130 CHAPTER8. RECTANGULARPOTENTIALS
Butinthiscase
R =
|B|^2
|A|^2
= 1 (8.36)
whichmeansthat all incidentparticlesarereflected; noparticles aretransmitted
towardsx→+∞,asintheclassicalcase. Theresultingwavefunctionisshownin
Fig.[8.13].Unliketheclassicalcase,however,thereisafiniteprobabilitytofindthe
particleatanyfinitedistancex> 0
P∆x(x>0)=|C|^2 e−^2 q^2 x/ ̄h∆x (8.37)Onceagain,thereisnowayofinterpretingthisresultinclassicalterms. Classically,
forceisproportionaltothederivativeofthepotential,sothereisonlyaforceonthe
particleatx=0.Thereisnothingtopreventaparticleatx> 0 tocontinuemoving
totheleft. Quantummechanically,althoughthereisafiniteprobabilitytofindthe
particleatx>0,thatprobabilitydecreasesexponentiallywithx,andanyparticle
ofenergyE<V isultimatelyreflected.
8.4 The Finite Square Well: Bound States
Anattractivepotentialwhichisfiniteatinfinitynormallyhasadiscrete(finiteorinfi-
nite)numberofboundstates,andaninfinitenumberofunboundstateswithenergies
intherange[Emin,∞]. Thesimplestexampleisthefinitesquarewellpotential^3
V(x)=
0 x<−a
−V 0 −a≤x≤a
0 x>a(8.38)
showninFig.[8.14].
For particleenergiesE > 0 theentirereallineisclassically allowed; theseare
theunboundstates. ForE < 0 regionsIandIIareclassicallyforbidden,andthe
wavefunctionofaphysicalstatemustfallexponentiallytozeroasx→±∞. These
aretheboundstates.
(^3) Itisconventionaltocallita”square”wellalthoughtheshapemayjustaswellberectangular.
Thepotentialforaparticleinatubeisknownasthe”infinitesquarewell.”