QM: Is this a Spherical Harmonic Identity?

[maverick280857]

Hi,

I came across the following expression in Landau and Lifgarbagez's Quantum Mechanics (Non-relativistic Theory) book:

$$\left(\cos\theta\frac{\partial}{\partial r} - \frac{\sin\theta}{r}\frac{\partial}{\partial\theta}\right)R_{nl}(r)Y_{l0}(\theta,\phi) = -\frac{i(l+1)}{\sqrt{4(l+1)^2-1}}\left[R_{nl}'-\frac{l}{r}R_{nl}\right]Y_{l-1,0} + \frac{i l}{\sqrt{4 l^2-1}}\left[R_{nl}' + \frac{l+1}{r}R_{nl}\right]Y_{l-1,0}$$

I don't see how this can be proved...is it using some cool recurrence relation or something, because I don't get it if I write

$$\frac{\partial}{\partial \theta} = \frac{1}{2\hbar}(e^{-i\phi} \hat{L}_{+} - e^{i\phi}\hat{L}_{-})$$

Any $\partial/\partial\theta$ can only change the m value, but not the l value, because it involves $\hat{L}_{\pm}$.

How does the l value change, and the m value remain constant, as an outcome of the particular differential operator $\partial/\partial z$?

Thanks in advance.

 

[tom.stoer]

w/o going into details: I guess one must express the cosine in terms of spherical harmonics and use a formula involving the products of spherical harmonics

 

[maverick280857]

w/o going into details: I guess one must express the cosine in terms of spherical harmonics and use a formula invoving the products oof spherical harmincs

Ahhh...yes, of course, thanks Tom! This never occurred to me.

 

[tom.stoer]

your welcome (and sorry for the terrible typos)