"Addition of angular momentum 2" Lx(f) = i hbar (sin(phi) d(f,theta) + cos(phi) cos(theta) / sin(theta) d(f,phi)) Ly(f) = i hbar (-cos(phi) d(f,theta) + sin(phi) cos(theta) / sin(theta) d(f,phi)) Lz(f) = -i hbar d(f,phi) L(f) = (Lx(f),Ly(f),Lz(f)) L2(f) = -hbar^2 (d(f,theta,theta) + cos(theta) / sin(theta) d(f,theta) + d(f,phi,phi) / sin(theta)^2) Psi = psi(theta,phi) check(Lx(Lx(Psi)) + Ly(Ly(Psi)) + Lz(Lz(Psi)) == L2(Psi)) Sx(f) = 1/2 hbar dot(((0,1),(1,0)),f) Sy(f) = 1/2 hbar dot(((0,-i),(i,0)),f) Sz(f) = 1/2 hbar dot(((1,0),(0,-1)),f) S(f) = (Sx(f),Sy(f),Sz(f)) S2(f) = Sx(Sx(f)) + Sy(Sy(f)) + Sz(Sz(f)) Jx(f) = Lx(f) + Sx(f) Jy(f) = Ly(f) + Sy(f) Jz(f) = Lz(f) + Sz(f) J(f) = (Jx(f),Jy(f),Jz(f)) J2(f) = Jx(Jx(f)) + Jy(Jy(f)) + Jz(Jz(f)) 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)) l = 3 m = 2 "Check equation 1" Psi = Y(l,m) (1,0) check(Lz(Psi) == m hbar Psi) check(L2(Psi) == l (l + 1) hbar^2 Psi) check(Sz(Psi) = 1/2 hbar Psi) check(S2(Psi) == 3/4 hbar^2 Psi) check(Jz(Psi) == (m + 1/2) hbar Psi) "ok" "Check equation 2" Psi = Y(l,m) (0,1) check(Lz(Psi) == m hbar Psi) check(L2(Psi) == l (l + 1) hbar^2 Psi) check(Sz(Psi) = -1/2 hbar Psi) check(S2(Psi) == 3/4 hbar^2 Psi) check(Jz(Psi) == (m - 1/2) hbar Psi) "ok" Run