Orbital Angular Momentum and the Quantum Numbers
Angular momentum is quantised in both size and direction, so only certain projections along a chosen axis are allowed. The shapes labelled s, p and d are the angular solutions drawn out.
What a learner can do afterwards
- States the allowed values of orbital angular momentum and its projection for a given level
- Connects the angular quantum numbers to the shapes of the orbitals
- Relates the orbital magnetic moment to the splitting of lines in a magnetic field
1 · Read
Solving hydrogen quantises the electron circulation. The orbital number l fixes the magnitude of orbital angular momentum at the square root of l times l plus 1, times h bar. Only whole values l equals 0, 1, 2 and so on occur, which is why both the size of circulation and its allowed tilts come in steps.
The magnetic number m fixes the projection on one chosen axis at m times h bar, with m running from minus l to plus l in whole steps. That gives 2l plus 1 tilts for each l. For n equals three, l is zero, one, or two, so the largest projection is two h bar and l equals two holds five tilts.
The angular shapes labelled s, p, and d are these rules drawn out: s is round for l equals 0, p is a dumbbell for l equals 1, d is cloverlike for l equals 2. A circling electron is a tiny current loop, so its magnetic moment stays locked to its angular momentum: bigger circulation means a stronger magnet.
Because the moment is quantised, a magnetic field shifts each sublevel by m times one fixed step, so a level splits into 2l plus 1 sublevels. Zeeman splitting proves the tilts are real. Ignoring spin, a jump changes m by at most one, so each line splits into three, whatever l is.
Size and tilt both come in steps, and magnets reveal the steps as split lines.
Rewritten readings were written for this dataset.
2 · Watch
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Where it sits
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.