Chapter 15
Identical Particles
Iwouldn’tbelongtoanyclubthatwouldhaveMEasamember!
- GrouchoMarx
Inordinaryexperience, no twoobjects areexactlyidentical. ”Identical” twins
aresimilar,ratherthanidentical;”identical”ping-pongballscanbealwaysbedis-
tinguished(atleastunderamicroscope)bysomesmalldifferenceinsize, shapeor
surfacemarkings.
It’sdifferentwithelementaryparticles.Asfaraswecantell,therearenoscratches,
marks,orpersonalitytraitsthatwoulddistinguishoneelectronfromanotherelectron,
oroneprotonfromanother.Whenwespeakofidenticalparticlesinquantumphysics,
wemeantheparticlesareidentical,notjustveryverysimilar. Thefactthatthere
areobjectsintheworldwhichseemtobeexactlyidenticalleadstoaveryinteresting
(almostmetaphysical!) questionofprincipleregardingphysicalstates.
Tostartout,imaginethatwehavetwoidenticalspin-0particles.Thepositionof
oneparticleisdenotedbyx 1 ,andthatoftheotherparticlebyx 2 .Considerthestate
whereparticle 1 isatthepointa,andparticle 2 isatthepointb,sothestatewould
be^1
ψ(x 1 ,x 2 )=δ(x 1 −a)δ(x 2 −b) (15.1)
Nowsupposesomeonecomesalongandinterchangesthetwoparticles(Fig. [15.1]).
Theparticleswouldthenbeinadifferentphysicalstate
ψ′(x 1 ,x 2 ) = δ(x 1 −b)δ(x 2 −a)
= ψ(x 2 ,x 1 ) (15.2)However...iftheparticlesareexactlyidentical,thenthereisnowayofeverdistinguish-
ingbetweenthestatesψ(x 1 ,x 2 ),andψ′(x 1 ,x 2 ).Ameasurementcouldonlydetermine
(^1) Strictlyspeaking,adeltafunctionisnotaphysicalstate,becauseitisn’tnormalizedtoone.So,
tobemorerigorous,justreplacethedeltafunctionsbygaussianswhicharenarrowlypeakedaround
pointsaandb.Thisdoesn’taffecttheargumentatall.