Verify by brute force that the three functions cos(θ), sin(θ)eiφ and sin(θ)e−iφ are all eigenfunctions of L2 and Lz.
I know that Lz = -iћ(∂/∂φ)
I also know that an eigenfunction of an operator if, when the operator acts, it leaves the function unchanged apart from a multiplicative factor (the eigenvalue)
The Attempt at a Solution
So, Lzcos(θ) = -iћ(∂/∂φ)cos(θ)
But I think that equals zero. There's no component of cos(θ) in the φ direction. There different variables. So I think that when the differential operator acts on it, it makes 0. But then the function isn't left unchanged. Can somebody help me resolve this in my head?