QMGreensite_merged

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