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28 CHAPTER2. ORIGINSOFQUANTUMMECHANICS

Thisimpliesthatthemoreaccuratelywe measuretheparticle’sposition(∆xvery
small),themoreuncertaintythereisinitsmomentum(∆pverylarge),andviceversa.
Aconcrete,althoughmuchoversimplifiedarrangementformeasuringparticlepo-
sitionisknownas”Heisenberg’smicroscope”;thesetupisshowninFig. [2.5]. The
ideaistomeasurethepositionofaparticlemovingalongthex-axiswithaknown
momentumpx,toanaccuracy∆x. Thisisdonebyshininglightontheparticle. The
particlescatterstheincidentlight,andthisscatteredlightisbroughttoafocalpoint
byalensofdiameterD.However,theabilityofalenstodeterminepositionislimited
bytheresolutionofthelens.
ConsidertwopointsourcesoflightadistanceLfromthelens,whichareseparated
byaninterval∆x. Resolutionofthesetwopointsrequiresresolutionofanangular
separation

φ=

∆x

L

(2.24)

Ontheotherhand,accordingtotheRayleighcriterion,thesmallestangularsepara-
tionswhichcanberesolvedbyalensofdiameterDisoftheorder

φmin≈

λ

D

(2.25)

Equatingthesetwoangles,wefindthatthesmallestpossibleseparationbetweentwo
pointswhichcanberesolvedbythelensis

∆x≈

λ

D/L

≈

λ

sinθ

(2.26)

Ifalenscannotresolvepointsourcesseparatedbyintervalslessthan∆x,itisalso
unabletoresolvethepositionofasinglepointsourcetoanaccuracybetterthan∆x.
Becausealens focusesall lightatacertain imagepoint,wecannot determine
atexactlywhatanglethelightwasscattered,relativetothey-axis. Scatteredlight
willreachthelensatanyscatteringanglebetween 0 andθ. Butthatmeansthatthe
x-componentofthemomentumofthescatteredphotonisuncertainbyapproximately

∆px≈psinθ=

h

λ

sinθ (2.27)

Multiplying(2.24)by(2.27)givestheuncertaintyrelation

∆x∆px≈h (2.28)

asbefore.
Aphysicalstateissupposedtobedeterminedbymeasurement,butfromthere-
lation(2.28)weseethatmeasurementsofpositionandmomentumcannotbemade
simultaneously, toarbitrarilyhighprecision,iflightisusedtoobservetheparticle
position. Thisisthefirsthintthatsomethingmaybewrongwiththeclassicalview