138 CHAPTER8. RECTANGULARPOTENTIALS
Inthiscase
T= 1 and R= 0 (8.83)
i.e.thereisnoreflectionatall.Aparticleofthisenergyiscertaintobetransmitted
pastthepotentialwell. Thecondition(8.82)hasasimpleinterpretationintermsof
deBrogliewavelengths.Themomentumoftheparticleinsidethewellis
P=
√
2 m(E−V)=
√
2 m(E+V 0 ) (8.84)
sothecondition(8.82)is
2 a
P
h ̄
= nπ
2 a
P
h
2 π = nπ
2 a = n
λ
2
(8.85)
whereweusethedeBroglierelationp=h/λ.Sincethewidthofthepotentialwellis
2 a,itmeansthatthereisnoreflectionwheneverthewidthofthewellisexactlyan
integernumberofhalf-wavelengthsacross,whichisalsotheconditionforastanding
waveonanintervaloflength 2 a.
ThereisarelatedphenomenoninatomicphysicsknownastheRamsauereffect,
inwhich low energy electrons arescattered by athin film of some material. At
certainwell-defined energies, the scatteringof theelectronsis minimized. Inthis
case,anatomcanbeviewedasanattractivepotentialwellfortheelectrons. Since
anatomiselectricallyneutral,thepotentialiszerountiltheincomingelectronsmove
past the electronsorbiting the nucleus. As the electron approachescloser tothe
nucleus, thereis ofcourse an attractive force,since thenucleusis positive. The
analysisof the scatteringismore complicatedthan for thefinitesquare well,the
motionisinthreedimensionsratherthanonedimension, andthepotentialwellis
notrectangular.Nevertheless,thesolidisnearly”transparent”forincomingelectrons
ofcertaindefiniteenergies;aneffectwhichiscloselyanalogoustothefinitesquare
wellcase.
8.5 Tunnelling
Nextweconsiderarepulsivesquarewellpotential
V(x)=
0 x<−a
V 0 −a≤x≤a
0 x>a
(8.86)
Sincethispotentialdiffersfromtheattractivesquarewellofthelastsectiononlyby
thesigninfrontofV 0 ,thereisnoneedtorepeatthealgebraofthelastsection. We