195
whichhasthegeneralsolution
u(ρ)=Ae−ρ/^2 +Beρ/^2 (12.14)
Thesecondterm,proportionaltotheconstantB,isnon-normalizable,thereforeB=
0 forphysicalstates.Hence,wefindthat
u(ρ)→Ae−ρ/^2 as ρ→∞ (12.15)
Next,considertheρ→ 0 regime. Thentermsproportionaltoaconstant,or 1 /ρ,
arenegligiblecomparedtothetermproportionalto 1 /ρ^2 , sotheradialequationis
approximately
d^2 u
dρ^2
−
l(l+1)
ρ^2
u= 0 (12.16)
which,onecaneasilycheck,issolvedby
u(ρ)=Cρl+1+Dρ−l (12.17)
Foranyvaluel> 0 thewavefunction(proportionaltou/r)isnon-normalizablefor
D+=0,andevenatl= 0 theratiou/risnon-differentiableatr= 0 ifD+=0.Sowe
willsetD= 0 also.
Sincewenowknowthatu(ρ)∼e−ρ/^2 atlargeρ,andu(ρ)∼ρl+1atsmallρ,we
canguessthattheexactsolutionmighthavetheform
u(ρ)=e−ρ/^2 ρl+1F(ρ) (12.18)
whereF(ρ)issomeunknownfunction,whichgoestoaconstantasρ→0,andgrows
slowerthanan exponentialasρ→∞. Thiskindof educatedguessregardingthe
formofthesolutionofan equationiscalledan”ansatz”. Substituting(12.18)in
(12.11)givesanequationforF(ρ)
[
ρ
d^2
dρ^2
+(2l+ 2 −ρ)
d
dρ
−(l+ 1 −λ)
]
F(ρ)= 0 (12.19)
ThenextstepistowriteF(ρ)asapowerseries
F(ρ)=
∑∞
j=0
cjρj (12.20)
andinsertinto(12.19)toget
ρ
∑
j
cjj(j−1)ρj−^2 +
∑
j
(2l+ 2 −ρ)cjjρj−^1 −
∑
j
(l+ 1 −λ)cjρj = 0
∑
j
{cjj(j−1)+cjj(2l+2)}ρj−^1 −
∑
j
{cjj+cj(l+ 1 −λ)}ρj = 0 (12.21)