# Definition of wigner D matrix

1. Mar 30, 2012

### sunjin09

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
$$u(\theta,\phi)=\sum_ma_mY_l^m(\theta,\phi)$$
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.

2. Relevant equations

3. The attempt at a solution