Quantum operators — UPSC CSE 2007 Physics PYQ

(i) Define angular momentum, Express it in operator form and show that \vec{L} \times \vec{L} = i\hbar \vec{L} Explain the physical significance of this relation. (10) (ii) Let Y_{lm} be an eigenstate of L^2 and L_z with eigenvalues l(l + 1)\hbar^2 and m\hbar, respectively. Show that \phi = (L_x + i L_y) Y_{lm} is likewise an eigenstate of L^2 and L_z, and determine the eigenvalues. (10)