# rotation of spherical harmonics using Wigner D-matrix

A very stupid question as I am very confused: I have a surface charge density which is a function of spherical harmonics $$\sigma_{l,m}=Y_{lm}$$ (only the real part). Now I need to rotate the particle, or rotate the surface charge, using the Euler angles. How can I properly rotate spherical harmonics using Wigner D-matrix in Mathematica?

As stated here in Wikipedia https://en.wikipedia.org/wiki/Wigner_D-matrix#Relation_to_spherical_harmonics_and_Legendre_polynomials

should I set the second index (here $$m_0$$, where we have summation over it) to zero to make the Wigner matrix proportional to the spherical harmonics? something like this:

σ0 = (Sqrt[(4 π)/(2 l + 1)]  )^-1
σlm[l_,m_] := σ0  Sum[WignerD[{l, -m0, -m}, α1, β1, γ1] , {m0, 0, 0}]


or should I keep it general:

σ0 = (Sqrt[(4 π)/(2 l + 1)]  )^-1
σlm[l_,m_] := σ0  Sum[WignerD[{l, -m0, -m}, α1, β1, γ1] , {m0, -l, l}]


Suppose I would like to rotate $$\sigma_{l,m}=Y_{21}$$, which one is it the correct way of rotation?

• For reasons unclear to me, WignerD[] is using a different convention for the Euler angles. In any event, corresponding to the notation in Wiki, $D_{l,m}^j(\alpha,\beta,\gamma)$ is WignerD[{j, l, m}, -α, -β, -γ], and $d_{l,m}^j(\beta)$ is WignerD[{j, l, m}, -β]. This gives the relation between SphericalHarmonicY[] and WignerD[]. Jun 7, 2020 at 2:05
• @J.M.'stechnicaldifficulties, Thank you so much for your reply. I think your comment have some "technical difficulties" ;) Are there any differences, in Mathematica, between making the $m$ values in Wigner matrix negative or making the Euler angles negative? Do you mean for rotating the spherical harmonics using Mathematica, I always should set the $m_0=0$ in Wigner matrix? I think I can understand the process on paper, but Mathematica is different. Jun 7, 2020 at 6:33
• * I mean making $m$ and $l$ values negative and in second part, $m_0=l=0$ Jun 7, 2020 at 6:43