23.2. THEPROBLEMOFMEASUREMENT 361
aneigenstateoftherelevantobservable,thenthecorrespondingeigenvalueissome-
howregisteredbythedetector.Onceagain,takethemeasurementofelectronspinas
anexample. ThedetectorisdesignedtomeasureSz. Thismeansthatiftheinitial
stateoftheparticle-detectorsystemis
|Ψ 0 >=|up>|ready> (23.16)where|up>meanstheelectronisinaspin-upstate,and|ready>meansthatthe
detectorhasbeenswitchedon,thentheinteractionbetweentheelectronanddetector
leavesthecombinedsysteminthefinalstate
|Ψf>=|up>|red> (23.17)where|red>indicatesthatredlightonthedetectorison. Itisassumedthat the
transitionfromtheintialtofinalstateisadequatelyexplainedbytheSchrodinger
equation,andthuswewrite,schematically,
|up>|ready>−→|up>|red> (23.18)Similarly,iftheparticlestartsinaspin-downstate,
|down>|ready>−→|down>|green> (23.19)Nowforthecrucialpoint. TheSchrodingerequationisalinearequation. This
meansthatif
ψa(x,t) and ψb(x,t) (23.20)
arebothsolutionsoftheSchrodingerequation,andiftheintialstateis
Ψ 0 =Aψa(x,0)+Bψb(x,0) (23.21)thenthecorrespondingfinalstate(attimet=T,say)is
Ψf=Aψa(x,T)+Bψb(x,T) (23.22)Suppose,then,thattheelectronenteringthedetectorisinthespinstate
|ψ>=A|up>+B|down> (23.23)sothattheinitialparticle-detectorstateis
|Ψ 0 > = |ψ>|ready>
= A|up>|ready>+B|down>|ready> (23.24)Then, from the linearity of the Schrodinger equation, we can conclude that the
particle-detectorsystemmustendupintheentangledfinalstate
|Ψ>=A|up>|red>+B|down>|green> (23.25)