Given a basis of angular momentum eigenstates | l , m >, where m is the component of
angular momentum along the z-axis and l is angular momentum operator:
(a) Show that the matrix elements < l', m' | L± | l, m> are non-zero only if l' = l and m' = m ± 1. (L± is the raising and lowering angular momentum operators)
(b) Using the relation L±L∓ = L2 − Lz2 ± (hbar)Lz, show that < l, m ± 1| L± | l, m> = hbar[l(l + 1) − m(m ± 1)]1/2
(c) Using the relation L± = Lx± iLy, derive the matrix representations for Lx = Sx, Ly = Sy, and Lz = Sz for a spin 1/2 system. Combine these operators together to get a matrix representation for L2 = S2 .