96 CHAPTER6. ENERGYANDUNCERTAINTY
anditremainsstaticbecausetheforceontheparticlevanishes,i.e.
F=−
∂V
∂x= 0 (6.67)
Ontheotherhand,inclassicalmechanicsthekineticenergyofastationarystate
iszero. Thisisnotthecaseinquantummechanics. Fortheexampleoftheparticle
inatube,theexpectationvalueofmomentumvanishesinanenergyeigenstate
<p>n =∫L0dxφ∗n(x)p ̃φn(x)=
2
L
∫L0dxsin(nπx
L)
(
−i ̄h∂
∂x)
sin(nπx
L)
= 0 (6.68)
However,asrequiredbytheUncertaintyPrinciple,theuncertaintyinmomentumis
notzero:
(∆pn)^2 =∫L0dxφ∗n(x)(p ̃−<p>n)^2 φn(x)=
2
L
∫L0dxsin(nπx
L)
(
−i ̄h∂
∂x) 2
sin(nπx
L)
=
n^2 ̄h^2 π^2
L^2(6.69)
Theenergyeigenvalues(inthisexample)aresimplygivenbythekineticenergywhich
isduetothisuncertainty
En=(∆pn)^2
2 m=
n^2 π^2 ̄h^2
2 mL^2(6.70)
Thereisnoadequateclassicalpicture,eitherintermsofaparticlefollowingatra-
jectory(in whichcase
casethekineticenergyvanishes),oftheenergyeigenstates. Thesearebestpictured
asstandingwaves,which,atamorefundamentallevel,representthecomponentsof
vectorsinHilbertspace. Nevertheless,whenstandingwavesofdifferentenergiesare
superimposed,itispossibletoproduceawavepacketwhich,aswehaveseeninthe
lastlecture,roughlyfollowsaclassicaltrajectoryaccordingtoEhrenfest’sPrinciple.
Thisisanimportantpointtokeepinmind: Dynamics,non-stationarity,changeof
anysortinquantummechanics,impliesanuncertaintyinthevalueofenergy. Ifthe
valueoftheenergyiscertain,thesystemisinanenergyeigenstate,andifthesystem
isinanenergyeigenstate,thesystemisstationary.