190 CHAPTER11. ANGULARMOMENTUM
andthenjustexpressing∇^2 insphericalcoordinates
∇^2 =(
1
r
∂^2
∂r^2
r)+
1
r^2
(
1
sinθ
∂
∂θ
sinθ
∂
∂θ
+
1
sin^2 θ
∂^2
∂φ^2
) (11.122)
Comparing∇^2 toeq. (11.64),weseethatitcontainstheoperatorL ̃^2 ,althoughthe
reasonforthisisnotsoclear. Theappearanceoftheangularmomentumoperator,
ontheotherhand,isquiteobviousfromeq.(11.112).
Havingfoundtheeigenstatesofangularmomentum,wewrite
φ(r,θ,φ)=Rklm(r)Ylm(θ,φ) (11.123)
wherethe indexk isintroducedto distinguish between statesof differentenergy,
whichhavetheeigenvaluesofL^2 andLz.SubstituteintotheSchrodingerequation
[
1
2 m
p ̃^2 r+
1
2 mr^2
L ̃^2 +V(r)
]
Rklm(r)Ylm(θ,φ) = EklmRklm(r)Ylm(θ,φ)
Ylm
[
1
2 m
p ̃^2 r+
h ̄^2
2 mr^2
l(l+1)+V(r)
]
Rklm(r) = EklmRklm(r)Ylm(θ,φ)(11.124)
CancellingYlmonbothsidesoftheequation,wenotethatneitherthepotentialnorthe
differentialoperatordependonm,sowecanwritethatRklm=RklandEklm=Ekl.
Wethenhaveanequationwhichinvolvesonlytheradialcoordinate
1
2 m
[
p ̃^2 r+
̄h^2 l(l+1)
r^2
]
Rkl(r)+V(r)Rkl(r)=EklRkl(r) (11.125)
which,using
p ̃^2 r = − ̄h^2
1
r
∂^2
∂r^2
r
= − ̄h^2
[
∂^2
∂r^2
+
2
r
∂
∂r
]
(11.126)
becomes
d^2 Rkl
dr^2
+
2
r
dRkl
dr
+
[
2 m
h ̄^2
{Ekl−V(r)}−
l(l+1)
r^2
]
Rkl(r)= 0 (11.127)
Thisistheradialequationforenergyeigenstatesinasphericalpotential.