3.3. DOELECTRONSTRAVELALONGTRAJECTORIES? 41
TheBornInterpretation
Denote the wavefunction associated with an electron by ψ(x,y,z,t).
Then theprobabilityP∆V attime t,offinding theelecroninsideasmall
volume∆V centeredatpoint(x,y,z)is
probability P∆V(x,y,z)=ψ∗(x,y,z,t)ψ(x,y,z,t)∆V (3.32)Ifanelectronexists,thenthetotalprobabilitythatitissomewhereintheUniverse
is100%. Therefore, acorrollaryof theBorninterpretationistheNormalization
Condition ∫
dxdydzψ∗(x,y,z,t)ψ(x,y,z,t)= 1 (3.33)
whichmustbesatisfiedbyanywavefunctionassociatedwithanelectron.
3.3 Do Electrons Travel Along Trajectories?
Thedouble-slitexperimentisarrangedinsuchawaythatequalnumbersofelectrons,
persecond,passthroughslitsAandB.ThiscanbecheckedbyclosingslitB,counting
thetotalnumberofelectronspersecondreachingthescreen,andthenclosingslitA,
andcountingagain.Ifthecounts/secareequal,then,whenbothslitsareopen,the
chancethatanelectronreachingthescreenpassedthroughslitAis50%,withan
equal50%chancethatitpassedthroughslitB.
WecandetermineexperimentallytheprobabilityPA(y)thatanelectron,having
passedthroughslitA,willlandonadetectorplacedatpointyonthescreen.Close
slitB,leavingAopenandcountthenumberofparticlesnythatlandonthedetector,
andthetotalnumbernT landinganywhereonthescreen. Then
PA(y)=ny
nT(3.34)
Inasimilarway,byopeningslitBandclosingslitA,wecandeterminetheprobability
PB(y). Then, withbothslits open, the probabilitythat anelectron that passed
throughthebarrierwilllandonthedetectoratyis
P(y) = (prob. ofcomingfromslitA)×PA(y)
+ (prob. ofcomingfromslitB)×PB(y)=1
2
[PA(y)+PB(y)] (3.35)Clearly,ifeitherPA(y),orPB(y)(orboth)arenon-zero,thentheprobabilityP(y)is
alsonon-zero.