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.