Y(l,m) = (-1)^m sqrt((2 l + 1) / (4 pi) (l - m)! / (l + m)!) * P(l,m) exp(i m phi) -- associated Legendre of cos theta (arxiv.org/abs/1805.12125) P(l,m,k) = test(m < 0, (-1)^m (l + m)! / (l - m)! P(l,-m), (sin(theta)/2)^m sum(k, 0, l - m, (-1)^k (l + m + k)! / (l - m - k)! / (m + k)! / k! * ((1 - cos(theta)) / 2)^k)) Lz(f) = -i hbar d(f,phi) -- Y is (trivially) an eigenfunction of Lz -- Lz Y(l,m) = m hbar Y(l,m) Lz(Y(2,2)) == 2 hbar Y(2,2) Lz(Y(2,1)) == hbar Y(2,1) Lz(Y(2,0)) == 0 Lz(Y(2,-1)) == -hbar Y(2,-1) Lz(Y(2,-2)) == -2 hbar Y(2,-2) Run