QMGreensite_merged

196 CHAPTER12. THEHYDROGENATOM

Sincethisequationhastobetrueforeveryvalueofρ,itfollowsthatthecoefficient
ofeachpowerofρmustvanish.Thecoefficientinfrontofρjis

cj+1(j+1)j+cj+1(2l+2)(j+1)−cjj−cj(l+ 1 −λ)= 0 (12.22)

or
cj+1(j+1)(2l+ 2 +j)−cj(l+ 1 +j−λ)= 0 (12.23)

Thisequationallowsustocomputethecoefficients{cj}iteratively:

cj+1=

j+l+ 1 −λ

(j+1)(j+ 2 l+2)

cj (12.24)

However,wehavealsorequiredthatF(r)growmoreslowlythananexponential,
inorderthatu(ρ)∼exp[−ρ/2]asρ→∞.Itturnsoutthatthiscanonlybetruefor
veryspecialvaluesoftheconstantλ.Recallthattheasympoticbehaviorofasolution
oftheSchrodingerequationcanbeeitherexp[+ρ/2]orexp[−ρ/2]. Supposeitturns
outthatF(ρ)∼exp(ρ)asρ→∞.Thenwewouldendupwiththenon-normalizable
solutionforu(r). Infact,foralmostallvalues ofλ,thisisexactlywhathappens.
Accordingto(12.24),atlargeindexj

cj+1≈

cj

j+ 1

(12.25)

whichimplies

cj≈

const.

j!

(12.26)

Thismeansthatatlargeρ,wherethepowerseriesinρisdominatedbythetermsat
largeindexj,

F(ρ) =

∑

j

cjρj

≈ const.×

∑

j

1

j!

ρj

≈ const.×eρ (12.27)

inwhichcase
u(ρ)≈ρl+1eρ/^2 (12.28)

whichisnon-normalizable.So,eventhoughwestartedwithanansatz(12.18)which
seemedtoincorporatetheasympoticbehaviorwewant,theequationforF endsup
generating,foralmostanychoiceofλ,theasympoticbehaviorthatwedon’twant.
ThissituationisalreadyfamiliarfromthediscussioninLecture8.Typically,solutions
oftheSchrodingerequationforE< 0 arenon-normalizableforalmostallvaluesofE.
Onlyatadiscretesetofbound-stateenergiesisitpossibletohavethewavefunction
falltozeroasymptotically.