128 CHAPTER8. RECTANGULARPOTENTIALS
ThechoiceofAandDisachoiceof theinitialstate. IfD= 0 thentheparticle
is initially approaching the potentialstep fromthe left; if A = 0 the particle is
approachingfromtheright.IfbothAandDarenon-zero,thentheinitialwavepacket
issplitintotwopieces,oneapproachingfromtheleftandonefromtheright. Itis
possibleto havea initialwavepacket whichis discontinuous inthisway, but itis
uncommon. Thestandardexperimental setupwould be to have aparticle source
locatedtooneside ortheotherofthescatteringregion. Wewillsupposethatthe
particlesarecominginfromtheleft,andsetD=0.Inthatcasetheremainingpieces
ofthewavefunctionhavetheinterpretationastransmittedandreflectedwaves
φref(x) = Be−ip^1 x/ ̄h (x<0)
φtrans(x) = Ceip^2 x/ ̄h (x>0) (8.25)
Todeterminethereflectionandtransmissioncoefficients,weneedBandCinterms
ofA.
ThesolutionoftheSchrodingerequationmustbeacontinuousfunction,andits
firstderivativemustalsobecontinuous. Ifthiswerenottrue,thesecondderivative
wouldbeinfinite,whichviolatestheSchrodingerequationiftheenergyandpotential
arefinite.^2 Imposingcontinuityofthewavefunctionφanditsfirstderivativeφ′at
x= 0 givetheconditions
φ 1 (0)=φ 2 (0) =⇒ A+B=C
φ′ 1 (0)=φ′ 2 (0) =⇒ p 1 (A−B)=p 2 C (8.26)
ThesetwoequationsallowustosolveforBandC intermsofA:
B =
p 1 −p 2
p 1 +p 2
A
C =
2 p 1
p 1 +p 2
A (8.27)
TheresultingwavefunctionissketchedinFig.[8.12].
Recallthatintensityisproportionaltothe(group)velocity,I=vφ∗φ,andinthis
casethevelocityofthetransmittedwavev 2 =p 2 /misdifferentfromthevelocityof
theincidentandreflectedwavesv 1 =p 1 /m. Thetransmissioncoefficientistherefore
T =
v 2 |C|^2
v 1 |A|^2
=
p 2
p 1
4 p^21
(p 2 +p 1 )^2
(8.28)
(^2) Anexceptionisifthepotentialbecomesinfiniteatsomepoint,asinthecaseofaparticleina
tube.Insuchsituations,theslopeofthewavefunctioncanbediscontinuous.