is a spin-orbit effect: the $3p$ first excited state in sodium can have $j=1/2$ or $j=3/2$, depending on the orientation of the electron spin relative to its orbit.

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In a magnetic field, the energy degeneracy among the various $m_j$ is broken, and the two lines in the sodium doublet split in ways that reveal their multiplicity.  This diagram suggests that the $\frac12 \to \frac12$ transition subdivides into two doublets.
The $\frac32\to\frac12$ transition divides into six pieces, rather than eight, because the transition from $m_j=-3/2$ to $m_j=+1/2$ would require the photon to carry away at least two units of angular momentum.
(It’s not immediately obvious to me why the six transitions in the $\frac32\to\frac12$ transition should be equally spaced, as sketched, but I’m prepared to believe they are.)
Magnetic splitting was actually discovered by Zeeman in this sodium transition, but this transition is an example of the “anomalous Zeeman effect.” One description.
