Does operator L^2 commute with spherical harmonics?

[Feelingfine]

Homework Statement

Does operator L^2 commute with spherical harmonics?

Relevant Equations

[L^2 , Y_lm] = ?
Y_lm are the spherical harmonics.

My teacher said me this commutator is zero because the spherical harmonics are eigenfunctions of L^2. Actually, he said that any operator must commute with its eigenfunctions.

I tried an example: [L^2,Y_20] expressing L^2 on spherical coordinates and I determined this commutator is not zero.

[L^2, Y_20]F = L^2(Y_20F) - Y_20(L^2F) = Y_20(L^2F) + F(L^2Y_20) - Y_20(L^2F) = F(L^2Y_20) , so [L^2, Y_20] = (L^2Y_20) what it's not equal to zero.

I think what he said it's wrong, actually I think it's almost obvious. I don´'t see any reason an operator commutes whit its eigenfunctions (acting like operators).

Can anybody help me with this problem?

 

[PeroK]

Homework Statement:: Does operator L^2 commute with spherical harmonics?
Relevant Equations:: [L^2 , Y_lm] = ?
Y_lm are the spherical harmonics.

My teacher said me this commutator is zero because the spherical harmonics are eigenfunctions of L^2. Actually, he said that any operator must commute with its eigenfunctions.

I tried an example: [L^2,Y_20] expressing L^2 on spherical coordinates and I determined this commutator is not zero.

[L^2, Y_20]F = L^2(Y_20F) - Y_20(L^2F) = Y_20(L^2F) + F(L^2Y_20) - Y_20(L^2F) = F(L^2Y_20) , so [L^2, Y_20] = (L^2Y_20) what it's not equal to zero.

I think what he said it's wrong, actually I think it's almost obvious. I don´'t see any reason an operator commutes whit its eigenfunctions (acting like operators).

Can anybody help me with this problem?

It's obviously nonsense. A better counterexample is to look at $L_z$, which is effectively $\frac{\partial }{\partial \phi}$. And take $Y_1^1 = \sin \theta e^{i\phi}$.

We can ignore the $\sin \theta$. And it's easy to see that $[L_z, e^{i\phi}I] \ne 0$.

 

[troglodyte]

To proof whether two operators A,B commute you have to check [A,B]=0.
so,what is your mistake in this case?
If you can't see the mistake you have to check the algebraic rules of the commutator once more .

troglodyte