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Raising momentum operator acting on spherical harmonics

  1. Dec 7, 2009 #1
    1. The problem statement, all variables and given/known data

    What is the result of raising momentum ladder operator (L+) acting on spherical harmonics Y04 ([tex]\theta[/tex],[tex]\phi[/tex])



    2. Relevant equations



    3. The attempt at a solution

    I was expecting Y14 ([tex]\theta[/tex],[tex]\phi[/tex])

    I applied L+ on Y04 ([tex]\theta[/tex],[tex]\phi[/tex]) and ended up with Y14 ([tex]\theta[/tex],[tex]\phi[/tex]) multiplied by [tex]\sqrt{20}[/tex]h-bar.


    Thanks in advance
     
  2. jcsd
  3. Dec 7, 2009 #2
    That is what you should get:

    [tex]
    L_+Y_\ell^m=\hbar\sqrt{(\ell-m)(\ell+m+1)}Y_\ell^{m+1}
    [/tex]

    with [itex]\ell=4[/itex] and [itex]m=1[/itex], you get

    [tex]
    L_+Y_4^0(\theta,\phi)=\hbar\sqrt{20}Y_4^1(\theta,\phi)
    [/tex]
     
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