237
Theonlyphysicalstateswhichareleftunchangedbyaninterchangeofcoordinates
andmomenta,wouldbethosespecialpointsinphasespaceforwhich
%x 1 =%x 2 %p 1 =%p 2 (15.6)Thisconstraintwouldimply,e.g.,thatallelectronsintheuniversearelocatedatthe
samepoint,andaremovingwiththesamevelocity!
Clearly,theInterchangeHypothesisistoomuchtoaskofclassicalphysics.Quan-
tumphysics,however,isalittlemoretolerant,duethesuperpositionprinciple. Al-
thoughneitherstate(15.1)norstate(15.2)satisfies theInterchangehypothesis, a
superpositionofthetwostates
ΨS(x 1 ,x 2 )=δ(x 1 −a)δ(x 2 −b)+δ(x 1 −b)δ(x 2 −a) (15.7)clearlydoes(the”S”standfor”symmetric”).Unliketheclassicalstatessatisfyingthe
InterchangeHypthesis,thequantumstateΨS(x 1 ,x 2 )allowsonetofindtwoidentical
particlesattwodifferentplaces,whilebeingsymmetricinthepositionsx 1 andx 2.
However,thereisoneotherstatethatbeconstructedfromstates(15.1)and(15.2)
whichalsosatisfiestheInterchangeHypothesis,namely
ΨA(x 1 ,x 2 )=δ(x 1 −a)δ(x 2 −b)−δ(x 1 −b)δ(x 2 −a) (15.8)(the”A”standsfor”antisymmetric”). Inthiscase,thechangeincoordinatesgivesa
changeinsign
ΨA(x 2 ,x 1 )=−ΨA(x 1 ,x 2 ) (15.9)
Itisimportanttounderstand,at thispoint,thattwowavefunctionsthatdiffer
onlybyanoverallsign,oringeneral,twowavefunctionsthatdifferonlybyanoverall
phase,i.e.
ψ and eiδψ (15.10)
correspondtothesamephysicalstate,becausetheconstanteiδfactordropsoutofall
expectationvaluesandprobabilities. Inparticular,ΨAand−ΨAcorrespondtothe
samephysicalstate.Soingeneral,wearelookingfor2-particlewavefunctionswhich
satisfy
ψ(x 2 ,x 1 )=eiδψ(x 1 ,x 2 ) (15.11)
Whatpossiblevaluescantherebe,ineq. (15.11),forthephaseeiδ?
Letsdefineanexchangeoperatorthatoperatesonwavefunctionsbyinterchang-
ingthecoordinates,and,iftheparticlehasspin,alsothespinindices,i.e.
PEψ(z 1 ,z 2 )=ψ(z 2 ,z 1 ) (15.12)where
zi≡{%xi,siz} (15.13)