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.