Question

From the cannonical commutation relations [x_l,p_m]=i(hbar)δ_lm (l,m = 1,2,3 or x,y,z) and thefact that the orbital angular momentum operator is defined L=r x p,prove the following commutation relations.

(a) [L_x,L_y]=i(hbar)L_z

(b)[L²,L_x]=[L²,L_y]=[L²,L_z]=0,whereL²=L_x²+L_y²+L_z²

(c) [L², L±]=0 (L± =L_x ± iL_y)

(d) [L± , L_z] = (-/+)(hbar)L±

(e) [L+, L-] = 2(hbar)L_z

Hint: Just use Commutator Algebra Rules [A,B]=-[B,A] and [AB,C]= A[B,C] + [A,C]B