"Eigenvalues of angular momentum" L2(f) = -hbar^2 (d(f,theta,theta) + cos(theta) / sin(theta) d(f,theta) + d(f,phi,phi) / sin(theta)^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) Lp(f) = Lx(f) + i Ly(f) Lm(f) = Lx(f) - i Ly(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 psi = Y(l,m) "Verify commutation relations" check(Lx(Ly(psi)) - Ly(Lx(psi)) == i hbar Lz(psi)) check(Ly(Lz(psi)) - Lz(Ly(psi)) == i hbar Lx(psi)) check(Lz(Lx(psi)) - Lx(Lz(psi)) == i hbar Ly(psi)) check(L2(Lx(psi)) - Lx(L2(psi)) == 0) check(L2(Ly(psi)) - Ly(L2(psi)) == 0) check(L2(Lz(psi)) - Lz(L2(psi)) == 0) "ok" "Verify ladder operators" check(Lz(Lp(psi)) - Lp(Lz(psi)) == hbar Lp(psi)) check(Lz(Lm(psi)) - Lm(Lz(psi)) == -hbar Lm(psi)) check(Lm(Lp(psi)) == L2(psi) - Lz(Lz(psi)) - hbar Lz(psi)) check(Lp(Lm(psi)) == L2(psi) - Lz(Lz(psi)) + hbar Lz(psi)) "ok" "Verify eigenvalues" check(L2(psi) == l (l + 1) hbar^2 psi) check(Lz(psi) == m hbar psi) "ok" Run