QMGreensite_merged

360 CHAPTER23. THEPROBLEMOFMEASUREMENT

Theinterferencetermis

<AB>int=

∑

i(=j

c∗icj<ψi|A|ψj><φi|B|φj> (23.12)

anditisnonzeroprovidingthematrices

Aij≡<ψi|A|ψj> and Bij≡<φi|B|φj> (23.13)

arebothnon-diagonal.

Fromthisdiscussionweextractthefollowingconclusions:

1.Inprinciple,itisalwayspossibletodistinguishexperimentallybetweenamixture

andapurestate;

2. Todistinguish betweenpureandmixedstatesofcompositesystems,onemust
   makemeasurementsoneachofthecomponents.

Thus, tothequestion,”whatdoesameasurementdo?,”ourtentativeresponse
is: ”itleavesthemeasuredsysteminamixedstate.”Moreprecisely,ifthedetector
makes ameasurement ofobservable Qwhose eigenstates are{φi}, andifthe the
particleasitentersthedetectorisintheone-particlepure-state

ψ=

∑

i

ciφi (23.14)

thentheparticle,immediatelyafterdetection,willbeinthemixedstate

M={particleinoneofthestates φi withprobability |ci|^2 } (23.15)

23.2 The Problem of Measurement

Theproblemofmeasurementistheproblemthatnoclosedsystem,evolvingaccording
totheSchrodingerequation,canevermakeatransitionfromapurestatetoamixed
state.Ameasurement,aswehavedescribedit,canthereforeneveroccurifallrelevant
objects(thedetectoraswellasthedetectedobject)obeytheSchrodingerequation.
Inordertoexplainhowadetectorworks, onewouldbeginbyanalyzingthein-
teraction betweenthedetector andtheparticle, anddescribe theevolutionofthe
particle-detectorsystemintermsofsome(probablyverycomplicated)Hamiltonian,
involvingthedegreesoffreedomofboththeparticleandthedetector.Buttounder-
standwhythereisaproblemconnectedwiththemeasurementprocess,onedoesn’t
needtoknowanythingaboutthiscomplicatedHamiltonian. Itisenoughtosimply
assumethatanappropriateHamiltonianexistssuchthat,iftheparticlestartsoutin