[Li,Lj] = i ̄hǫijkLk
[L^2 ,Li] = 0.We have shown that angular momentum is quantized for a rotor with asingle angular variable. To
progress toward the possible quantization of angular momentum variables in 3D, wedefine the
operatorL+and its Hermitian conjugateL−.
L±≡Lx±iLy.SinceL^2 commutes withLxandLy, it commutes with these operators.
[L^2 ,L±] = 0The commutator withLzis.
[L±,Lz] = [Lx,Lz]±i[Ly,Lz] =i ̄h(−Ly±iLx) =∓ ̄hL±.From the commutators [L^2 ,L±] = 0 and [L±,Lz] =∓ ̄hL±, we can derive the effect of the operators
(See section 14.4.5) L±on the eigenstatesYℓm, and in so doing, show thatℓis an integer greater
than or equal to 0, and thatmis also an integer
ℓ= 0, 1 , 2 ,...
−ℓ≤m≤ℓ
m=−ℓ,−ℓ+ 1,...,ℓ
L±Yℓm= ̄h√
ℓ(ℓ+ 1)−m(m±1)Yℓ(m±1)Therefore,L+raises thezcomponent of angular momentum by one unit of ̄handL−lowers it by
one unit. The raising stops whenm=ℓand the operation gives zero,L+Yℓℓ= 0. Similarly, the
lowering stops becauseL−Yℓ−ℓ= 0.
l=0 l=1 l=2 l=3 l=40
1
2
-1
3
4
-2
-3
-4
mL+L+
L--L--L+
L+
L+
L+
L+
L+
L+
L+ L--L--L--L--L--L--L--L--Solution to radial equation depends on l.L
Angular momentum is quantized. Any measurement of a component of angular momentum will
give some integer times ̄h. Any measurement of the total angular momentum gives the somewhat
curious result
|L|=
√
ℓ(ℓ+ 1) ̄h