188 CHAPTER11. ANGULARMOMENTUM
andtheenergyeigenstatesareangularmomentumeigenstates
φlm(θ,φ)=Ylm(θ,φ) Elm=
1
2 I
l(l+1) ̄h^2 (11.107)
Wenotethattheenergyeigenvaluesaredegenerate,asexpected
Elm=Elm′ −l≤m,m′≤l (11.108)
11.5 The Radial Equation for Central Potentials
Ithasalreadybeennotedthattheangularmomentumoperatorsinvolveonlyθand
φ,soaneigenstateofL ̃^2 andLzhasthegeneralform
φ=f(r)Ylm(θ,φ) (11.109)
wheref(r)isanarbitraryfunctionofr.ForcentralpotentialsV(r,θ,φ)=V(r),the
Hamiltoniancommuteswiththeangularmomentumoperators,sobythecommutator
theoremH ̃,L ̃^2 ,L ̃zhaveacommonsetofeigenstatesoftheform(11.109). However,
ifφisaneigenstateofH,thentheradialfunctionf(r)isnolongerarbitrary,but
isdeterminedfromanordinarydifferentialequationinthevariablerknownasthe
”RadialEquation”.
LetusbeginwiththeclassicalHamiltonian
H=
%p·%p
2 m
+V(r) (11.110)
The momentum vectorp%, insphericalcoordinates, can beexpressed as a sumof
twovectors,oneofwhich(%pr)isparalleltotheradialdirection,andtheother(%p⊥)
perpendiculartotheradialdirection,i.e.
%p = %pr+%p⊥
%pr ≡ pcos(θrp)eˆr=
1
r^2
(%r·%p)%r
|%p⊥| ≡ psin(θrp)=
|%r×%p|
r
=
|%L|
r
(11.111)
sothat
H=
1
2 m
[
p^2 r+
L^2
r^2
]
+V(r) (11.112)
where
pr=|%pr|=
%r·%p
r