Spherical Harmonics/Angular Momentum

[atomicpedals]

Homework Statement

Given that L_z(x+iy)^m=m\hbar(x+iy)^m. Show that L₊=(x+iy)^m.

2. The attempt at a solution

I'm probably grasping at straws here, but when I see the expression for L_z I instantly go to L_z|lm>=m\hbar|lm>. This then leads me to suspect that |lm>=(x+iy)^m. Is this correct, and how on Earth does it get me any closer to what I want to show for the raising operator?

 

[vela]

Not exactly. What \hat{L}_z(x+iy)^m=\hbar m(x+iy)^m tells you is that (x+iy)^m \propto \langle x,y | l~m \rangle. You need to keep the distinction between kets and the representation of the ket in some basis. Writing (x+iy)^m = |l~m\rangle is, at best, a terrible abuse of notation.

I have no idea what \hat{L}_+ = (x+iy)^m is supposed to mean.