QMGreensite_merged

15.1. TWO-ELECTRONSTATES 241

Letsbeginwiththespinpart ofthe wavefunction. Inthelast chapter, itwas
foundthatspin^12 combinationsthatgives= 1 are

ΦS 11 = χ^1 +χ^2 +

ΦS 10 =

1

√

2

[χ^1 +χ^2 −+χ^1 −χ^2 +]

ΦS 1 − 1 = χ^1 −χ^2 − (15.29)

Thesuperscript”S”hasbeenaddedtodrawattentiontothefactthatthesestates
aresymmetricwithrespecttotheinterchange 1 ↔2.Butthespin-statisticstheorem
onlysaysthatthetotalwavefunction,includingthespinandthespatialparts,must
changesignunderthe 1 ↔ 2 interchange. Soifthespinpartissymmetric,thespatial
partmustbeantisymmetric,i.e.

eipa·x^1 / ̄heipb·x^2 / ̄h−eipa·x^2 / ̄heipb·x^1 / ̄h (15.30)

andthepossiblewavefunctionsfors= 1 are

ψtriplet=(eipa·x^1 / ̄heipb·x^2 / ̄h−eipa·x^2 / ̄heipb·x^1 / ̄h)







ΦS 11 = χ^1 +χ^2 +

ΦS 10 = √^12 [χ^1 +χ^2 −+χ^1 −χ^2 +]

ΦS 1 − 1 = χ^1 −χ^2 −

(15.31)

Theotherpossibilityiss=0.Inthiscase,thespinwavefunctionis

ΦA 00 =

1

√

2

[χ^1 +χ^2 −−χ^1 −χ^2 +] (15.32)

Sincethispartofthewavefunctionisantisymmetricunderthe 1 ↔ 2 interchange,
thespatialpartmustbesymmetric,andtherefore

ψsinglet = (eipa·x^1 / ̄heipb·x^2 / ̄h+eipa·x^2 / ̄heipb·x^1 / ̄h)ΦA 00

= (eipa·x^1 / ̄heipb·x^2 / ̄h+eipa·x^2 / ̄heipb·x^1 / ̄h)×

1

√

2

[χ^1 +χ^2 −−χ^1 −χ^2 +] (15.33)

Finally,letsaskifitspossibletoprepare 2 electronsinastatewherebothelectrons
havethesamemomentumpa=pb=p,andbothparticleshavethesamespin,e.g.
botharespinup.Itsnothardtoseethattheonlywavefunctionwhichwoulddescribe
suchasystemis
ψ=eip·x^1 / ̄heip·x^2 / ̄hχ^1 +χ^2 + (15.34)

Butthisisasymmetricstate,therefore,itcannotexistinnature!Twoelectronscan
neverbepreparedinastate,orfoundtobeinastate,withthesamemomentaand
thesamespin. Suchastate,incidentally,isans= 1 state,andonecanseefromeq.
(15.32)thatthes= 1 statesvanishforpa=pb. ThisisafirstexampleofthePauli
ExclusionPrinciple.