QMGreensite_merged

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.