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