QMGreensite_merged

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

∼

1

√

N

(4.62)