QMGreensite_merged

40 CHAPTER3. THEWAVE-LIKEBEHAVIOROFELECTRONS

Butinorderforanelectrontopassthroughasmallregionofthescreensometime
intheintervalt=t 0 andt=t 0 +∆t,anelectronmovingwithvelocityvmusthave
beenlocatedinasmallvolume

∆V=∆A×v∆t (3.30)

infrontofthesurface∆A(seeFig.[3.3]). Therefore,theprobabilitythataparticle
passestheareaelement∆Aintime∆t,isequaltotheprobabilitythattheparticle,
atagiventimet 0 ,islocatedinasmallvolumeinfrontofthescreensize∆V,i.e.

prob.tobein∆V attimet = prob.tocross∆A/sec×∆t

=

N(y)∆A

Ntotal

∆t

=

1

vNtotal

N(y)∆V

∝ ψ∗(y,t)ψ(y,t)∆V (3.31)

Weseethattheprobabilityforanelectrontobeinacertainvolume∆V ispropor-
tionaltothesquareofthemodulusofthedeBrogliewave,timesthevolumeelement.
NowifψisasolutionofthedeBrogliewaveequation(3.16),soisψ′=const.×ψ;
thisfollowsfromthelinearityofthewaveequation.Thereforewecanalwayschoose
asolutionofthewaveequationsuchthat theproportionalitysign ineq. (3.31)is
replacedbyanequalssign. Withthischoice,wearriveattheinterpretationofthe
deBrogliewavefunctionproposedbyMaxBorn: