Chapter 5
Dynamics of the Quantum State
Historyisjustonedamnthingafteranother.
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Theclassicalmotionofaparticleisrepresentedbyatrajectoryin3-dimensional
space,whilethequantum-mechanicalmotionofaparticle,accordingtothelastchap-
ter,issupposedtobedescribedbyacurveontheunitsphereofaninfinitedimen-
sionalspace.Howcanclassicalphysicsbeinanysenseanapproximationofquantum
physics,ifthearenaofmotionissoverydifferentinthetwotheories?
Theansweristhis:itisnotthephysicalstateoftheparticle,butratherourobser-
vationsofitsposition,thatshould,insomelimit,approximateaclassicaltrajectory.
Andalthoughthe quantumstateof aparticle doesn’tcorrespond,ingeneral,toa
definitepointin3-space,itisstilltruethattheexpectationvalueofparticleposition
attimet
<x>≡∫
dxxψ∗(x,t)ψ(x,t) (5.1)tracesa trajectory throughordinary space. Givenan equationof motionforthe
quantumstateψ(x,t),itshouldbepossibletoderivelawsofmotionfor
comparethemwiththeclassicallawsofmotionforx(t). Thesimplestpossibilityis
thatthesetwosetsoflawslookthesame. Thisisknownas”Ehrenfest’sPrinciple”.
5.1 Ehrenfest’s Principle
Let{qa,pa}bethe generalized coordinatesandmomentaof amechanical system.
Ehrenfest’sPrincipleisthestatementthatHamilton’sequationsofmotionarevalid
asexpectationvalueequationsinquantummechanics: