QMGreensite_merged

4.4. EXPECTATION,UNCERTAINTY,ANDTHEQUANTUMSTATE 57

ThedeviationoforderNp−^1 /^2 ineq. (4.61)hasthesameorigin,andispresentinany
setofmeasurementswhichinvolverandomprocesses. Fromnowonwewillsuppose
thatNpissolargethatthisdeviationisignorable.
InsteadofsimultanouslyobservingaswarmofNpparticles,eachofwhicharein
thesamestate,onecouldinsteadperformanexperimentinwhichasingleparticle
isplacedinaparticularstateψ(x,t),itspositionisobserved,andthentheidentical
experimentisrepeatedNp times. Ifnxisthenumberofexperimentsinwhichthe
particlelandsintheinterval∆Laroundpointx,thenthepredictionaccordingtothe
BornInterpretationisagaineq. (4.61).Infact,thisiswhatisactuallyhappeningin
theelectron2-slitexperiment.Electronscanbemadetopassonebyonethroughthe
slits,andthewavefunctionofeachelectronatthelocationofthescreenisidentical.
The statistical prediction for the number of electrons reaching aparticular small
regionofthescreenistheneq. (3.28)ofthelastlecture.
Letusimaginemakingaseriesofexperimentsonthemotionofaparticlemoving
inone dimension, in which the particle issomehow initializedto be ina certain
quantumstate|ψ>,andthenthepositionoftheparticle ismeasured. Thereare
two particularly useful numbers whichcan bepredicted by quantum theory, and
measuredby theseriesofexperiments. Onenumberistheaveragevaluexofthe
particleposition. Letxibethepositionoftheparticleinthei-thexperiment,and
NE be the total number of experiments. Then the averagevalueof the particle
positionisdefinedas

x≡

1

NE

N∑E

i=1

xi (4.63)

Wecanalsodefinetheaveragevalueofx^2 as

x^2 ≡

1

NE

∑NE

i=1

(xi)^2 (4.64)

Theotherusefulnumberistheaveragesquaredeviationδx^2 oftheobservedpositions
xifromtheiraveragevaluex. Theaveragesquaredeviationisdefinedas

δx^2 ≡

1

NE

∑NE

i=1

(xi−x)^2 (4.65)

or,expandingthequantityinparentheses

δx^2 =

1

NE

∑NE

i=1

[x^2 i− 2 xix+x^2 ]

= x^2 − 2 xx+x^2

= x^2 −x^2 (4.66)

The”root-mean-square”deviationδxisjustthesquare-rootofδx^2 .Thesetwonum-
bers,xandδx,canbereportedasresultoftheseriesofmeasurements.