QMGreensite_merged

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)