QMGreensite_merged

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)