QMGreensite_merged

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)