8.5. TUNNELLING 139
cansimplytaketheformulasforthescatteringstateoftheattractivewell,replace
V 0 everywhereby−V 0 ,andthesewillbethecorrespondingformulasfortherepulsive
well.Thenagaindenote
k=
√
2 mE
̄h
and q=
√
2 m(E−V 0 )
h ̄
(8.87)
sothat
φIII(x) = Aeikx+Be−ikx
φII(x) = Ceiqx+De−iqx
φI(x) = Eeikx (8.88)
Classically, ifE > V 0 , the incoming particlewill travel through the welland
continuemovingtotheright.IfE<V 0 theincomingparticleisreflectedatx=−a,
andtravelsbacktotheleft.
Inquantummechanics, ifE >V 0 , thesituationisqualitativelymuchlike the
casefortheattractivepotential: someoftheincomingwaveisreflected,andsomeis
transmitted.ItisworthnotingthatnomatterhowlargeEiscomparedtoV 0 ,there
isalwayssomefiniteprobabilitythattheparticleisreflected.Quantum-mechanically,
ifabulletisfiredatafixedtargetoftissuepaper,thereisalwayssome(exceedingly
small)probabilitythatthebulletwillbounceoffthepaper.
IfE<V 0 ,thenthequantity
q=
√
2 m(E−V 0 )
̄h
(8.89)
isimaginary.Substituting
q=iQ and sin(2qa)=−isinh(2Qa) (8.90)
intoequation(8.80)givesus
T =
1
cosh^2 (2Qa)+^14 (Qk+Qk)^2 sinh^2 (2Qa)
R =
1
4 (
Q
k−
k
Q)
(^2) sinh^2 (2Qa)
cosh^2 (2Qa)+^14 (Qk+Qk)^2 sinh^2 (2Qa)
(8.91)
Ifthewidthofthewellislargeenough,sothat
2 a>>
1
Q
=
̄h
√
2 m(V 0 −E)
(8.92)
then
sinh^2 (2Qa)≈cosh^2 (2Qa)≈
1
4
e^2 Qa (8.93)