6.3. THEENERGYOFENERGYEIGENSTATES 95
state”). Energyeigenstatesofhigherenergiesareknownas”excitedstates.”The
factthatthegroundstateenergyoftheparticleinthetubeE 1 isgreaterthanzero
isanotherexampleoftheUncertaintyPrincipleatwork. Sincethethe particleis
confinedinaregionoflengthL,itmeansthat∆x≤L,andtherefore
∆p>̄h
2 L(6.61)
Assuming
=0,thiswouldgivealowerboundtotheenergyofroughly
E≈
(∆p)^2
2 m>
̄h^2
8 mL^2(6.62)
andthislowerboundisinfact smaller,bysome numerical factors, than thetrue
groundstateenergyE 1 givenin(6.59),inagreementwiththeUncertaintyPrinciple.
Notethatthedependenceonh, ̄m,LofthelowerboundisthesameasthatofE 1.
Again, itisnotaquestion of ”theobservationdisturbing theobserved.” Thereis
simplynophysicalstateofaparticleinatubewhichhasanenergylowerthanE 1.
Energyeigenstatesarestationarystates,inthesensethatthetime-dependenceof
thewavefunctionisentirelycontainedinanoverallphasefactor
ψα(x,t)=φα(x)e−iEαt/ ̄h (6.63)andthisphasecancelsoutwhencomputingexpectationvaluesofanyfunctionofx
andp,e.g.
<x> =∫
dxψα∗(x,t)xψα(x,t)=∫
dxφ∗α(x)xφα(x)<p> =∫
dxψα∗(x,t)p ̃ψα(x,t)=∫
dxφ∗α(x)p ̃φα(x) (6.64)Therefore
∂t<x> = 0<−∂V
∂x> = 0 (6.65)
wherethesecondequationfollowsfrom∂t
=0,andEhrenfest’sPrinciple.Note
that theseequations aresimilar to thecorresponding expressionsfor astationary
stateinclassicalmechanics:theparticleisstatic,so
∂tx= 0 (6.66)