QMGreensite_merged

126 CHAPTER8. RECTANGULARPOTENTIALS

Inanyrealscatteringexperiment,ofcourse, thereisalwayssome finiteuncer-
taintyinthemomentumoftheincomingparticle,thewavepacketisalwaysfinitein
extent,andthatmeansthateventuallytheincidentwaveiscompletelyreplacedby
thescatteredwaves.Nevertheless,tocomputethequantitiesofinterestinascattering
experiment,itisconvenienttoconsiderthe∆p= 0 limit,andworkwithstationary
statesinwhichtheincomingandreflectedwavesalwaysoverlap.The”quantitiesof
interest” inascatteringexperimentarethe ratioof intensitiesoftheincidentand
scatteredwaves,becausesuchratioscanbedirectlyrelatedtoobservablequantities.
Inconnectionwiththe2-slitexperimentdiscussedinLecture3,welearnedthat
ifwehaveabeamofnp particlesofaveragevelocityv=

/m,withthewave-
functionofeachparticlerepresentedatsometimetbyφ(x),thentheintensityofthe
beamatpointxattimetisgivenby

I = npvφ∗(x)φ(x)

= averageno.ofparticles

passingpointxperunittime (8.14)

Thegoalofscatteringtheoryistofindtheintensityofthescatteredbeamascompared
totheintensityoftheincomingbeam.Inthesimplecaseofanincomingparticleof
momentump 0 approachingtheendofatube,

Iinc =

p 0

m

φ∗inc(x)φinc(x)=

p 0

m

|A|^2

Iref =

p 0

m

φ∗ref(x)φref(x)=

p 0

m

|B|^2 (8.15)

and,sinceA=−B,
Iref=Iinc (8.16)

Physically,thismeansthatifthefluxofincomingparticlespastsome pointx 0 is,
say, 1000 particlespersecond,thenthefluxofreflectedparticlespastthatpointis
also 1000 particlespersecond;i.e.everyincidentparticleisreflected,noparticlesare
transmittedpasttheinfinitepotentialbarrieratx=0.Inthesecondexampleshown
inFig. [8.10]wehave,inadditiontoIincandIref,anintensityforthetransmitted
wave,representingparticleswhichhavegonethroughthepotentialbarrier

Itrans=

p 0

m

φ∗trans(x)φtrans(x)=

p 0

m

|C|^2 (8.17)

Thequantitiesweneedtocalculate,whichcanthenbecomparedwithexperiment,
aretheReflectionCoefficient

R =

Iref

Iinc

=

|B|^2

|A|^2

=

no.ofparticles/secreflected

no.ofparticles/secincident

(8.18)