"Angular momentum 2" Dx(f) = sin(theta) cos(phi) d(f,r) + cos(theta) cos(phi) / r d(f,theta) - sin(phi) / (r sin(theta)) d(f,phi) Dy(f) = sin(theta) sin(phi) d(f,r) + cos(theta) sin(phi) / r d(f,theta) + cos(phi) / (r sin(theta)) d(f,phi) Dz(f) = cos(theta) d(f,r) - sin(theta) / r d(f,theta) px(f) = -i hbar Dx(f) py(f) = -i hbar Dy(f) pz(f) = -i hbar Dz(f) Lx(f) = i hbar (sin(phi) d(f,theta) + cos(theta) cos(phi) / sin(theta) d(f,phi)) Ly(f) = i hbar (-cos(phi) d(f,theta) + cos(theta) sin(phi) / sin(theta) d(f,phi)) Lz(f) = -i hbar d(f,phi) LL(f) = -hbar^2 (d(f,theta,theta) + cos(theta) / sin(theta) d(f,theta) + 1 / sin(theta)^2 d(f,phi,phi)) "Verify operators" W = psi(r,theta,phi,t) x = r sin(theta) cos(phi) y = r sin(theta) sin(phi) z = r cos(theta) check(y pz(W) - z py(W) == Lx(W)) check(z px(W) - x pz(W) == Ly(W)) check(x py(W) - y px(W) == Lz(W)) check(Lx(Lx(W)) + Ly(Ly(W)) + Lz(Lz(W)) == LL(W)) "ok" "Verify commutation relations" check(Lx(Ly(W)) - Ly(Lx(W)) == i hbar Lz(W)) check(Ly(Lz(W)) - Lz(Ly(W)) == i hbar Lx(W)) check(Lz(Lx(W)) - Lx(Lz(W)) == i hbar Ly(W)) check(Lx(LL(W)) - LL(Lx(W)) == 0) check(Ly(LL(W)) - LL(Ly(W)) == 0) check(Lz(LL(W)) - LL(Lz(W)) == 0) "ok" "Verify Griffith's formula" er = (sin(theta) cos(phi), sin(theta) sin(phi), cos(theta)) etheta = (cos(theta) cos(phi), cos(theta) sin(phi), -sin(theta)) ephi = (-sin(phi), cos(phi), 0) check(cross(er,er) = 0) check(cross(er,etheta) = ephi) check(cross(er,ephi) = -etheta) D(f) = d(f,r) er + d(f,theta) / r etheta + d(f,phi) / (r sin(theta)) ephi L(f) = -i hbar cross(r er, D(f)) check(L(W) == (Lx(W), Ly(W), Lz(W))) "ok" Run