QMGreensite_merged

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)