Relationship between angular momentum and spherical harmonics

[atomicpedals]

I'm having a hard time grasping the logical flow from orbital angular momentum to spherical harmonics. It feels like it's just sort of been sprung out of nowhere from both my lecture notes and the textbook. Can anyone help fill in the gaps that clearly must link them somehow?

How did I get from eigenvectors of L² and L_z being functions that depend only on theta and psi to the spherical harmonics $$|lm>=Y^{m}_{l}(\theta,\phi)$$?

 

[atomicpedals]

Digging around the internet has netted me that spherical harmonics are the normalized common eigenfunction of L² and L_z.

 

[dextercioby]

I beg your pardon ? Using the abstract Dirac formalism will get you the |lm> things. But don't forget that

$$Y^{m}_{l}(\theta,\phi) = \langle r,\theta,\phi|lm\rangle$$

That is going from an abstract Hilbert space H to L²(R³,d³x) and then to spherical coordinates.

 

[atomicpedals]

How did I get from orbital angular momentum to spherical harmonics in the first place?

 

[dextercioby]

The spherical harmonics are eigefunctions for the squared orbital angular momentum operator in the position representation.

 

[atomicpedals]

Got it, thanks!