QMGreensite_merged

242 CHAPTER15. IDENTICALPARTICLES

TheExclusionPrincipleitselfisderivedfromtheSpin-StatisticsTheormbythe
following argument: Any stateof Nelectronsisasuperposition ofproductsof N
1-particlestates,witheach1-particlestateoftheform

ψa(xi)=

[

ψa+(xi)

ψa−(xi)

]

(15.35)

Forexample,fortheHydrogenatomtheindexacouldbea=(nlmssz),sothat,e.g.

ψnlmssz(xi)=Rnl(ri)Ylm(θi,φi)χisz (15.36)

Nowsupposewetrytoprepareastateinwhichoneelectronisinstatea,asecondis
instateb,athirdinstatec,andsoon.Becauseofthespin-statisticstheorm,sucha
statemustbeantisymmetricundertheinterchangeofanytwoindicesi↔j. Then
theonlypossibilityis

Ψ=

∑N

na=1

∑N

nb=1

...

∑N

nz=1

Dnanb...nzψa(xna)ψb(xnb)...ψz(xnz) (15.37)

whereDnanb...nzisantisymmetricunderinterchangeofanytwoindices,i.e.

Dnanbnc...nz=−Dnbnanc...nz (15.38)

Butsupposethatanytwostatesarethesame,e.g. a=b. Thenforanyterminthe
sum,e.g.
Dijk...ψa(xi)ψa(xj)ψc(xk).... (15.39)

thereisanequalterminthesum,oftheoppositesign,i.e.

Djik...ψa(xj)ψa(xi)ψc(xk)... (15.40)

whereoppositesignofthistermisduetotheantisymmetryproperty(15.38),and
so the twocontributions (15.39)and (15.40)cancel. Thismeans that ifanytwo
1-electronstates arethesame, theN-particlewavefunctioniszero. Thereiszero
probabilitytofindtwoelectronsinthesamequantumstate.ThisisthePauliExclu-
sionPrinciple,whoseconsequencesforatomicphysicsareprofound.

15.2 The Helium Atom

Beforetalkingaboutatomsingeneral,let’sdiscussoneinparticular: Helium.Thisis
obviouslythenextstepupincomplexityfromHydrogenand,sinceithastwoelectrons
ratherthanone,isagoodplacetoapplyourknowlegeof2-electronwavefunctions.
TheHamiltonianforthesystemis

H =

{

p^21

2 m

−

2 e^2

r 1

}

+

{

p^22

2 m

−

2 e^2

r 2

}

+

e^2

r 12

= H 0 (1)+H 0 (2)+V′ (15.41)