236 CHAPTER15. IDENTICALPARTICLES
thatoneparticlewasfoundatpointa,andoneparticlewasfoundatpointb. The
particlesthemselvesdon’tcarrylabelstellingwhichisparticle1,andwhichisparticle
2.Butiftherearedifferentphysicalstateswhichareexperimentallyindistiguishable,
thenthedifferencebetweenthemamountstoakindof”hidden”information,which
isnotaccessibletohumanknowledge.
Now,thereisreally noobviousreason,apriori,that wavefunctionscan’tcarry
hiddeninformation. Perhapsthisinformationisreallythere,butonlyaccessibleto
theLordandHisAngels.Inasimilarvein,theexampleoftheHeisenbergmicroscope
inLecture 2 suggestedthatthereisnoexperimentalwayto measurepositionand
momentumsimultaneouslytoarbitraryprecision. Thatdoesn’tlogicallyimplythat
aparticlecan’thaveadefinitepositionandmomentum,atanygiventime,butwe
foundinfactthattherearenophysicalstateswiththisproperty. Therefore,letus
tentativelytakethepointofviewthatphysicalstatesdonotcontainanyexperimen-
tallyinaccessibleinformation,andseewhereitleads. Wehavetwoparticles,with
particle 1 atpositionx 1 andparticle 2 atpositionx 2 .Nowinterchangetheparticles,
as inFig. [15.1], movingparticle 2 to positionx 1 , and particle 1 topositionx 2.
Mathematically,thismeansrelabelingx 1 →x 2 andx 2 →x 1 ;i.e.itisaninterchange
ofparticlecoordinates. Quantummechanically,ifthetwo-particlestateisdenoted
ψ(x 1 ,x 2 ),theinterchangeofcoordinatesbringsustoanewstatewhichdependson
thevariablesx 1 andx 2 inadifferentway,namely
ψ′(x 1 ,x 2 )=ψ(x 2 ,x 1 ) (15.3)whichisexperimentallyindistinguishablefromtheoldstate. Nowwewillmakethe
following
Interchange Hypothesis:
Interchangingthepositionsoftwoidenticalparticlesdoesnotchangethephysical
state.
Thishypothesisimposesaverystrongconstraintonphysicalstates. Itsinteresting
tofirstseewhatitwouldimplyinclassicalphysics. Theclassicalstateofasystem
oftwoidenticalparticles,accordingtothediscussioninthefirstchapter,isgivenby
apointinthe12-dimensionalphasespace
{%x 1 ,%p 1 ,%x 2 ,%p 2 } (15.4)Inclassical physics, an interchange should involve interchanging the positionand
momentumofeachparticle.Suchaninterchangewouldlead,ingeneral,toadifferent
(althoughexperimentallyindistinguishable)pointinphasespace
{%x 2 ,%p 2 ,%x 1 ,%p 1 } (15.5)