3.2. THEDOUBLE-SLITEXPERIMENT 39
whereθistheangle,relativetothex-axis,fromthecenterofthebarriertothepoint
yonthescreen.
Nowtheintensityofawaveisproportionaltothesquareofitsamplitude. The
generalizationtocomplexamplitudesisthattheintensityisproportionaltothesquare
ofthemodulusoftheamplitude,i.e.
I∝ψ∗ψ (3.24)TheintensityofthedeBrogliewavearrivingatpointyonthescreenistherefore
I(y)∝cos^2(
pdsinθ
2 ̄h)
(3.25)Finallywecanmakecontactwithparticlemotion,because”intensity”isaconcept
whichappliesnotonlytowaves,butalsotoabeamofparticles.Intensityistheenergy
passingthroughasurface,perunitarea,perunittime.Let
N(y)=no.ofelectronsperunitareaperunittime (3.26)whichreachthescreenintheneighborhoodofpointy. Sinceeachelectronhasan
energyE=p^2 / 2 m,itfollowsthat
I(y)=N(y)p^2
2 m(3.27)
Comparingthisexpressionto(3.24)and(3.25),wehaveapredictionfromdeBroglie’s
theorythat
N(y) ∝ ψ∗ψ∝ cos^2(
pdsinθ
2 ̄h)
(3.28)whichcanbecomparedtoexperiment,simplybycountingthenumberofelectronsar-
rivingpersecondatdifferentregionsonthescreen.TheresultoftheDavisson-Germer
experiment,andotherexperimentscloselyanalogoustothedouble-slitexperiment,is
thatdeBroglie’spredictionisconfirmed.Justasphotonshaveparticle-likeproperties,
electronsundoubtablyhavewave-likeproperties.
SupposethatNtotalelectronsgetpast thebarrier. Letsaskthequestion: what
istheprobabilitythatanyoneoftheseelectronswillpassthroughasmallarea∆A
ofthescreen,centeredatpointy,inonesecond? Assumingthattheparticlesmove
independently,thisprobabilityis
prob.tocross∆A/sec=N(y)∆A
Ntotal