"Addition of angular momentum" Lx(f) = -i hbar (y d(f,z) - z d(f,y)) Ly(f) = -i hbar (z d(f,x) - x d(f,z)) Lz(f) = -i hbar (x d(f,y) - y d(f,x)) L(f) = (Lx(f),Ly(f),Lz(f)) L2(f) = Lx(Lx(f)) + Ly(Ly(f)) + Lz(Lz(f)) LxL(f) = (Ly(Lz(f)) - Lz(Ly(f)), Lz(Lx(f)) - Lx(Lz(f)), Lx(Ly(f)) - Ly(Lx(f))) Sx(f) = hbar / 2 dot(((0,1),(1,0)),f) Sy(f) = hbar / 2 dot(((0,-i),(i,0)),f) Sz(f) = hbar / 2 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)) SxS(f) = (Sy(Sz(f)) - Sz(Sy(f)), Sz(Sx(f)) - Sx(Sz(f)), Sx(Sy(f)) - Sy(Sx(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)) JxJ(f) = (Jy(Jz(f)) - Jz(Jy(f)), Jz(Jx(f)) - Jx(Jz(f)), Jx(Jy(f)) - Jy(Jx(f))) # theta and phi are the spin direction chi = (c1(theta,phi), c2(theta,phi)) Psi = psi(x,y,z) chi check(LxL(Psi) == i hbar L(Psi)) check(SxS(Psi) == i hbar S(Psi)) check(JxJ(Psi) == i hbar J(Psi)) "Verify equation (1)" LS(f) = contract(L(S(Psi))) check(J2(Psi) == L2(Psi) + S2(Psi) + 2 LS(Psi)) "ok" "Verify commutation relations" check(J2(L2(Psi)) - L2(J2(Psi)) == 0) check(J2(S2(Psi)) - S2(J2(Psi)) == 0) check(J2(Jx(Psi)) - Jx(J2(Psi)) == 0) check(J2(Jy(Psi)) - Jy(J2(Psi)) == 0) check(J2(Jz(Psi)) - Jz(J2(Psi)) == 0) check(J2(Lx(Psi)) - Lx(J2(Psi)) == 2 i hbar (Ly(Sz(Psi)) - Lz(Sy(Psi)))) check(J2(Ly(Psi)) - Ly(J2(Psi)) == 2 i hbar (Lz(Sx(Psi)) - Lx(Sz(Psi)))) check(J2(Lz(Psi)) - Lz(J2(Psi)) == 2 i hbar (Lx(Sy(Psi)) - Ly(Sx(Psi)))) check(J2(Sx(Psi)) - Sx(J2(Psi)) == -2 i hbar (Ly(Sz(Psi)) - Lz(Sy(Psi)))) check(J2(Sy(Psi)) - Sy(J2(Psi)) == -2 i hbar (Lz(Sx(Psi)) - Lx(Sz(Psi)))) check(J2(Sz(Psi)) - Sz(J2(Psi)) == -2 i hbar (Lx(Sy(Psi)) - Ly(Sx(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) 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(S2(Sx(Psi)) - Sx(S2(Psi)) == 0) check(S2(Sy(Psi)) - Sy(S2(Psi)) == 0) check(S2(Sz(Psi)) - Sz(S2(Psi)) == 0) check(Sx(Sy(Psi)) - Sy(Sx(Psi)) == i hbar Sz(Psi)) check(Sy(Sz(Psi)) - Sz(Sy(Psi)) == i hbar Sx(Psi)) check(Sz(Sx(Psi)) - Sx(Sz(Psi)) == i hbar Sy(Psi)) "ok" Run