QMGreensite_merged

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)