Angular momentum: definition and commutation

1. Angular momentum: definition and commutation • classically, this is a central concept (no pun intended)! • if the (conservative) force is centralL is conserved • the quantum mechanical implications are profound indeed • if the situation is that one body ‘orbits’ around another one, do the usual reduction to the equivalent ‘one-body’ problem, with m the reduced mass replacing m in all formulae • study of the commutation relations is most revealing • all coordinates commute, and all momentum components commute

2. Now for some non-zero commmutators

3. The key non-zero commmutators • most interesting of all is angular momentum with itself • one can simply move around operators that commute, preserving the order otherwise

4. Expressing components of L in sphericals I • this is in a succinct and ‘hybrid’ notation

5. Expressing components of L in sphericals II • for z, things proceed a bit differently and it is a lot simpler! • we can now construct the angular momentum operators

6. Expressing components of L in sphericals III, and an astounding eigenresult • Sweet!! The F functions are eigenfunctions of the Lz operator, with eigenvalue mħ!! • therefore, the spherical harmonics are eigenfunctions as well with that same eigenvalue

7. Working out the square of L: another astounding eigenresult • compare this with Legendre’s equation: exactly the same thing • evidently, one can simultaneously determine both the magnitude of the angular momentum, and its z component • claim: [L2, Lz] = 0 proof: you do it on paper! • claim: [L2, L] = 0 L2 commutes with any component of L • we’ve already established that the components don’t commute

8. Two new angular momentum operators, built from Lx and Ly, and a bizarre identity • interesting commutation relations • since [L2, L] = 0  [L2, L± ] • a fantastic and bizarre operator identity:

9. Learning about raising and lowering • now to ask: suppose some function Y is an eigenfunction of both L2 (with eigenvalue l) & Lz (with eigenvalue m) • what is the effect on Y of L± ? first, test it with L2: • second, test it with Lz : • we see how the raising and lowering takes place: l is unchanged, while m is raised or lowered by one unit of angular momentum • there is therefore a ‘ladder’ of states for a given l • but once the z component of Lzgets as big as (or nearly as big as) L itself the process must stop: there must be a’top’ state Ytop

10. Trying to raise the top, or lower the floor • we therefore have • now lower the states one by one with the lowering operator • same l each time, but m is knocked down by ħ • not yet clear what the multiplicative factor might be… • finally we arrive at the ‘unlowerable’ bottom state Ybot

11. Sorting our the relationship between top and bot • two m must differ by some integer nћ • in principle n may be odd or even • it may be shown (Griffiths problem 4.18) • the spherical harmonics are not eigenfunctions of the raising and lowering operators