56 CHAPTER4. THEQUANTUMSTATE
Aphysicalstatehasthe propertythat,giventhestateat sometimet,wecan
determinethestateataslightlylatertimet+!.Inclassicalmechanics,theruleis
qa(t+!)=qa(t)+!
(
∂H
∂pa
)
t
pa(t+!)=pa(t)−!
(
∂H
∂qa
)
t
(4.58)
Thephysicalstate|ψ>inquantummechanics alsohasthisproperty. Giventhe
wavefunction ψ(x,t)at anyparticular timet, thewavefunctionat aslightlylater
timet+!isdetermined,foraparticlemovingfreelyinonedimension,fromthewave
equationfordeBrogliewaves
ψ(x,t+!)=ψ(x,t)+i!
̄h
2 m
∂^2
∂x^2
ψ(x,t) (4.59)
TheBornInterpretationtellsushowtousethewavefunctionψ(x,t)tomakecer-
tainexperimentalpredictions.Unlikethepredictionsofclassicalmechanics,however,
whichspecify theoutcomeof anymeasurementon asinglesystemwithcertainty,
giventhephysicalstate{qi,pi}ofthesystem,thepredictionsofquantummechanics
arestatisticalinnature.TheBornInterpretationtellsusthattheprobabilitytofind
theparticleinasmallintervalaroundthepointx,oflength∆L,is
P∆L(x)=ψ∗(x,t)ψ(x,t)∆L (4.60)
Themeaningofthisprediction,inpractice, isas follows: Supposewehaveavery
largenumberofparticlesNp whichareinidentical physical statesdescribedbya
certain wavefunction ψ(x,t). If we measurethe position of all of these particles
simultaneously,thenumbernxwhichwillbeobservedtobeintheinterval∆Laround
pointxispredicted,accordingtotheBornInterpretation,tobe
nx
Np
= P∆L(x) + O(Np−^1 /^2 )
= ψ∗(x,t)ψ(x,t)∆L + O(Np−^1 /^2 ) (4.61)
ThetermoforderNp−^1 /^2 isstatisticalerror,whichcanbemadearbitrarilysmallby
choosingasufficientlylargenumberofparticlesNp.
Tounderstandtheoriginofthestatisticalerror,considerflippingacoinNtimes,
whereNisaverylarge,evennumber.Onlyveryrarelydoesthecoincomeupheads
exactlyN/ 2 times. UsuallythenumberofheadsdeviatesfromN/ 2 byaamounton
theorderof∆N∼
√
N. Theratioofthedeviation∆N tothetotalnumberofcoin
flipsN varies eachtimeonedoesthefullsetofN coinflips,but itisgenerallyof
order
∆N
N