240 CHAPTER15. IDENTICALPARTICLES
15.1 Two-Electron States
Supposewehavetwospin-zeroidenticalparticles(suchaspions)whicharepreparedin
suchawaythatoneparticlehasmomentumpa,andtheotherparticlehasmomentum
pb.Whatisthewavefunctionofthesystem?
Wereitnotforthespinstatisticstheorem,thewavefunctioncouldbe(uptoa
normalizationconstant)
ψ(x 1 ,x 2 )=eipa·x^1 / ̄heipb·x^2 / ̄h or eipa·x^2 / ̄heipb·x^1 / ̄h (15.23)oranylinearcombination
ψ(x 1 ,x 2 )=Aeipa·x^1 / ̄heipb·x^2 / ̄h+Beipa·x^2 / ̄heipb·x^1 / ̄h (15.24)However,thespin-statisticstheormtellsusthattheonlypossiblecombinationisthe
onewhichissymmetricwithrespecttotheinterchangeofcoordinatelabels 1 ↔2.
Thusthereisonlyonepossiblestateinthiscase
ψS(x 1 ,x 2 )=eipa·x^1 / ̄heipb·x^2 / ̄h+eipa·x^2 / ̄heipb·x^1 / ̄h (15.25)Nextsupposetherearetwoelectrons, onewithmomentumpa, theotherwith
momentumpb. Thistime, we alsohave to say something about the spinstates.
First,supposetheparticle withmomentumpa hasspinup, andthe particlewith
momentumpbhasspindown. Again,withoutknowingthespin-statisticstheorem,
thestatescouldbe
ψ=eipa·x^1 / ̄hχ^1 +eipb·x^2 / ̄hχ^2 − or eipa·x^2 / ̄hχ^2 +eipb·x^1 / ̄hχ^1 − (15.26)oranylinearcombination(χ^1 +means”particle 1 hasspinup,etc.). Butthespin-
statisticstheoremtellsusthat,uptoanormalizationconstant,onlytheantisymmet-
riccombination
ψA=eipa·x^1 / ̄hχ^1 +eipb·x^2 / ̄hχ^2 −−eipa·x^2 / ̄hχ^2 +eipb·x^1 / ̄hχ^1 − (15.27)willoccurinnature.
The2-electronstate(15.27)isnotaneigenstateofthetotalelectronspin
S^2 =(S 1 +S 2 )^2 =S 12 +S 22 + 2 S 1 ·S 2 (15.28)becauseχ^1 +χ^2 −isnotaneigenstateofS 1 ·S 2. Butwesaw,inthelastchapter,that
itispossibleto combine the spinangularmomentumoftwospin^12 particlesinto
eigenstates oftotalspins= 0 ands=1. Whatisthenthewavefunctionifone
particlehasmomentumpa,theotherpb,andthetotalspinangular momentumis
s=1?