量子力学英文格里菲斯Chapter2课件

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您的内容打在这里，或者通过复制您的文本后。+++整体概况2InChapter1,wehavestudiedalotaboutthe

wavefunction

andhowyouuseitto

calculatevariousquantitiesofinterest.Question:

Howdoyouget(x,t)

inthefirstplace?

Howdoyougoabout

solvingtheSchrödingerequation?3InthisChapter，weassumethatthepotential(orpotentialenergyfunction),V(x,t)=V(x),ofthesystem,isindependentof

time

t

!!!4Nowtheleftsideisafunctionoftalone,andtherightsideisafunctionofxalone.5

Theonlywaythiscanbepossiblybetrueisifbothsidesareinfactconstant,weshallcalltheseparation

constant

E.Then

Separationofvariableshasturneda

partial

differentialequationintotwoordinary

differentialequations(Eq.[2.3]and[2.4]).

6Thefirstequation[2.3]iseasytosolve,thegeneralsolutionofEq.[2.3]is

Thesecondequation[2.4]iscalledthetime-independentSchrödingerequation,wecangoonfurtherwithituntilthepotential

V(x)isspecified.

Therestofthischapterwillbedevotedtosolvingthetime-independentSchrödingerequation[2.4],foravarietyofsimplepotentials.Butbeforewegettothatwewouldliketoconsiderfurtherthequestion:

7What’ssogreataboutseparablesolution

?

可分离的解（即(x,t)=(x)f(t)）为何如此重要？Afterall,most

solutionstothe(time-dependent)Schrödingerequationdonottaketheform(x)f(t).Wewillofferthreeanswers—twoofthemphysicalandonemathematical:8NOTE:fornormalizablesolutions,E

mustbereal(seeProblem2.la).

9NothingeverhappensintheStationaryState(x,t)

!

1011Pleasedistinguishtheoperatorwith“hat”(^)toitsdynamicalvariableinEq.[2.12].

12Conclusion:Aseparablesolutionhasthepropertythat

everymeasurementofthetotalenergyiscertaintoreturnthevalueE.

(That’swhywechosethatletterEfortheseparationconstant.)

3.Thegeneralsolutionisa

linearcombinationofseparablesolutions13Nowthe(time-dependent)Schrödingerequation(Eq.1.1)hasthepropertythatany

linearcombination5ofsolutionsisitselfasolution.14Itsohappensthatevery

solutiontothe(time-dependent)Schrödingerequationcanbewritteninthisform—itissimplyamatteroffindingtherightconstants(c1,c2,c3,c4,…）soastofittheinitialconditionsfortheproblemathand.

Oncewehavefoundtheseparablesolutions,then,wecanimmediatelyconstructamuchmoregeneralsolution,oftheform

15You’llseeinthefollowingsectionshowallthisworksoutinpractice,andinChapter3we’llputitintomoreelegantlanguage,butthemainpointisthis:

Onceyou’vesolvedthetime-independentSchrödingerequation,you’reessentiallydone;gettingfromtheretothegeneralsolutionofthetime-dependentSchrödingerequationissimpleandstraightforward.

16Briefsummaryofsection2.1V(x,t)=V(x)boundaryconditions1718ThefirstexampletosolvetheSchrödingerequationistheinfinitesquarewell19Aparticleinthispotentialiscompletelyfree,exceptatthetwoends(x=0andx=a),whereaninfiniteforcepreventsitfromescaping.

Outsidethewell,(x,t)=0(theprobabilityoffindingtheparticlethereiszero).Insidethewell,whereV=0,thetime-independentSchrödingerequation(Equation2.4)reads

20Equation[2.17]isthe(classical)simpleharmonicoscillator

equation;thegeneralsolutionis

Typically,theseconstantsarefixedbytheboundaryconditionsoftheproblem.Whataretheappropriateboundaryconditionsfor(x)?

21Fortheinfinitesquarewell,both(x)andd(x)/dxarecontinuousatthetwoends(x=0

and

x=a)!NOTE:onlythefirstconditionoftheseisappliedsincethepotentialgoestoinfinityhere!Continuityof(x)requiresthat

2223Curiously,theboundaryconditionatx=adoesnotdeterminetheconstantA,butrathertheconstant

k,andhencethepossiblevalues

of

EcanbeobtainedfromEq.[2.17]and[2.22]:Insharpcontrasttotheclassicalcase,aquantumparticleintheinfinitesquarewellcannothavejustanyoldenergy—onlythesespecialallowedvalues.24Aspromised,thetime-independentSchrödingerequationhasdeliveredaninfinitesetofsolutions,oneforeachintegern.ThefirstfewoftheseareplottedinFigure2.2:

25theylookjustlikethestandingwavesonastringoflengtha.

1,whichcarriesthelowestenergy,iscalledthegroundstate;theothers,whoseenergiesincrease

inproportionto

n2,arecalledexcitedstates.

26Thewavefunctions

n(x)havesomeinterestingandimportantproperties:

2728Notethat:

thisargumentdoesnotworkifm＝n(canyouspotthepointatwhichitfails?);inthatcasenormalizationtellsusthattheintegralis1.Infact,wecancombineorthogonalityandnormalizationintoasinglestatement

:Wesaythatthe’sareorthonormal.294.Theyarecomplete

Inthesensethatany

otherfunction,f(x),canbeexpressedasalinearcombinationofthemn(x):

“Any”functioncanbecxpandedinthiswayissometimescalledDirichlet’s(狄利克雷)theorem.Theexpansioncoefficients(cn)canbeevaluated—foragivenf(x)-

byamethodcalledFourier’strick(技巧),whichbeautifullyexploitstheorthonormalityof{n(x)}:

MultiplybothsidesofEquation[2.28]bym*(x),andintegrate.

30Thusthemthcoefficientintheexpansionoff(x)

isgivenby

31Thesefourpropertiesareextremelypowerful,andtheyarenotpeculiar(特有的)totheinfinitesquarewell.

Thefirstistruewheneverthepotentialitselfisanevenfunction;

Thesecondisuniversal,regardlessoftheshapeofthepotential.

Orthogonalityisalsoquitegeneral-we’1showyoutheProofinChapter3.

Completenessholdsforallthepotentialsyouarelikelytoencounter,buttheProofstendtobenastyandlaborious;I’mafraidmostphysicistssimplyassumecompletenessandhopeforthebest.

32Thestationarystates(Equation2.6)fortheinfinitesquarewellareevidently

byusingofEq.[2.23]and[2.24]

:Themostgeneralsolutiontothe(time-dependent)SchrödingerequationisalinearcombinationofstationarystatesEq.[2.31]:

33Ingeneral,whent=0,accordingtoEquation[2.32],wecanfit

anyprescribedinitialwavefunction,(x,0),byappropriatechoiceofthecoefficientscn:Thecompletenessofthe(x,0)’s(confirmedinthiscasebyDirichlet’stheorem)guaranteesthatwecanalwaysexpress(x,0)inthisway,andtheirorthonormalitylicensestheuseofFourier’stricktodeterminetheactualcoefficients.Forexample,infinitesquarewell,wehave

34Giventheinitialwavefunction,(x,0)Wefirstcomputetheexpansioncoefficientscn，byusingofEquation[2.33]ThenplugtheseintoEquation[2.32]toobtainΨ(x,t)

Armedwiththewavefunction,weareinapositiontocomputeanydynamicalquantitiesofinterest,usingtheproceduresinChapter1.Andthissameritualappliestoanypotential——theonlythingsthatchangearethefunctionalformofthe’sandtheequationfortheallowedenergies.35Homework:

Example2.1,Example2.2

Problem2.7,Problem2.3736

Theparadigmforaclassicalharmonicoscillatorisamassmattachedtoaspring(弹力)offorceconstantk.ThemotionisgovernedbyHooke’slaw:37Ofcourse,there’snosuchthingasaperfectsimpleharmonicoscillator—ifstretchittoofarthespringisgoingtobreak,andtypicallyHooke’slawfailslongbeforethatpointisreached.

Butpracticallyany

potentialisapproximately

parabolic(抛物线的),intheneighborhoodofalocalminimum(Figure2.3).

3839That’swhythesimpleharmonicoscillator

issoimportant:Virtually(实际上;事实上)

anyoscillatorymotionisapproximatelysimpleharmonic,aslongastheamplitudeissmall.40Intheliteratureyouwillfindtwoentirelydifferentapproachestothisproblem.

The

QuantumproblemistosolvetheSchrödingerequationforthepotentialAswehaveseen,itsufficestosolvethetimeindependentSchrödingerequation:41Thesecondisadiabolically(魔鬼似地)clever(聪明的)

algebraic(代数的)

technique,usingso-calledladder(阶梯)operators.Wewillstudythealgebraicmethodfirstly,becauseitisquickerandsimpler(andmorefun).

Thefirstisastraightforward“bruteforce”(强力)solutiontothedifferentialequation,usingthemethodofpowerseriesexpansion;ithasthevirtue(优点)thatthesamestrategy(策略)canbeappliedtomanyother

potentials(infact,wewilluseitinChapter4totreattheCoulombpotential).

42Theideaistofactortheterminsquarebrackets.Ifthesewerenumberseasy:

Tobeginwith,let’srewriteEquation[2.39]inamoresuggestiveform:

Here,however,it’snotquitesosimple,becauseuandvareoperators,andoperatorsdonot,ingeneral,commute(i.e.uv≠vu).Still,thisdoesinviteustotakealookattheexpressions,fromEq.[2.40],weassume

43Warning:Operators

canbeslipperytoworkwithintheabstract,andyouareboundtomakemistakesunlessyougivethema“testfunction”,f(x),toacton.

Attheendyoucanthrowawaythetestfunction,andyou’llbeleftwithanequationinvolvingtheoperatorsalone.“product”积44Inthepresentcase,wehave

45ByusingofEq.[2.42],theSchrödingerequation[2.40]becomes

Noticethattheorderingofthefactora+andaisimportanthere!46Thesameargument,witha+ontheleft,yields

Thus,Eq.[2.42]

–

Eq.[2.44],wehave47Now,herecomesthecrucial(关键的)step:

48Here,then,isawonderfulmachineforgrindingoutnewheresolutions,withhigherandlower

energies—ifwecanjustfind

onesolution,togetstared!49Wecalleda±

ladder

operators,becausetheyallowustoclimbupanddowninenergy;a+iscalledtheraisingoperator,andathe

loweringoperator.

The“ladder”ofstatesisillustratedinFigure2.4.

50Butwait!

Whatifweapplytheloweringoperator

arepeatedly?Eventuallywe’regoingtoreachastatewithenergylessthanzero,which(accordingtothegeneraltheoreminProblem2.2)doesnotexist!Atsomepointthemachinemustfail.Howcanthathappen？WeknowthataisanewsolutiontotheSchrödingerequation,butthereisnoguarantee

thatitwillbe

nonmalleable—itmightbezero,oritssquareintegralmightbeinfinite.Problem2.11rulesoutthelatterpossibility.

51Conclusion:Theremustoccura“lowestrung(阶梯)”(let’scallit0）suchthat(seeEq.[2.41])52Todeterminetheenergyofthis

state,weplugitintotheSchrödingerequation(intheformofEquation2.46)

53Withourfootnowsecurelyplantedonthebottomrung(thegroundstateofthequantumoscillator),wesimplyapplytheraisingoperatortogeneratetheexcitedstate:Thismethoddoesnotimmediatelydeterminethenormalizationfactor

An,whichwillbeworkedoutbyyourselfinProblem2.12.54Wewouldn’twanttocalculate50

inthisway,butnevermind:Wehavefoundalltheallowedenergies,andinprinciplewehavedeterminedthestationarystates—therestisjustcomputation.55Homework:

Example2.556WereturnnowtotheSchrödingerequationfortheharmonicoscillator(Equation[2.39]):

Weintroducethedimensionless

variable57OurproblemistosolveEquation[2.56],andintheprocessobtainthe“allowed”valuesofK

(andhenceofEfromEq.[2.57]).

58NotethattheBterminEq.[2.59]isclearlynotnormalizable

(itblowsupas|x|→∞);thephysicallyacceptablesolutions,then,havetheasymptoticform

Thissuggeststhatwe“peeloff”theexponentialpart

59sotheSchrödingerequation(Eq.[2.56])becoms

6061This

recursion(递推)formulaisentirelyequivalenttotheSchrödingerequationitself.

FromEq.[2.65]:

givena0,Eq.[2.65]enablesus(inprinciple)togenerate

a2,a4,a6,…

givena1,Eq.[2.65]generates

a3,a5,a7,

….Letuswrite62Thusequation[2.65]determinesh(ξ)intermsoftwoarbitrary(a0anda1)—whichisjustwhatwewouldexpect,forasecond-orderdifferentialequation.However,notallthesolutionssoobtainedarenormalizable.Foratverylargej,therecursionformula[2.65]becomes(approximately)63Now,ifhgoeslikeexp(ξ2),then(remember?—that’swhatwe’retryingtocalculate)goeslikeexp(ξ2/2)(Equation2.61),whichispreciselytheasymptoticbehaviorwedon’twant.

64Thereisonlyonewaytowiggleoutofthis:Fornormalizablesolutionsthe

powerseriesmustterminate.Theremustoccur

some“highest”j(callitn)suchthattherecursionformulaspitsoutan+2=0.

Conclusion:theseries

hoddwillbetruncatedatsomehighestn;theserieshevenmustbezerofromthestart.

Forphysicallyacceptablesolutions,then,wemusthave

forsomepositiveintegern,whichistosay(referringtoEquation[2.57])thatthe

energy

mustbeoftheform

65Thus,werecover,byacompletelydifferentmethod,thefundamentalquantizationcondition

wefoundalgebraicallyinEquation[2.50].(Eq.[2.68]isobtainedbyputtingK=2n+1intoEq.[65])6667Ingeneral,hn(ξ)willbeapolynomialofdegreeninξ,involvingevenpowersonly:ifnisaneveninteger,andoddpowersonly,ifnisanoddinteger.Apartfromtheoverallfactor(a0ora1)theyaretheso-calledHermitepolynomials,Hn(ξ).ThefirstfewofthemarelistedinTable2.1.

68Bytradition,thearbltrarymultiplicativefactorischosensothatthecoefficientofthehighestpowerofξis2n.Withthisconvention,thenormalizedstationarystatesforthe

harmonicoscillatorareTheyareidentical(ofcourse)totheonesweobtainedalgebraicallyinEquation[2.50].

InFigure2.5awecanplotn(x)forthefirstfewn’s.6970Thequantumoscillatorisstrikingly(醒目地)differentfromitsclassicalcounterpart—notonlyaretheenergiesquantized,butthepositiondistributionshavesomebizarre(怪诞的)features.

Forinstance:

Theprobabilityoffindingtheparticleoutsidetheclassicallyallowedrange(thatis,withxgreaterthantheclassicalamplitudefortheenergyinquestion)isnotzero(seeProblem2.15).Inalloddstatestheprobabilityoffindingtheparticleatthecenterofthepotentialwelliszero.Onlyatrelativelylargendowebegintoseesomeresemblance(相似)totheclassicalcase.

71InFigure2.5b,wehavesuperimosedtheclassicalpositiondistributiononthequantumone(forn=100);ifyousmoothedoutthebumpsinthelatter,thetwowouldfitprettywell.Fig.2.5(b)Graphof|ψ100|2,withtheclassicaldistributionsuperimposed.7273Weturnnexttowhatshouldhavebeenthesimplestcaseofall:thefreeparticle[V(x)=0everywhere].Asyou’llseeinamoment,thefreeparticle

freeparticleisinfactasurprisingly(令人惊讶地)subtle(难以捉摸的)andtricky(难处理的)example.Thetime-independentSchrödingerequationreads74Sofar,it’sthesameasinsidetheinfinitesquarewell(Equation[2.17]),wherethepotentialisalsozero.Thistime,however,weprefertowritethegeneralsolutionofthefreeparticle

inexponentialform(insteadofsinesandcosines)forreasonsthatwillappearinduecourse:Unliketheinfinitesquarewell,therearenoboundaryconditionstorestrictthepossiblevaluesofk(andhenceofE

asshownbyEq.[2.75])

75Thefreeparticlecancarryany(positive)energy.Tackingon(添加)thestandardtimedependence,

InEquation[2.77]:thefirsttermrepresentsawavetravelingtotheright,thesecondtermrepresentsawave(ofthesameenergy)goingtotheleft.Bytheway,sincetheyonlydifferbythesigninfrontofkinEq.[2.77],wemightaswellwrite

76Thespeedofthese

waves(thecoefficientoft

overthecoefficientofx)is

Ontheotherhand,theclassical

speedofafreeparticlewithenergyEisgivenbyE=(1/2)mv2(purekinetic,sinceV＝0),so

77Evidentlythequantummechanicalwavefunctiontravelsathalfthespeedoftheparticle.It(themovementofwaveinquantummechanics)issupposed(假想)torepresent(描绘)!

Thereisanevenmoreseriousproblemweneedtoconfrontfirst:Thiswavefunctionisnotnormalizable!For

Note:Thisparadox(悖论)willbediscussedattheendofthissection(readitcarefullybyyourself).78Inthecaseofthefreeparticle,then,theseparablesolutionsdonotrepresentphysicallyrealizablestates.Afreeparticlecannotexistinastationarystate;or,toputitanotherway,thereisnosuchthingasafreeparticlewithadefiniteenergy．Thegeneralsolutiontothetime-dependentSchrödingerequationisstillalinearcombinationofseparablesolutions(onlythistimeit’sanintegraloverthecontinuousvariablek,insteadofasumoverthediscreteindexn):Butthatdoesn’tmeantheseparablesolutionsareofnousetous.Fortheyplayamathematicalrolethatisentirelyindependentoftheirphysical

interpretation.

79Nowthiswavefunctioncanbenormalized.Butitnecessarilycarriesarangeofk’s,andhencearangeofenergiesandspeeds.Wecallitawavepacket.

ThisisaclassicprobleminFourieranalysis;theanswerisprovidedbyPlancherel’stheorem(see

Problem2.20):80F(k)iscalledtheFouriertransformoff(x);

f(x)istheinverseFouriertransformofF(k).(theonlydifferenceisinthesignoftheexponent)Thereis,ofcourse,somerestrictionontheallowablefunctions:Theintegrals

haveto

exist.Forourpurposesthisisguaranteedbythephysicalrequirementthat(x,t)itselfbenormalized.Sothesolutiontothegenericquantumproblem,forthefreeparticle,isequation[2.83],with

8182Supplement8384858687Wehaveencounteredtwoverydifferentkindsofsolutionstothetime-independent

Schrödingerequation:

(1)Fortheinfinitesquarewellandtheharmonicoscillatorthey(i.e.wavefunction)arenormalizable,andlabeledbyadiscreteindexn.

(2)Forthefreepaticletheyarenon-normalizable,andlabeledbyacontinuousvariablek.

88Inbothcasethegeneralsolutiontothetime-dependentSchrödingerequationisalinearcombinationofstationarystates—forthefirsttypethiscombinationtakestheformofasum(overn),whereasfortheseconditisanintegral(overk).

Theformer(i.e.wavefunction)representphysicallyrealizablestatesintheirownright,thelatterdonot.89

theparticleis“stuck”inthepotentialwell—itrocksbackandforthbetweentheturningpoints,butitcannot

escape.Wecallthisa

boundstate.

ClassicalMechanicsAone-dimensionaltimeindependentpotential

V(x)cangiverisetotworatherdifferentkindsofmotion:

(i)IfV(x)riseshigherthantheparticle’stotalenergy

Eoneitherside(Figure2.7a)90thentheparticlecomesinfrom“infinity”,slowsdownorspeedsupundertheinfluenceofthepotential,andreturnstoinfinity.Wecallthisascatteringstate.

(ii)Iftheparticle’stotalenergy

EexceedsV(x)

ononeside(orboth)asshownbyFigure2.7b91

QuantumMechanicsThetwokindsofsolutionstotheSchrödingerequationcorrespondpreciselytoboundandscatteringstates.Thedistinctionisevencleanerinthequantumdomain,becausethephenomenonoftunnelingallowstheparticleto“leak”throughany

finitepotentialbarrier,sotheonlythingthatmattersisthepotentialatinfinity(Figure2,7c):92In“reallife”most

potentialsgotozeroatinfinity,inwhichcasethecriterionsimplifiesevenfurther:

Becausetheinfinitesquarewellandharmonicoscillator

potentialsgotoinfinityasx→±∞,theyadmitboundstatesonly.Becausethefreeparticle

potentialiszeroeverywhere,itonlyallowsscatteringstates.Thus93Inthissection(andthefollowingone)weshallexplorepotentialsthatgiverisetobothkindsofstates.The

Diracdeltafunction,

(x),isdefinedinformallyasfollows:Itisaninfinitelyhigh,infinitesimallynarrowspikeattheorigin,whosearea

is1(Figure2.8).94Technically,it’snotafunctionatall,sinceitisnot

finiteatx=0.Mathematicianscallitageneralizedfunction,ordistribution.Nevertheless,itisanextremelyusefulconstructintheoreticalphysics.95Noticethat(x-a)wouldbeaspikeofarea1atthepointa.Ifyoumultiply(x-a)byanordinaryfunctionf(x),it’sthesameasmultiplyingbyf(a):

becausetheproductiszeroanywayexceptatthepointa.Inparticular,fromEq.[2.94]wehaveThat’sthemostimportantpropertyofthedeltafunction:Undertheintegralsignitservesto“pickout”thevalueoff(x)atthepointa.96Now,let’sconsiderapotentialoftheformwhereαissomeconstant.deltapotentialbarrierdeltapotentialwell97TheSchrödingerequationreads

Thispotentialyieldsbothboundstates(E<0)andscatteringstates(E>0);we’lllookfirstattheboundstates.98(Eisnegative,byassumption,sok

isrealandpositive.)ThegeneralsolutiontoEquation[2.98]isbutthefirsttermblowsupasx→–∞,sowemustchooseA=0:99Intheregionx>0,V(x)isagainzero,andthegeneralsolutionisoftheformFexp(-kx)+Gexp(kx);thistimeit’sthesecondtermthatblowsup(asx→+∞),so

Itremainsonlytostitchthesetwofunctionstogether,usingtheappropriateboundaryconditionsatx=0.Wequotedearlierthestandardboundaryconditionsfor：100ThefirstboundaryconditiontellsusthatF=B,so

combiningEq.[2.101]andEq.[2.102],wehaveThewavefunctiion(x)Eq.[2.104]isplottedbyFigure2.9.101It’sclearfromthegraph(seeFig.2.9)thatthewavefunction

(x)hasakink(扭折)atx=0.Thesecondboundaryconditiontellsus:thedeltafunctionmustdeterminethediscontinuity

inthederivativeof,atx=0.

We’llshowyounowhowthisworks,andasabyproductwe’llseewhyd/dxisordinarilycontinuous.102Theideaistointegratethetime-independentSchrödingerequation,from–εto+ε,

andthentakethelimit

asε→0.

103Ordinary,thelimitontherightisagainzero,andhenced/dxiscontinuous.ButwhenV(x)isinfiniteattheboundary,thatargumentfails.

Inparticular,ifV(x)=–(x),thepropertyofthedeltafunction

Eq.[2.95]yields104105Finally,wenormalize(x):Evidentlythedelta-functionwell,regardlessofit’s“strength”,hasexactlyoneboundstatewhen

E<0:106Whataboutscatteringstates,withE>0?

For

x<0,theSchrödingerequationreads

isrealandpositive.ThegeneralsolutionisAndthistimewecannotruleouteitherterm,sinceneitherofthemblowsup.Similarly,forx>0,107108Havingimposedtheboundaryconditions,weareleftwithtwo

equations(Eqs.[2.115]and[2.117])infourunknowns(A,B,F,andG)—five,ifyoucountk.

109Normalizationwon’thelp—thisisnotanormalizablestate.Perhapswe’dbetterpause,then,andexaminethephysicalsignificanceofthesevariousconstants.Recallthatexp(ikx)

givesrisetoawavefunctionpropagatingtotheright,andexp(-ikx)leadstoawavepropagatingtotheleft.

110Itfollowsthat:A(inEq.[2.113])istheamplitudeofawavecominginfromtheleft,Bistheamplitudeofawavereturningtotheleft,F(inEq.[2.114])istheamplitudeofawavetraveling

offtotheright,Gistheamplitudeofawavecominginfromtheright(seeFig.2.10).111112Inatypicalscatteringexperimentparticlesarefiredinfromonedirection—let’ssay,fromtheleft.Inthatcasetheamplitudeofthewavecominginfromthe

right

willbezero:Aisthentheamplitudeofthe

incidentwave,Bistheamplitudeofthereflectedwave,andFistheamplitudeofthe

transmittedwave.

SolvingEqs.[2.115]and[2.117]forBandF,wefind113Now,theprobabilityoffindingtheparticleataspecifiedlocationisgivenby||2,sotherelative

probabilitythatanincidentparticlewillbereflectedbackisRiscalledthereflectioncoefficient.

Meanwhile,theprobabilityoftransmissionisgivenbythetransmissioncoefficient

114NoticethatRandTarefunctionsof，andhence(Eqs.[2.112]and[2.117])ofE:

FromEq.[2.123],itcanbefoundthatthehighertheenergy,thegreatertheprobabilityoftransmission