15.2. THEHELIUMATOM 243
wherer 1 (r 2 )isthedistanceofelectron 1 (electron2)tothenucleus,andr 12 isthe
distancebetweenthetwoelectrons.WehaveneglectedallothertermsintheHamil-
tonianthat involve spin; nevertheless,due tothe spin-statisticstheorem,electron
spinturnsouttobeveryimportantindeterminingthepossibleelectronstates.
ThefirstthingtodoisneglectV′also,andconstructeigenstatesof
H 0 (12)=H 0 (1)+H 0 (2) (15.42)
Thisiseasybecausethevariablesareseparable,andthesolutionsof
H 0 Ψ(x 1 ,x 2 )=EΨ(x 1 ,x 2 ) (15.43)arejust
Ψnlm,n′l′m′ = φnlm(r 1 ,θ 1 ,φ 1 )φn′l′m′(r 2 ,θ 2 ,φ 2 )
Enlm,n′l′m′ = En+En′ (15.44)wheretheφnlm(r,θ,φ)andEnarejusttheHydrogenatomwavefunctions,withthe
onemodification thatthechargee^2 inthoseexpressionsarereplacedby 2 e^2 ,since
theHeliumnucleushastwoprotons. Ofcourse,ifthequantumnumbers(nlm)and
(n′l′m′)aredifferent,thenanylinearcombination
Ψ=Aφnlm(r 1 ,θ 1 ,φ 1 )φn′l′m′(r 2 ,θ 2 ,φ 2 )+Bφnlm(r 2 ,θ 2 ,φ 2 )φn′l′m′(r 1 ,θ 1 ,φ 1 ) (15.45)isalsoan eigenstateof H 0 (12),withenergy En+En′. Inparticular,we canform
normalized,symmetricandantisymmetriccombinations
ΨSnlm,n′m′l′ =
1
√
2
[φnlm(r 1 ,θ 1 ,φ 1 )φn′l′m′(r 2 ,θ 2 ,φ 2 )+φnlm(r 2 ,θ 2 ,φ 2 )φn′l′m′(r 1 ,θ 1 ,φ 1 )]ΨAnlm,n;l′m′ =1
√
2
[φnlm(r 1 ,θ 1 ,φ 1 )φn′l′m′(r 2 ,θ 2 ,φ 2 )−φnlm(r 2 ,θ 2 ,φ 2 )φn′l′m′(r 1 ,θ 1 ,φ 1 )](15.46)
Sofarwehaveonlywrittendownthespatialpartoftheenergyeigenstates,and
ignoredthespinpart.ButHcommuteswiththetotalspinoperatorsS^2 , Sz,sowe
shouldbeabletoconstructenergyeigenstateswhicharealsototalspineigenstates.
Theonlypossibilitiesfortotalspiniss=1,whichhasatripletofsymmetricspin
wavefunctions,ands=0,whichhasasingleantisymmetricspinwavefunction. To
satisyspin-statistics,s= 1 mustgowithanantisymmetricspatialwavefunction,and
s= 0 withasymmetricspatialwavefunction,soallinall
Ψtrnliplm,net′l′m′ =1
√
2
[φnlm(r 1 ,θ 1 ,φ 1 )φn′l′m′(r 2 ,θ 2 ,φ 2 )−φnlm(r 2 ,θ 2 ,φ 2 )φn′l′m′(r 1 ,θ 1 ,φ 1 )]
Φ 11
Φ 10
Φ 1 − 1
Ψsingnlm,nlet′l′m′ =1
√
2
[φnlm(r 1 ,θ 1 ,φ 1 )φn′l′m′(r 2 ,θ 2 ,φ 2 )+φnlm(r 2 ,θ 2 ,φ 2 )φn′l′m′(r 1 ,θ 1 ,φ 1 )]Φ 00 (15.47)