8.4. THEFINITESQUAREWELL:BOUNDSTATES 135
- Theenergiesarenon-degenerate;
- Eachenergyeigenstateiseitherevenparityoroddparity,andthisisaconsequence
ofthesymmetryofthepotentialV(x)=V(−x),andthefactthattheenergies
arenon-degenerate. - Althoughtheparticleisboundtothewell,thewavefunctionoftheparticlemay
benon-negligiblefaroutsidethewell,where(inclassicalphysics)thereareno
forcesatallactingontheparticle.
ScatteringStates E> 0
ForE>0,thesolutionofthetime-independentSchrodingerequationinallthree
regionsis
φIII(x) = Aei
√
2 mEx/ ̄h+Be−i
√
2 mEx/ ̄h
φII(x) = Cei
√
2 m(E+V 0 )x/ ̄h+De−i
√
2 m(E+V 0 )x/ ̄h
φI(x) = Eei
√
2 mEx/ ̄h+Fe−i
√
2 mEx/ ̄h (8.68)
Asexplainedintheintroduction,thewavefunctioninregionsIandIIIcanbethought
of asasuperpositionof incomingandoutgoingwavepackets, inthe limitthatthe
uncertaintyinmomentumofthewavepacketsgoestozero.Themomentumiseither
p=
√
2 mE,inwhichcasethepacketismovingfromlefttoright,orp=−
√
2 mE,
inwhichcasethepacketismovingfromrighttoleft.Therefore,wecaninterpretthe
componentsofφIIIandφIasrepresenting:
Aei
√ 2 mEx/ ̄h
anincomingwavepacketfromtheleft
Be−i
√
2 mEx/ ̄h anoutgoingwavepackettotheleft
Eei
√ 2 mEx/ ̄h
anoutgoingwavepackettotheright
Fe−i
√
2 mEx/ ̄h anincomingwavepacketfromtheright
Nowsupposeweareinterestedinanexperimentalsituationinwhichaparticleofa
fairlyprecisemomentumissenttowardsthepotentialwellfromtheleft.Inthatcase
thereisnoincomingwavepacketfromtheright,sowecansetF=0.Theincoming
wavepacketeventuallyreachesthepotentialwell. Someoftheincomingwavepacket
isreflectedtowardstheleft;someistransmittedtowardstheright.Sowenowhave
φIII(x) = Aei
√
2 mEx/ ̄h+Be−i
√
2 mEx/ ̄h
φII(x) = Cei
√
2 m(E+V 0 )x/ ̄h+De−i
√
2 m(E+V 0 )x/ ̄h
φI(x) = Eei
√
2 mEx/ ̄h (8.69)