How Can the Creation Operator L+ Be Applied to Y_l^m in Quantum Mechanics?

[rubertoda]

I have: $$Y_l^m= Ne^{im\varphi}P_l^m(cos\theta)$$ where

$$P_l^m(cos\theta)$$ is the associated legendre polynomials: $$P_l^m(cos\theta)=(-1)^m(sin\theta)^m(\frac{d^m}{d (cos\theta)^m})$$

The problem is that i want to use this expression to apply on it the creation operator for the orbital angular momentum operator i. e to make this function with m = -l; $$Y_{-l}^l \rightarrow Y_{-l+1}^l$$

When i attempt this, i get a very complicated set of derivatives etc, because i haven't specified the "l".
Now, my question is: can i prove this for the general case or do i have to use a specific case, for example l = 1?

The creation operator is: $$L_+ = L_x + iL_y$$

thanks!

 

[vela]

You can probably show it for the general case by using recurrence relations the associated Legendre polynomials satisfy.

 

[rubertoda]

Thank you very much vela. but i think i should use the expression for the ladder operator?or do u have any idea how to start with the recurrance?