4.4. EXPECTATION,UNCERTAINTY,ANDTHEQUANTUMSTATE 55
Asanexample,weprovethethirdoftheseidentitiesusingtheformulaforinte-
grationbyparts:
∫∞−∞dxf(x)d
dxδ(x−y) = lim
L→∞∫∞−∞dxf(x)d
dxδL(x−y)= lim
L→∞[
f(x)δL(x−y)|xx==∞−∞−∫∞−∞dxdf
dxδL(x−y)]= lim
L→∞∫∞−∞dx[
−df
dxδL(x−y)]=
∫∞−∞dx[
−df
dx]
δ(x−y) (4.56)wheretheboundarytermsaredroppedbecauseδL(±∞)=0.
Problem- Provetheotherthreedelta-functionidentitiesabove,inthesenseofeq.
(4.55)
4.4 Expectation, Uncertainty, and the Quantum
State
Inclassicalmechanics,thephysicalstateofasystemisspecifiedbyasetofgeneralized
coordinatesandmomentum{qi,pi},whichisapointinthephasespaceofthesystem.
Inthecourseoftime,thephysicalstatetracesatrajectorythroughthephasespace. In
thecaseofasingleparticlemovinginthreedimensions,thephysicalstateisdenoted
{%x,%p},andthephasespaceis6-dimensional. Theprojectionofthetrajectoryinthe
6-dimensionalphasespaceontothethreedimensionalsubspacespannedbythex,y,
andzaxes,orinotherwords,thepath%x(t),isthetrajectorywhichwecanactually
seetheparticlefollow.
Inquantummechanics,thephysicalstateofapointlikeparticle,movinginone
dimension, isspecifiedateachmomentoftimebyawavefunctionψ(x,t). Atany
giventimet, thiswavefunction isafunctiononlyofx, andcan be regardedas a
vector|ψ>inHilbertspace.Becauseofthenormalizationconditionimposedbythe
BornInterpretation
<ψ|ψ>=∫
dxdydzψ∗(x,y,z,t)ψ(x,y,z,t)= 1 (4.57)|ψ >isnecessarilyavector ofunitlength. Inthecourseof time,|ψ>followsa
paththroughHilbertspace. Inroughanalogytothemotionofunitvectorsinfinite
dimensional spaces,onecouldimaginethatthe tipofthe unitvector |ψ >traces
apathon thesurface ofaunitsphere,althoughinthiscasethe spaceisinfinite-
dimensional.