58 CHAPTER4. THEQUANTUMSTATE
TheBornInterpretationenablesustopredictthevaluesofthesetwonumbers,
giventhequantumstate|ψ>.Thesepredictedvaluescomefromprobabilitytheory.
SupposewemakeaseriesofmeasurementsonthequantityQ,whichcanonlytake
onthepossiblevaluesQ 1 ,Q 2 ,Q 3 ...,QnQ;andsupposethecorrespondingprobabilities
of findingthesevalues,inany givenmeasurement, isP 1 ,P 2 ,P 3 ....,PnQ. Then the
expectedaveragevalue,or”ExpectationValue”ofQisdefinedas
=
∑nQ
n=1
QnPn (4.67)
wherenQisthenumber(whichcouldbeinfinite)ofpossiblevaluesoftheobservable
Q.Thistheoreticalexpectationvalueshouldequal(uptostatisticalerror)theaverage
value
Q=
1
NM
∑nQ
n=1
NnQn (4.68)
whereNnisthenumberofmeasurementsthatfoundthevalueQ=Qn,andNM is
thetotalnumberofmeasurements
NM=
∑nQ
n=1
Nn (4.69)
(One shouldbecarefulnot toconfuse theindexiineq. (4.63), whichlabels the
experiment, with theindex nin eq. (4.67), whichlabelsthe possible values the
observableQ.)ThereasonthatandQshouldequaloneanotheristhatthe
fractionoftimesthatthevalueQ=Qnisobservedshouldequaltheprobabilityof
observingQninanygivenmeasurement,i.e.
Pn=
Nn
NM
(4.70)
Insertingthisexpressionfortheprobabilityinto(4.67),onefinds=Q.
Inthecaseofaparticlemovinginonedimension,oneofthepossibleobservables
isthepositionx.Sincexcantakeanyvalueinacontinousrange,thesumin(4.67)is
replacedbyanintegral,andtheprobabilityofobservingaparticleinaninfinitesmal
intervaldxaroundthepointxis
Pdx(x)=ψ∗(x,t)ψ(x,t)dx (4.71)
Withthesemodifications,theformulaforexpectationvaluespredicts
<x> =
∫∞
−∞
xPdx(x)
=
∫∞
−∞
xψ∗(x,t)ψ(x,t)dx (4.72)