197
Forthehydrogenatom,theonlywaytohaveanormalizablesolutionisifoneof
thecjcoefficientsvanishes,i.e. forsomej=js,wehavecjs+1=0. Then,fromeq.
(12.24),
cj= 0 forall j>js (12.29)
and
F(ρ) =
∑js
j=0
cjρj
∼ cjsρjs as ρ→∞ (12.30)
SinceF(ρ)isapolynomial,R(r)=u(ρ)/rwillbenormalizable.
Fromeq. (12.24),weseethattheconditionforcjs+1tovanish,givencjs+=0,is
that
λ=n=js+l+ 1 (12.31)
wherenisaninteger.Therefore,thebound-stateenergiesoftheHydrogenatomare
deducedfromtheconditionthat
λ=
1
ka 0
=n (12.32)
Theintegernisknownasthe”principalquantumnumber.”Using
k=
√
2 m|E|
̄h
and a 0 =
̄h^2
me^2
(12.33)
thebound-stateenergies,labeledbytheintegern,are
En=−
me^4
2 n^2 ̄h^2
(12.34)
ThisisthesameresultasthatobtainedbyNielsBohr; itisinagreementwiththe
experimentalresult(eq. 2.29and2.30ofChapter2)fortheHydrogenspectrum.
Wecannowwrite
ρ= 2 kr=
2 r
na 0
(12.35)
Puttingeverythingtogether,theenergyeigenstatesare
φnlm(r,θ,φ) = NnlRnl(r)Ylm(θ,φ)
= Nnl
unl(2r/na 0 )
r
Ylm(θ,φ)
ρ =
2 r
na 0
unl(ρ) = ρl+1e−ρ/^2 Fnl(ρ)