6.2. THEHEISENBERGUNCERTAINTYPRINCIPLE 89
TheHeisenbergUncertaintyPrinciple
It is impossible, by any measurement process, to simultaneously de-
terminethepositionand momentumofaparticletoanaccuracygreater
than
∆x∆p=̄h
2(6.25)
Ifinfactwecoulddeterminethepositionandmomentumtoanaccuracygreater
than(6.25),thentheparticlewouldbeleftinaphysicalstatewith∆x∆p< ̄h/2.But
accordingtoeq.(6.24)therearenosuchphysicalstates. Therefore,measurementsof
thatkindareimpossible.
TheUncertaintyPrincipleisanunavoidableconsequenceofquantumtheory,and
ifonecoulddesignameasurementprocesswhichwouldbemoreaccuratethanthe
bound(6.25),thenquantumtheorywouldbewrong.Wehavealreadydiscussedone
attempttomeasurexandpsimultaneously,intheexampleoftheHeisenbergmi-
croscope. Inthatcaseitwasfoundthatthephotoncompositionoflight,combined
withtheRayleighcriterionof resolution,resultsin∆x∆p≈ ̄h, whichisinagree-
mentwiththeUncertaintyPrinciple.Therehavebeenotheringeniousproposalsfor
measurements whichwouldviolatetheUncertainty Principle,especiallyduetoAl-
bertEinsteininhisdiscussionswithNielsBohr.^1 Acarefulstudyoftheseproposals
alwaysrevealsaflaw.Inmuchthesamewaythattheexistenceofperpetualmotion
machinesisruledoutbytheSecondLawofThermodynamics,theexistenceofan
apparatuswhichwouldgiveaprecisedeterminationofpositionandmomentumisin
conflictwiththenatureofphysicalstatesinquantummechanics.
However,theexampleofHeisenberg’smicroscopeoftenleadstoamisunderstand-
ingthatthe UncertaintyPrincipleissimplysayingthat”theobservation disturbs
whatisbeingobserved.”Itistruethatanobservationusuallychangesthephysical
stateoftheobservedsystem. ButitisnottruethatthisisfullcontentoftheUn-
certaintyprinciple. Ifposition-momentumuncertaintywereonlyamatterof light
disturbingtheobservedparticle, thenwewould be freetoimaginethat aparticle
reallyhasadefinitepositionanddefinitemomentumateverymomentintime,but
thatthephysicalpropertiesoflightpreventtheiraccuratesimultaneousdetermina-
tion. Thisinterpretationiswrong,becauseifaparticlehadadefinitepositionand
momentumateverymomentoftime,thentheparticlewouldfollowadefinitetra-
jectory. Wehavealreadyseenthattheassumptionthatparticlesfollowtrajectories
isinconsistent withelectron interference. TheHeisenbergprincipleisbest under-
stood,notasaslogan”theobservationdisturbswhatisobserved,”butratherasa
consequenceofthenatureofphysicalstatesinquantummechanics,whichcannotbe
simultaneouslyeigenstatesofpositionandmomentum. Thefactthatuncertainties
(^1) c.f.NielsBohr,”DiscussionswithEinstein,”inAtomicPhysicsandHumanKnowledge,available
onloaninmyoffice.