QMGreensite_merged

198 CHAPTER12. THEHYDROGENATOM

Fnl(ρ) =

n−∑l− 1

j=0

cjρj

cj+1 =

j+l+ 1 −λ

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

cj

c 0 = 1 (12.36)

withthenormalizationconstantsNnldeterminedfromthenormalizationcondition

1 =

∫∞

0

drr^2

∫π

0

dθsin(θ)

∫

dφφ∗nlm(r,θ,φ)φnlm(r,θ,φ)

= Nnl^2

∫∞

0

drr^2 R^2 nl(r)

= Nnl^2

∫∞

0

dru^2 nl(2r/na 0 ) (12.37)

andcorrespondingtoanenergyeigenvalue

En=−

me^4

2 n^2 ̄h^2

(12.38)

Thechoicec 0 = 1 isarbitrary. Adifferentchoicewouldsimplyleadtoadifferent
normalizationconstant,andthewavefunctionwouldendupexactlythesame.
Itisclearthattheenergyeigenvaluesaredegenerate.Tospecifyauniqueeigen-
stateφnlm(r,θ,φ),itisnecessarytospecifythreeintegers:n,l,m.Ontheotherhand,
theenergyeigenvalueEndependsonlyononeofthoseintegers.Todeterminethede-
greeofdegeneracy,i.e.thenumberoflinearlyindependenteigenstateswiththesame
energyEn, weobservethatthemaximumindexjsofthe(non-zero)cj coefficients
satisfies
js=n−l− 1 ≥ 0 (12.39)

andtherefore
l= 0 , 1 , 2 ,...,n− 1 (12.40)

Also,thereare 2 l+ 1 valuesofm,

m=−l,−l+ 1 ,......,l− 1 ,l (12.41)

correspondingtoagivenvalueofl.Therefore,thetotalnumberofl,mcombinations
thatcanexistforagivenintegernare

degeneracyof En =

n∑− 1

l=0

(2l+1)

=

(

2

n∑− 1

l=0

l

)

+n

= n^2 (12.42)