8.3. THESTEPPOTENTIAL 127
andtheTransmissionCoefficient
T =
Itrans
Iinc=
|C|^2
|A|^2
=
no.ofparticles/sectransmitted
no.ofparticles/secincident(8.19)
Ofcourse, theratiosB/AandC/Acan onlybe obtainedby actuallysolvingthe
time-independentSchrodingerequation,whichwewilldo,inthislecture,forsimple
rectangularpotentials.
8.3 The Step Potential
ConsideraparticlemovinginapotentialwhichrisessuddenlyfromV(x)=0,at
x<0,toV(x)=V,atx≥0,asshowninFig. [8.11]. IfE>V thentheentire
reallineisaclassicallyallowedregion;anparticlemovingtotherightwillpassthe
potentialbarrierandcontinuemovingtotheright.Ontheotherhand,ifE<V,the
half-linex> 0 isaclassicallyforbiddenregion;anincomingparticlewillbereflected
atx= 0 andmoveawaytotheleft.Wewillconsiderthesetwocasesseparately:
Energies E>V
TheSchrodingerequationinregionI(x<0)istheequationforafreeparticle−
̄h^2
2 m∂^2 φ 1
∂x^2=Eφ 1 (8.20)withtheusualfreeparticlesolution
φ 1 (x)=Aeip^1 x/ ̄h+Be−ip^1 x/ ̄h where p 1 =√
2 mE (8.21)InregionII(x>0)theequationis
−
̄h^2
2 m∂^2 φ 2
∂x^2=(E−V)φ 2 (8.22)withthesolution
φ 2 (x)=Ceip^2 x/ ̄h+De−ip^2 x/ ̄h where p 2 =√
2 m(E−V) (8.23)Thescatteringregion isat x= 0,thisis where∂V/∂x+= 0. Thepart of the
wavefunctionthatrepresentsa(verylong)wavepacketmovingtowardsthescattering
regionis
φinc(x)={
Aeip^1 x/ ̄h x< 0
De−ip^2 x/ ̄h x> 0