whereℓis an integer.
Note that we can easily write the components of angular momentum interms of the raising and
lowering operators.
Lx =1
2
(L++L−)
Ly =1
2 i(L+−L−)
We will also find the following equations useful (and easy to compute).
[L+,L−] = i[Ly,Lx]−i[Lx,Ly] = ̄h(Lz+Lz) = 2 ̄hLz
L^2 = L+L−+L^2 z− ̄hLz.- See Example 14.5.1:What is the expectation value ofLzin the stateψ(~r) =R(r)(
√
2
3 Y^11 (θ,φ)−i
√
1
3 Y^1 −^1 (θ,φ))?*- See Example 14.5.2:What is the expectation value ofLxin the stateψ(~r) =R(r)(
√
2
√^3 Y^11 (θ,φ)−
1
3 Y^10 (θ,φ))?*
14.3 The Angular Momentum Eigenfunctions
The angular momentum eigenstates areeigenstates of two operators.
LzYℓm(θ,φ) =m ̄hYℓm(θ,φ)
L^2 Yℓm(θ,φ) =ℓ(ℓ+ 1) ̄h^2 Yℓm(θ,φ)All we know about the states are the two quantum numbersℓandm. We have no additional
knowledge aboutLxandLy since these operators don’t commute withLz. Theraising and
lowering operatorsL±=Lx±iLyraise or lowerm, leavingℓunchanged.
L±Yℓm= ̄h√
ℓ(ℓ+ 1)−m(m±1)Yℓ(m±1)The differential operators take some work to derive (See section 14.4.4).
Lz=̄h
i∂
∂φ
L±= ̄he±iφ(
±
∂
∂θ+icotθ∂
∂φ