QMGreensite_merged

Chapter 6

Energy and Uncertainty

Wehaveseenthatforagaussianwavepacket

φ(x)=

1

(πa^2 )^1 /^4

e−(x−x^0 )

(^2) / 2 a 2
eip^0 x/ ̄h (6.1)
theuncertaintyinpositionis∆x=a/

√

2,whichmeansthatthisuncertaintyvanishes
asa→0. Inthislimit,φ∗φapproachesaDiracdeltafunction,andalltheprobability
Pdx(x)isconcentratedatthepointx=x 0. Ifaparticleisinsuchaquantumstate
when itspositionismeasured, itiscertain tobefoundat thepointx= x 0 ; this
quantumstateisknownasanEigenstateofPosition.

Ontheotherhand,asa→∞,thegaussianwavepacket(6.1)approachesaplane
wave. AccordingtodeBroglie,theplanewaverepresentsaparticlemovingwitha
definitemomentump 0 ,thereforetheuncertaintyinmomentumshouldbe∆p= 0 in
thatlimit. Iftheparticleisinaplanewavestatewhenitsmomentumismeasured,
itiscertaintobefoundtohavemomentump=p 0 ;thisquantumstateisknownas
anEigenstateofMomentum.

Becausetheeigenstatesofpositionandmomentumareverydifferent,itisimpossi-
bleforaparticletobeinaphysicalstateinwhichbothitspositionanditsmomentum
arecertain;i.e. ∆x=∆p=0. Asimpleconsequenceisthatnomeasurementcan
determine,preciselyandsimultaneously, thepositionandmomentumofaparticle.
Ifsuchmeasurementswerepossible,theywouldleavetheparticleinaphysicalstate
having∆x=∆p=0,andnosuchquantumstateexists. Amorequantitativeexpres-
sionofthislimitationonmeasurementsisknownastheHeisenbergUncertainty
Principle.Toderiveit,weneedtobeabletocompute∆pforanarbitraryphysical
state.

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