Rotation

Angular momentum is something that took me a while to appreciate. But as one has to deal with it a lot (orbitals, rotating molecules, spectroscopy terms) it makes sense to take a closer look at it.

I think the problem is if one thinks too quantum mechanically. And then it's difficult to understand what's happening. Angular momentum can be represented by a vector that points in the axis of rotation, its length corresponds to the absolute value of the momentum. We can write:



Because of the uncertainty principle we cannot know all the components. In a spherical system, except of having all three bits of information, we have two: L² and L_z. This can be derived from the fact that the corresponding operators commute and must have a set of identical eigenfunctions. We cannot exactly determine L_x and L_y because their operators do not commute with L_z. In other words: We know how fast our particle rotates and we have partial information about the direction.

Actually quantum mechanics imposes a second restriction (but still: keep thinking about a rotating particle). L² and L_z are quantised, hence "quantum mechanics". We have



and



One thing you notice is that:



In other words there will always remain something for L_x and L_y and the total momentum cannot be clearly determined. But if you move to classical mechanics (large l), the difference will vanish.

In QM strong motion means a large derivative which in turn means nodal planes (if you consider real functions [1]). This works here, too: l is the number of nodal planes the wave function has on a given sphere. m is the number of nodal planes you pass when you move around the z-axis.

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[1] Complex functions can oscillate without nodal planes. e^ix always has the absolute value of 1 but a non-zero derivative. It's a rotation in the complex plane.