182 CHAPTER11. ANGULARMOMENTUM
becauseinsphericalcoordinates,thervariabledropsoutoftheangularmomentum
operators:
L ̃x = i ̄h
(
sinφ
∂
∂θ
+cotθcosφ
∂
∂φ
)
L ̃y = i ̄h
(
−cosφ
∂
∂θ
+cotθsinφ
∂
∂φ
)
L ̃z = −i ̄h∂
∂φ
L ̃^2 = −h ̄^2
[
1
sinθ
∂
∂θ
sinθ
∂
∂θ
+
1
sin^2 θ
∂^2
∂φ^2
]
(11.64)
andthereforeanyeigenstateofangularmomentumhastheform
φlm(x,y,z)=f(r)Ylm(θ,φ) (11.65)
where
L ̃^2 Ylm(θ,φ) = l(l+1) ̄h^2 Ylm(θ,φ)
L ̃zYlm(θ,φ) = m ̄hYlm(θ,φ) (11.66)
andwheref(r)isanyfunctionsuchthatthenormalizationcondition
1 =
∫
dxdydzφ∗(x,y,zφ(x,y,z)
=
∫∞
0
drr^2
∫π
0
dθ sinθ
∫ 2 π
0
dφf∗(r)f(r)Ylm∗(θ,φ)Ylm(θ,φ) (11.67)
issatisfied.ItisconventionaltonormalizetheYlmsuchthattheintegraloverangles
isalsoequaltoone:
∫π
0
dθ sinθ
∫ 2 π
0
dφYl∗m(θ,φ)Ylm(θ,φ)= 1 (11.68)
Withthisnormalization,theYlm(θ,φ)areknownas”SphericalHarmonics”.
Insphericalcoordinates,theraisingandloweringoperatorsare
L ̃+ = L ̃x+iL ̃y
= ̄heiφ
(
∂
∂θ
+icotθ
∂
∂φ
)
L ̃− = L ̃x−iL ̃y
= − ̄he−iφ
(
∂
∂θ
−icotθ
∂
∂φ
)
(11.69)