122 CHAPTER8. RECTANGULARPOTENTIALS
ticleintheclassicallyforbiddenregions,andthisfactgivesrisetosomeremarkable
phenomenawhichareinconceivableat theclassicallevel. Nevertheless, thewave-
functiondoesbehave quitedifferentlyintheclassicallyallowedandtheclassically
forbiddenregions,asonemightexpect,sinceclassicalbehaviorisanapproximation
toquantummechanics,andshouldbecomeexactinthe ̄h→ 0 limit.
Thetime-independentSchrodingerequationisa2ndorderdifferentialequation
d^2 φ
dx^2=−
2 m
̄h^2(E−V)φ (8.1)Ingeneral,thefirstderivativeofafunctionf′=df/dxistheslopeoff(x),thesecond
derivativef′′=d^2 f/dx^2 isthecurvature.Iff′′> 0 atpointx,thenf(x)isconcave
upwardsatthatpoint;similarly,iff′′<0,thenf(x)isconcavedownwards. From
eq. (8.1),wecanseethat:
E>V ClassicallyAllowedRegion
φ> 0 φ(x) concavedownwards
φ< 0 φ(x) concaveupwards(8.2)
Sincethewavefunctiontends tocurve downwardswhen φispositive, andupward
whenφisnegative,theresultisanoscillatingwavefunctionintheclassicallyallowed
regions,asshowninFig. [8.2].
E<V ClassicallyForbiddenRegion
φ> 0 φ(x) concaveupwards
φ< 0 φ(x) concavedownwards(8.3)
Intheclassicallyforbiddenregions,oscillatorybehaviorisimpossible,andthewave-
functiontendstoeithergrowordecayexponentially,asshowninFig. [8.3]
Wecan use this qualitative informationto sketcha roughpicture of whatan
energyeigenstateinthepotentialshowninFig. [8.1]mustlooklike. Letusbegin
withE > Vmax. Inthatcase, there areno classically forbidden regions,and the
functionoscillates everywhere. Thecurvature,andthereforetherateofoscillation,
dependsontheratio
φ′′
φ
=−
̄h^2
2 m(E−V) (8.4)
so that thegreaterE−V is, thefaster thewavefunctionoscillates. Thegeneral
behaviorisshowninFig.[8.4];anenergyeigenstatewilllooksomethinglikethisfor
anyenergyE>Vmax.