Expression for Yl,-l again

  • Thread starter rubertoda
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  • #1
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I have: [tex]Y_l^m= Ne^{im\varphi}P_l^m(cos\theta)[/tex] where

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

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; [tex]Y_{-l}^l \rightarrow Y_{-l+1}^l[/tex]

When i attempt this, i get a very complicated set of derivatives etc, because i havent 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: [tex] L_+ = L_x + iL_y [/tex]

thanks!

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  • #2
vela
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You can probably show it for the general case by using recurrence relations the associated Legendre polynomials satisfy.
 
  • #3
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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?
 

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