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)