[ajross]

It's actually more confusing IMHO, because these graphs overload the radial dimension to show probability as "distance from the origin". You have to multiply that by the radial function to get an actual probability distribution, which kinda/sorta looks like these pictures but not really.

Really the harmonics are best understood as something like "wave height on the surface of a sphere". They tell you how the electrons (or whatever) are going to distribute themselves radially, not where they're going in 3D space.

Also FWIW: the much harder thing to grok here (at least it was for me), and that no one tries to tackle, is why the "l" number corresponds directly to angular momentum. In particular "l==0" doesn't look like there's any rotation going on at all.

[momoschili]

Simply speaking, "l" describes the number of nodes. In the same sense that a particle in a box with sin(nx) wave function has more nodes the higher energy (or momentum) state it is in.

As for why l==0 has no rotation going on at all, one would say that this should be expected. Qualitatively, the symmetric sphere does not change with rotation, so how would we tell if it is rotating or not? And perhaps the next step is controversial, but if there is no way to tell, maybe there is no dependence? This is a similar argument to why the electric field of an infinite plane is constant with respect to distance from it.