# Spherical harmonics

1. Dec 11, 2007

### plmokn2

1. The problem statement, all variables and given/known data
Why is is the for physical applications of the spherical harmonics |m| must be less than or equal to l, with both being integers?

2. Relevant equations
Y(m,l)=exp(im phi)P{m,l}(cos theta)
Hopefully my notation is clear, if not please say.

3. The attempt at a solution
Well m must be integer so that the the exponential is single valued as we go through 2pi on the phi axis, and l must also be integer so that the associated Legendre equation is well behaved at the poles, but I'm not sure why we require |m| is less than or equal to l? I suppose it must be something to do with the behaviour of the P part of the solution but I can't see what.

Any hints/ help appreciated,
Thanks

2. Dec 12, 2007

### Avodyne

Well, it's possible to prove just from the commutation relations of the components of the angular momentum operator that m (the eigenvalue of Lz/hbar) must be less than or equal to l (where l(l+1) is the eigenvalue of L^2/hbar^2). Classically, Lz must be less than or equal to L^2, so this is sensible.

Edit: I meant to say that, classically, Lz^2 (not Lz) must be less than or equal to L^2.

Last edited: Dec 12, 2007
3. Dec 12, 2007

Thanks