"Addition of angular momentum 3" Y(l,m) = (-1)^m sqrt((2 l + 1) / (4 pi) (l - m)! / (l + m)!) * P(l,m) exp(i m phi) 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)) 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) = Lx(Lx(f)) + Ly(Ly(f)) + Lz(Lz(f)) 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)) "Verify eigenstates" for(l,0,3, j = l + 1/2, Psi = Y(l,l) (1,0), check(J2(Psi) == j (j + 1) hbar^2 Psi), Psi = Y(l,-l) (0,1), check(J2(Psi) == j (j + 1) hbar^2 Psi) ) for(l,1,3,for(m, -l, l - 1, A = sqrt((l + m + 1) / (2 l + 1)), B = sqrt((l - m) / (2 l + 1)), Psi = A Y(l,m) (1,0) + B Y(l, m + 1) (0,1), j = l + 1/2, check(J2(Psi) == j (j + 1) hbar^2 Psi), Psi = B Y(l,m) (1,0) - A Y(l, m + 1) (0,1), j = l - 1/2, check(J2(Psi) == j (j + 1) hbar^2 Psi) )) "ok" Run