Raising momentum operator acting on spherical harmonics

[FourierX]

Homework Statement

What is the result of raising momentum ladder operator (L₊) acting on spherical harmonics Y⁰₄ ($$\theta$$,$$\phi$$)

Homework Equations

The Attempt at a Solution

I was expecting Y¹₄ ($$\theta$$,$$\phi$$)

I applied L₊ on Y⁰₄ ($$\theta$$,$$\phi$$) and ended up with Y¹₄ ($$\theta$$,$$\phi$$) multiplied by $$\sqrt{20}$$h-bar.


Thanks in advance

 

[jdwood983]

That is what you should get:

$$L_+Y_\ell^m=\hbar\sqrt{(\ell-m)(\ell+m+1)}Y_\ell^{m+1}$$

with $\ell=4$ and $m=1$, you get

$$L_+Y_4^0(\theta,\phi)=\hbar\sqrt{20}Y_4^1(\theta,\phi)$$