8.1. AQUALITATIVESKETCHOFENERGYEIGENSTATES 123
Nowsupposetheenergy isintherangeE∈[0,Vmax]. Thentherewillbeclas-
sicallyforbidden regions,andintheseregionsthewavefunctionwillgrowordecay
exponentially. Intheclassicallyallowedregions,thewavefunctionoscillatesasshown
inFig.[8.5].
Finally,supposethatE<0. Then,exceptforafiniteintervalwhereV(x)< 0
andE>V,theentirereallineisaclassicallyforbiddenregion. Typicalsolutionsof
thetime-independentSchrodingerequationwouldblowupexponentiallyasx→±∞.
Suchwavefunctionsarephysicallymeaningless;theysaythattheparticleisinfinitely
moreprobabletobefoundatthe”point”x=±∞thananywhereelse,butinfinity,
asweknow,isnotapoint.Wavefunctionsofthiskindarecallednon-normalizable
anddonotbelongtotheHilbertspaceofphysicalstates.Non-normalizablesolutions
ofthetime-independentSchrodingerequationaresimplythrownawayinquantum
mechanics.Althoughsuchwavefunctionssatisfytheequationofinterest,theydonot
correspondtophysicalstates.
AlmostallsolutionswithE< 0 arethereforeexcluded, butnotquiteall. Itis
stillconsistentwiththebehaviour(8.3)forthewavefunctiontodecayexponentially
to zeroas x→ ±∞, as shownin Fig. [8.6]. Inwavefunctionsof this sort, the
probabilityoffindingtheparticleoutsideregionIIdropsexponentiallywithdistance,
soitisreasonabletosaythattheparticleistrappedbythepotentialwell,orinother
wordsthattheparticleisina”boundstate.”However,itisonlypossibletohavethe
wavefunctiondecaytozeroinregionsIandIIIonlyforcertainveryspecialvaluesof
theenergyeigenvalueE,forreasonswhichwillnowbeexplained.
Supposethevalueofthewavefunctionφ(x)anditsfirstderivativeφ′(x)arespec-
ified atsome pointx= a. Sincethe time-independentSchrodingerequationisa
2nd-order equation,thisisenough informationtosolvethe equationandfind the
wavefunctioneverywhereinspace. Nowforalmostallchoicesoffirstderivativeφ′at
x=a,thewavefunctionwilleventuallydivergeto φ→±∞asx→∞,asshown
inFig. [8.7]. Forφ′(a)>c,thewavefunctiondivergesto+∞,whileforφ′(a)<c,
thewavefunctiondiverges to−∞,wherecissomeconstant. However,preciselyat
φ′(a)=c,itispossibleforthewavefunctiontodecaysmoothlytozeroasx→∞,
aswewouldexpectforaboundstate.Evenso,wedonotyethaveaphysicalstate,
becauseingeneralthewavefunctionwillbedivergentasx→−∞. Butthereisone
moreparameterthat canbevaried,namely, theenergy eigenvalueE. Soimagine
varyingE, alwayschoosingφ′(a)so thatφ(x)→ 0 as x→∞. As Evaries, the
pointbinFig.[8.7]wherethewavefunctionstartstoblowupcanbemovedfurther
andfurthertotheleft. Eventually,foronespecialvalueofE,wegetb→−∞,and
thewavefunctiondecayssmoothlytozerobothinregionIandregionIII,asshown
inFig. [8.6]. Onlyforsuchan energydoesthewavefunctionrepresentaphysical
boundstate. Changingtheenergyslightlyresultsinanunacceptable,exponentially
divergentsolution.
Ingeneral thereismore thanonevalueofE, sometimesaninfinitenumberof
values,whichareassociated withboundstatesolutions. But sinceaninfinitesmal