8.2. UNBOUNDSTATESANDQUANTUMSCATTERING 125
Becauseφ(x)= 0 forx> 0 andbecausethewavefunctioniscontinuousatx=0,we
musthave
φ(x)= 0 ⇒ A=−B (8.9)
or
φ(x)=Nsin[p 0 x/ ̄h] (8.10)
Thestate(8.10)isastationarystate.Substituting(8.10)intothetime-independent
Schrodingerequation,thecorrespondingenergyeigenvalueisfoundtobe
E=p^20
2 m(8.11)
Despitethestationarity,thisenergyeigenstateisobviouslyjustthelimitofanon-
stationarysituation,inwhichanincomingwavepacketisscatteredbackwardsbyan
infinitepotential.Duringthe(long)intervalinwhichtheincomingwavepacket(with
∆pverysmall)reflectsfromtheendofthetube,thewavefunctionneartheendofthe
tubelooksverymuchliketheenergyeigenstate(8.10). Infact,inthe∆p→ 0 limit,
wecan easilyidentifythepart oftheeigenstatethat corresponds totheincoming
wavepacket(φinc)andthepartwhichcorrespondstothescatteredwavepacket(φref).
Thisis ageneral feature of unbound states: an unbound stationarystate can be
viewedas thelimitof adynamicalsituation, inwhichan incomingwavepacketof
averyprecisemomentumisscatteredbyapotential. Partoftheunboundstateis
identifiedastheincomingwave,otherpartsrepresentthescatteredwaves.
Asasecondexample,consideraparticlewavepacketincidentonthepotentialwell
ofFig. [8.1].TheincomingwavepacketisshowninFig.[8.10a]. Uponencountering
thepotential,thewavepacketsplitsintoareflectedwavepacket,movingbackwards
towardsx=−∞,andatransmittedwavepacketwhichhaspassedthroughthewell
andismovingontowardsx=+∞,asshowninFig. [8.10c].^1 Atsomeintermediate
time,theincomingandreflectedwavepacketsoverlap,andaswetake∆p→ 0 forthe
incomingwave,theincomingandreflectedwavesoverlapoveraverylargeregion..
ThelimitingcaseisshowninFig.[8.5].Thisisagainastationarystate,butthewave
totheleftofthepotentialwellisasuperpositionofincomingandreflectedwaves
φleft(x) = φinc(x)+φref(x)
= Aeip^0 x/ ̄h+Be−ip^0 x/ ̄h (8.12)whilethewavefunctiontotherightofthepotentialwellistheoutgoingtransmitted
wave
φright(x) = φtrans(x)
= Ceip^0 x/ ̄h (8.13)(^1) Incidentally,thisisanotherreasonwhyaparticleinquantummechanicscannotbetakenliterally
asawave.Althoughthewavepacketsplitsintoreflectedandtransmittedpackets,theparticleitself
doesnotsplit: thereissomeprobabilitythattheparticlewouldbefoundtoreflect,andsome
probabilitythattheparticleistransmitted,buttheparticleitselfneversplitsintoareflectedparticle
andatransmittedparticle.