QMGreensite_merged

8.4. THEFINITESQUAREWELL:BOUNDSTATES 135

3. Theenergiesarenon-degenerate;


4. Eachenergyeigenstateiseitherevenparityoroddparity,andthisisaconsequence
   ofthesymmetryofthepotentialV(x)=V(−x),andthefactthattheenergies
   arenon-degenerate.


5. 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)