Quantum angular momentum — UPSC CSE 2008 Physics PYQ

Write the commutation relations of angular momentum operator \hat{L}_x, \hat{L}_y and \hat{L}_z and calculate the commutators [\hat{L}_+, \hat{L}_z] and [\hat{L}_-, \hat{L}_z]. Show that \hat{L}_+ |l, m\rangle = \sqrt{l(l+1) - m(m+1)} |l, m+1\rangle, where |l, m\rangle is the state with definite values for L^2 and L_z.