136 CHAPTER8. RECTANGULARPOTENTIALS
where
Aei
√
2 mEx/ ̄h representstheincomingwavepacket
Be−i
√
2 mEx/ ̄h representsthereflectedwavepacket
Eei
√
2 mEx/ ̄h representsthetransmittedwavepacket
Thereflectionandtransmissioncoefficientswillthereforebe
R=
BB∗
AA∗
T=
EE∗
AA∗
(8.70)
sotheproblemistocomputeBandEintermsofA.
Weshouldpausetonotethatφ(x)+=±φ(−x). Thereasonisthat inthiscase
theenergyisdegenerate;aparticleapproachingthepotentialfromtheleftcanhave
exactlythesameenergyasaparticleapproachingthewellfromtheright.Inthissitua-
tion,anenergyeigenstatedoesnotnecessarilyhavethepropertythatφ(x)=±φ(−x);
althoughφ(x)=φ(−x)isanenergyeigenstate,itisnotnecessarilyequivalenttoφ(x)
iftheenergyisdegenerate.Forthewavefunctionφ(x)ofeq. (8.69),whichrepresents
aparticleapproachingthewellfromtheleftandthenscattering,φ(x)representsa
particleapproachingthepotentialfromtheright,andthenscattering.
Denote
k=
√
2 mE
h ̄
and q=
√
2 m(E+V 0 )
h ̄
(8.71)
sothat
φIII(x) = Aeikx+Be−ikx
φII(x) = Ceiqx+De−iqx
φI(x) = Eeikx (8.72)
Imposingcontinuityofthewavefunctionanditsfirstderivativeatx=agiveus
φI(a)=φII(a) =⇒ Eeika=Ceiqa+De−iqa
φ′I(a)=φ′II(a) =⇒ kEeika=q(Ceiqa−De−iqa) (8.73)
Thecorrespondingcontinuityconditionsatx=−aare
φIII(−a)=φII(−a) =⇒ Ae−ika+Beika=Ce−iqa+Deiqa
φ′III(−a)=φ′II(−a) =⇒ k(Ae−ika−Beika)=q(Ce−iqa−Deiqa) (8.74)
Wefirstsolveeq. (8.73)forC andDintermsofE:
C =
1
2
(1+
k
q
)Eei(k−q)a
D =
1
2
(1−
k
q
)Eei(k+q)a (8.75)