358 CHAPTER23. THEPROBLEMOFMEASUREMENT
exitstheapparatus,whichisenclosedinablackbox,inaneigenstateofSz(seeFig.
22.2).
Wheneverweknowthephysicalstateofanobject,theobjectissaidtobeina
”purestate.”Supposethattheelectronisina”purestate”enteringtheapparatus,
i.e.itsstateisknowntobe
φ=Aαz+Bβz (23.1)
where,forsimplicity,wedisregardthex,y,zdegreesoffreedomandconcentrateon
thespin. Then,ifthedetector hasbeenswitchedon,butbeforewelookinsidethe
blackbox,theelectronexitingthedetectormustbe
eitherinstate αz, withprobability |A|^2
orelseinstate βz, withprobability |B|^2(23.2)
Thisisanexampleof a”mixed state”or”mixture.”Ingeneral,ifthestateof
anobjectisnotknownwithcertainty,butitisknownthattheobjectisinoneofa
numberofpossiblestates,togetherwiththeprobabilityofbeingineachstate,then
theobjectissaidtobeinamixedstate.
John,whohassetupthisexperiment,hasleftthelabfortheday.Mary,knowing
John’sforgetful nature, goes to check that the detector inside theblack box has
beenswitchedoff,therebyconservingtheveryexpensive electronicsinside. Toher
consternation,shediscoversthattheboxhasalreadybeenlockedbythejanitor,who
hastheonlykey. Canshetell,withoutopeningthebox,whetherthedetectorinside
theboxisonofoff?
Itisalwayspossibletodistinguishbetweenapurestateandamixture. Suppose
Marymeasures, onthebeamof electronsemergingfromthebox, thevalueofthe
observableQ. Ifthedetectorisswitchedoff,thentheparticlesemergingfromthe
detectorremainintheinitalstateφ,sothat
<Q>pure = <φ|Q|φ>
= |A|^2 <αz|Q|αz>+|B|^2 <βz|Q|βz>
+A∗B<αz|Q|βz>+B∗A<βz|Q|αz> (23.3)whileforthemixture
mix = Prob.tofindspinup×<αz|Q|αz>+Prob.tofindspindown×<βz|Q|βz>
= |A|^2 <αz|Q|αz>+|B|^2 <βz|Q|βz> (23.4)Thedifferencebetweenthepureandmixed-stateresultsiscalledthe”interference
term”
<Q>int = <Q>pure−<Qmix>
= A∗B<αz|Q|βz>+B∗A<βz|Q|αz> (23.5)