Can (quantum) angular momentum $L$ be zero? I am trying to calculate the orbital magnetic moment, $\bar{\mu}$, for Sodium, which has an electron configuration of $1s^2 2s^2 2p^6 3s^1$. The full shells do not contribute to $\bar{L}$ and $\bar{S}$ so only the outer shell electron will contribute to them.
The $3s$ shell corresponds to $l=0$. And $\bar{L}=\hbar\sqrt{l(l+1)}$ which gives $\bar{L}=0$.
Is this possible? Doesn't that mean that the electron isn't "orbiting" the nucleus? I am thinking that $\bar{L}^2$ tells you if the electron is "orbiting" the nuclues, but then what is the significance of $\bar{L}$?
Extra question:
The z-component of the orbital magnetic moment is given by $\mu_z = -m_l\mu_b$. Since $m_l$ varies as $-l\le m_l\le l$ does that mean that there is more than one value for the z-component? Strange, no?
 A: Yes, the quantum angular momentum can be zero.
But it seems you were trapped by your intuition from classical mechanics.
In classical mechanics an orbit with zero angular momentum
means an orbit with a purely radial motion, i.e. kind of a degenerate
ellipse with eccentricity $=1$.
To show what I mean, here are elliptical orbits with eccentricity
$0.99$ and $1$. (The nucleus is marked as a red dot)

But in quantum mechanics an orbital with $l=0$ (also called an $s$-orbital)
is somewhat different. It is a wave function with spherical symmetry.

(image from LibreTexts Chemistry - 6.6: 3D Representation of orbitals)
To get a crude intuition, you may think of the $s$-orbital as a
superposition of the classical degenerate ellipses from above,
averaged over all possible radial directions.
A: I read your question as: how can an electron orbital have zero angular momentum?
For a classical orbit, zero AM means that the motion is strictly radial. If any motion can be associated with an s-orbital it should be strictly radial motion.
A: Zero value to angular momentum doesn't mean that the electron is not revolving around the nucleus. It means that the motion of the electron is such that there is no preferred direction in space. For example, in a spherically symmetric orbital and that's exactly what s orbital is. Since s orbital is spherically symmetric there is no particular direction in space, which the angular momentum vector is pointing to and thus its average value is 0.
