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.