QMGreensite_merged

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)