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Definition of wigner D matrix

  1. Mar 30, 2012 #1
    1. The problem statement, all variables and given/known data

    I'm not sure if this is the appropriate board, but quantum mechanics people surely know about spherical harmonics. I need to implement the Wigner D matrix to do spherical harmonic rotations. I am looking at
    http://en.wikipedia.org/wiki/Wigner_D-matrix#Wigner_.28small.29_d-matrix
    for the definition, since I'm not a quantum mechanics guy, I need to verify whether this definition of the Wigner D matrix correspond to the rotation of the spherical harmonics itself or the coordinate system? In other words, if
    [tex]
    u(\theta,\phi)=\sum_ma_mY_l^m(\theta,\phi)
    [/tex]
    then does
    [tex]
    u(\theta', \phi')=\sum_{m'}b_{m'}Y_l^{m'}(\theta', \phi')
    [/tex]
    where [itex]b_{m'}=\sum_md^l_{m',m}(\beta)a_m[/itex], correspond to the same function u under the new coordinates rotated around y axis by β? [itex]d^l_{m',m}(\beta)[/itex] is the small d matrix element.
    Further more, I assume the spherical harmonics Ylm is the standard definition where Y_l^(-m)=(-1)^mY_l^m*? Thank you for your patience.

    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
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