Is the Wigner D matrix definition applicable to spherical harmonic rotations?

[sunjin09]

Homework Statement

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
$$u(\theta,\phi)=\sum_ma_mY_l^m(\theta,\phi)$$
then does
$$u(\theta', \phi')=\sum_{m'}b_{m'}Y_l^{m'}(\theta', \phi')$$
where $b_{m'}=\sum_md^l_{m',m}(\beta)a_m$, correspond to the same function u under the new coordinates rotated around y-axis by β? $d^l_{m',m}(\beta)$ 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.

Homework Equations

The Attempt at a Solution

 

[member 553083]

I think the answer to your question is yes; the Wigner D matrix corresponds to the rotation of the spherical harmonics itself. The definition of the Wigner D matrix in the Wikipedia article you linked to states that it is "a matrix representation of the rotation group SO(3) in the space of angular momenta." Angular momentum is related to the spherical harmonics, so it makes sense that the Wigner D matrix would correspond to a rotation of the spherical harmonics. The spherical harmonics are defined such that Y_l^(-m)=(-1)^mY_l^m*. So, if b_{m'} is computed using d^l_{m',m}(\beta) and a_m, then b_{m'} should represent the same function u under the new coordinates rotated around the y-axis by β.