42 CHAPTER3. THEWAVE-LIKEBEHAVIOROFELECTRONS
NowtheprobabilitiesPA(y)andPB(y)arenearlyconstantfordetectorsplaced
inasizableregionofthescreen (seeFig. [3.4]),whichisconsistentwiththe fact
thatthemoduliofthedeBrogliewavesψAandψBare(nearly)constant.ThenP(y)
shouldbenearlyconstantaswell.
Infact,P(y)isnotconstantatall.Instead,itgoeslike
P(y) ∝ cos^2(
pdsinθ
2 ̄h)∝ ψ∗(y,t)ψ(y,t) (3.36)Inparticulartherearecertainpoints,satisfying
pdsinθ=2 ̄h(n+1
2
)π (3.37)whereP(y)= 0 eventhoughPA(y)andPB(y)arenon-zero! Inotherwords,ifone
closesslitB,leavingAopen,thereisafinitechancethattheelectronwillreachsuch
apoint. IfoneclosesAandleavesBopen,thereisafinitechancethattheelectron
willreachthatpoint.ButifbothslitsAandBareopen,theelectronwillneverreach
suchapoint,iftheanglefromtheslitstothepointsatisfies(3.37).Clearlythen,in
contrastto(3.35),
P(y) += (prob. ofcomingfromslitA)×PA(y)
+ (prob. ofcomingfromslitB)×PB(y) (3.38)Whenaconclusion(eq.(3.35)inthiscase)turnsouttobefalse,andthereasoning
whichledto the conclusion iscorrect, thentheremustbe somethingwrong with
thepremises. Thepremises inthiscasearethat theelectronsinthebeamtravel
independentlyofoneanother,andthat eachelectroninthe beampassedthrough
either slitA or slitB. More generally, we have assumed that the electron, as a
”pointlikeparticle”,followsatrajectorywhichtakesitthroughoneofthetwoslits
onitswaytothescreen.
Itiseasytocheckwhethertheinterferenceeffectisduetosomecollectiveinter-
actionbetweenelectronsinthebeam.Wecanuseabeamofsuchlowintensitythat
theelectronsgothroughthebarrieronebyone.Ittakesalittlelongertoaccumulate
thedataneededtocomputePA(y),PB(y),andP(y),buttheresultsareultimately
thesame. Sothisisnottheexplanation.
Thenperhapsweshouldstopthinkingoftheelectronasapoint-likeobject,and
startthinkingofitasliterallyadeBrogliewave?Awave,afterall,canpassthrough
bothslits; thatistheoriginofinterference. Theproblemwiththisideaisthatde
Brogliewavesexpand. AtslitsAandB,thewave islocalized justinthenarrow
regionoftheopenslits,butasthewavepropagatestowardsthescreenitexpands,
typically spanning(inarealelectrondiffractionexperiment)aregionontheorder
of 10 cm across. Then onewoulddetect ”parts”of an electron. Assuming ade