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38 CHAPTER3. THEWAVE-LIKEBEHAVIOROFELECTRONS

3.2 The Double-Slit Experiment

Weconsiderabeamofelectronswhich,afteraccellerationacrossapotentialdifference
V,aquiresamomentuminthex-direction

p=

√

2 meV (3.18)

Thebeamisdirectedatabarrierwithtwohorizontalslitsofnegligiblewidth,sep-
aratedbyadistanced. Thoseelectronswhichpassthroughtheslitseventuallyfall
onascreen, whichislinedwithdetectors(geigercountersorphotographicplates),
torecordthenumberofelectronsfallingonthescreen,asafunctionofthevertical
distanceyalongthescreen(seeFig.[3.1]).
Letus compute the amplitude of the de Broglie wave at the screen, without
concerningourselves, forthe moment,withtheconnection betweenthedeBroglie
waveandtheactualmotionoftheelectron. Beforeencounteringthebarrier,thede
Brogliewaveisaplanewavemovinginthex-direction,asshowninFig.[3.2].Onthe
othersideofthebarrier,thetwoslitsactascoherentsourcesofcylindricalwaves.To
computetheamplitudeatapointyonthescreen,wesimplysumtheamplitudeof
twowaves,oneoriginatingatslitA,andtheotheratslitB.Theamplitudeatpoint
y,ofthewavepassingthroughslitA,is

ψA(y,t)=Nexp[i(kLAy−ωt)]=Nexp[i(pLAy−Et)/h ̄] (3.19)

whereN isaconstant, pisthemagnitudeof theelectronmomentum,andLAyis
thedistancefromslitAtopointyonthescreen. Similarly, theamplitudeof the
contributionfromslitBis

ψB(y,t)=Nexp[i(kLBy−ωt)]=Nexp[i(pLBy−Et)/ ̄h] (3.20)

Bythesuperpositionprincipleofwavemotion^1 theamplitudeofthedeBrogliewave
atpointyis

ψ(y,t) = ψA(y,t)+ψB(y,t)

= 2 Nexp[i(pLav−Et)/ ̄h]cos

(

p∆L

2 ̄h

)

(3.21)

where

∆L = LBy−LAy

Lav =

1

2

(LAy+LBy) (3.22)

Ifthedistancetothescreenismuchgreaterthantheseparationdoftheslits,then
∆Lisapproximately
∆L=dsinθ (3.23)

(^1) whichactuallyfollowsfromthefactthatthewaveequation(3.16)isalinearequation.