Chapter 8
Rectangular Potentials
Mostoftheeffortinsolvingquantum-mechanicalproblemsgoesintosolvingthetime-
independentSchrodingerequation(243). Givenacompleteset ofsolutionstothe
time-independentequation,thegeneralsolution(248)ofthetime-dependentequation
followsimmediately.
Wehavealreadyseen thatinthecaseofafreeparticle, theenergyeigenvalues
couldtakeonanyvalueintherange[0,∞],whileinthecaseofaparticletrappedina
tube,theenergiescouldtakeononlycertaindiscretevalues.Thisdistinctionbetween
theenergiesofboundandunboundparticlesisverygeneral,andinfactthesolutions
toanytime-independentSchrodingerequation,withanypotentialV(x),arealways
oftwokinds: (i)boundstates,whichhavediscreteenergyeigenvalues,E 1 ,E 2 ,E 3 ,...;
and(ii)unboundor”scattering”states,whichhaveenergiesinacontinuous range
E ∈[Elow,∞]. Sowe beginthis lecturewithan explanationof why theenergies
of boundstates arealwaysdiscrete, and alsowhy theunbound states, whichare
energyeigenstatesandthereforestationary,havesomethingtodowiththescattering
ofparticlesbyapotential,whichisofcourseadynamicprocess.
8.1 A Qualitative Sketch of Energy Eigenstates
ConsideraparticleoftotalenergyEinthepotentialshowninFig. [8.1].Inclassical
physics,ifE<Vmax,theparticlecouldneverbefoundintheregionswhereV(x)>E,
becausethatwouldimplythatthekineticenergyKE=E−V wasnegative. Such
intervals arereferredto as the”classically forbidden” regions; intervals where
E>V(x)arethe”classicallyallowed”regions.AparticleinregionIIcouldnever
getout;itwouldbeaboundstatetrappedbythepotential,foreverbouncingbetween
pointsx 2 andx 3. Ontheotherhand,aparticleinregionIorIIIcouldneverenter
regionII,butwouldbounceoffthepotentialwallatpointsx 1 andx 4 ,respectively.
InquantummechanicsthephysicalstateisgovernedbytheSchrodingerequation,
andgenerallythereisnoreasonthatthewavefunctionmustbeexactlyzeroinregions
whereE<V(x).Thismeansthatthereisusuallyafiniteprobabilitytofindapar-