QM: Is this a Spherical Harmonic Identity?

  • #1
1,789
4

Main Question or Discussion Point

Hi,

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

[tex]
\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}
[/tex]

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

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

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

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

Thanks in advance.
 
Last edited:

Answers and Replies

  • #2
tom.stoer
Science Advisor
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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
 
Last edited:
  • #3
1,789
4
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.
 
  • #4
tom.stoer
Science Advisor
5,766
161
your welcome (and sorry for the terrible typos)
 

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