4.2. HILBERTSPACE 49
andthisistheendofthedigression.
Returningtothe”BornInterpretation”ofthevectorψ%(or”|ψ>”),weseethat
thetotalprobabilityfortheelectrontobefoundsomewhereinthetubeis
Ptotal = P 1 +P 2 +...+PN
= ψ 1 ∗ψ 1 +ψ∗ 2 ψ 2 +...+ψ∗NψN
= ψ%·ψ%
= <ψ|ψ> (4.13)Fromthefactthattheprobabilitytofindtheparticlesomewhereinthetubeis100%,
wecanconcludethat|ψ>isaunitvector,i.e.
<ψ|ψ>= 1 (4.14)Wecannowviewelectronmotioninthefollowingway: Thephysicalstateofthe
electronisrepresentedbyunitvectors|ψ>inacomplexN-dimensionalvectorspace.
SincethetipofaunitvectorfallsonthesurfaceofsphereofradiusR=1,inthe
N-dimensionalspace,themotionoftheelectroncorrespondstoatrajectoryalongthe
surfaceofaunitsphereinN-dimensionalspace. Inthisway,anelectroncantravel
smoothlyandcontinuously,withno”jumps”,frome.g. theinterval1,represented
byvector%e^1 totheinterval2,representedby%e^2 ,asshowninFig. [4.3].Ofcourse,
ingoingfrominterval 1 tointerval2,theelectronpassesthroughintermediatestates
suchas(4.4),wheretheelectroncannotbeviewedasbeingeitherininterval 1 orin
interval2.Buttheelectronisalwayslocatedatadefinitepointonthesurfaceofthe
unit-sphere,anditisthissurface,ratherthanthelineoflengthLalongthetube,
whichshouldberegardedasthearenaofelectronmotion.
Tocompletethisnewrepresentationofmotionweneedtotakethelimit!→0,
whichsendsthenumberof intervalsN →∞. Then thephysical states|ψ>are
vectorsof unitlength inan infinitedimensional vector space knownas Hilbert
Space.
4.2 Hilbert Space
Anyfunctionf(x)withthepropertyofsquareintegrability
∫∞
−∞dxf∗(x)f(x)<∞ (4.15)canberegardedasavector,andadeBrogliewavefunctionψ(x,t)can,inparticular,
beregardedasavectorofunitlength.
Recallthatavector%visrepresentedbyasetofnumbers(thecomponentsofthe
vector)labeledbyanindex. Thereisonenumber,denoted,viforeachintegervalue