量子力学薛定谔方程.docx
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量子力学薛定谔方程.docx
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量子力学薛定谔方程
Lecture10-11Schrödinger(薛定谔)equations
Priorto1925quantumphysicswasa“hodgepodge”ofhypotheses,principles,theoremsandrecipes.Itwasnotalogicallyconsistenttheory.
Onceweknowthiswavefunctionweknow“everything”aboutthesystem!
Outline
Part1DynamicEquations
Part2DynamicEquationofWavefunction-
Time-dependentSchrödingerequation(TDSE)
Part3StationarystateSchrödingerequation(TISE)
Part4ConditionsonwavefunctionandProbabilitycurrentdensity
Part1DynamicEquations
Ifweknowtheforcesactingupontheparticlethan,accordingtoclassicalphysics,weknoweverythingaboutaparticleatanymomentinthefuture.
Adifferentialequationbyitselfdoesnotfullydeterminetheunknownfunction
.
Part2DynamicEquationofWavefunction
----Schrödingerequations
用
描述的粒子,只能有5种动量取值,分别是
,对应的几率分别是
,这些几率总和应该为1。
Dowehavethesamerecipeforcalculationofaveragemomentumbyusingwavefunctioninpositionrepresentation?
Yes,ofcourse,wehave!
Tofindtheexpectation(average)valueofp,wefirstneedtorepresentpintermsofxandt.Considerthederivativeofthewavefunctionofafreeparticlewithrespecttox:
Wefindthat
Thissuggestswedefinethemomentumoperatoras
Theexpectationvalueofthemomentumis
So,wecannothavedefinitevaluesforthedynamicalvariables,suchasthemomentum,whenthestateofaparticleisdeterminedbythewavefunctionwithrespecttox.WehavetofindtheotherwaytodescribethedynamicalvariablesinQuantumMechanics.
ForeverydynamicalvariableoranyobservablethereisacorrespondingQuantumMechanicalOperator
PhysicalQuantitiesOperators
Operatorsareimportantinquantummechanics.
Allobservableshavecorrespondingoperators.
OperatorsSymbolsformathematicaloperation
✧Thepositionxisitsownoperator
.Done.Otheroperatorsaresimplerandjustinvolvemultiplication
.
✧ThepotentialenergyoperatorisjustmultiplicationbyV(x).
✧Themomentumoperatorisdefinedas
Eigenvalueequationofanoperator
DerivingtheSchrödingerEquationusingoperators:
Thiswasaplausibilityargument,notaderivation.WebelievetheSchrödingerequationnotbecauseofthisargument,butbecauseitspredictionsagreewithexperiments.
SchrödingerEquationNotes:
系综平均(EnsembleAverage)
TheSchrödingerEquationisTHEfundamentalequationofQuantumMechanics.Therearelimitstoitsvalidity.Inthisformitappliesonlytoasingle,non-relativisticparticle(i.e.onewithnon-zerorestmassandspeedmuchlessthanc.)
●Onthelefthandpicture 13 velocityvectorsofanindividualflyareshown;thechainofvectorsclosesso
●Ontherighthandpicturethesame 13velocityvectorsareassignedto 1 flyeachtodemonstratethattheensembleaverage yieldsthesameresult,i.e.
●i.e. timeaverage=ensembleaverage.Thenewsubscripts"e"and"r"denoteensembleandspace,respectively.Thisisasimpleversionofaveryfarreachingconceptinstochasticphysicsknownunderthecatchword"ergodichypothesis".
●Aslongaseveryflydoes-onaverage-thesamething,the vector averageovertimeoftheensembleisidenticaltothatofanindividualfly-ifwesumupafewthousandvectorsfor one fly,orafewmillionforlotsoffliesdoesnotmakeanydifference.However,wealsomayobtainthisaverageinadifferentway:
●Wedonotaverage oneflyintime obtaining
●Thismeans,wejustaddupthevelocityvectorsofallfliesatsomemomentintimeandobtain
=
a)Schrödingerequationisalinearhomogeneouspartialdifferentialequation.
b)TheSchrödingerequationcontainsthecomplexnumberi.Thereforeitssolutionsareessentiallycomplex(unlikeclassicalwaves,wheretheuseofcomplexnumbersisjustamathematicalconvenience.)
c)Thewaveequationhasinfinitenumberofsolutions,someofwhichdonotcorrespondtoanyphysicalorchemicalreality.
1.Foranelectronboundtoanatom/molecule,thewavefunctionmustbeeverywherefinite,anditmustvanishintheboundaries
2.Singlevalued
3.Continuous
4.Gradient(d/dr)mustbecontinuous
5.*disfinite,sothatcanbenormalized
d)Solutionsthatdonotsatisfytheseproperties(above)DONOTgenerallycorrespondtophysicallyrealizablecircumstances.
e)Conditionsonthewavefunction(波函数的三个基本条件——有限、单值、连续)
1.Inordertoavoidinfiniteprobabilities,thewavefunctionmustbefiniteeverywhere.
2.Thewavefunctionmustbesinglevalued.
3.Thewavefunctionmustbetwicedifferentiable.Thismeansthatitanditsderivativemustbecontinuous.(AnexceptiontothisruleoccurswhenVisinfinite.)
4.Inordertonormalizeawavefunction,itmustapproachzeroasxapproachesinfinity.
f)Onlythephysicallymeasurablequantitiesmustbereal.Theseincludetheprobability,momentumandenergy.
CanthinkoftheLHSoftheSchrödingerequationasadifferentialoperatorthatrepresentstheenergyoftheparticle?
ThisoperatoriscalledtheHamiltonianoftheparticle,andusuallygiventhesymbol
.Hamiltonianisalineardifferentialoperator.
Hencethereisanalternative(shorthand)formforthetime-dependentSchrödingerequation:
Part3Time-independentSchrödingerequation(TISE),i.e.stationarystate(定态)Schrödingerequation
Supposepotentialisindependentoftime
Lookforaseparatedsolution,substitute
into
•ThisonlytellsusthatT(t)dependsontheenergyE.Itdoesn’ttelluswhattheenergyactuallyis.Forthatwehavetosolvethespacepart.
•T(t)doesnotdependexplicitlyonthepotentialU(x).ButthereisanimplicitdependencebecausethepotentialaffectsthepossiblevaluesfortheenergyE.
Thisisthetime-independentSchrödingerequation(TISE)orso-calledstationarystateSchrödingerequation.
SolutiontofullTDSEis
Eventhoughthepotentialisindependentoftimethewavefunctionstilloscillatesintime.Butprobabilitydistributionisstatic
ForthisreasonasolutionoftheTISEisknownasaStationaryState(定态)
StationarystateSchrödingerEquationNotes:
•Inone-dimensionspace,theTISEisanordinarydifferentialequation(notapartialdifferentialequation)
•TheTISEisaneigenvalueequationfortheHamiltonianoperator:
Part4Probabilitycurrentdensityandcontinuityequation
Definitionofprobabilitycurrentdensity
Innon-relativisticquantummechanics,theprobabilitycurrent
ofthewavefunctionΨisdefinedas
inthepositionbasisandsatisfiesthequantummechanicalcontinuityequation
withtheprobabilitydensity
definedas
.
Ifoneweretointegratebothsidesofthecontinuityequationwithrespecttovolume,sothat
thenthedivergencetheoremimpliesthecontinuityequationisequivalenttotheintegralequation
wheretheVisanyvolumeandSistheboundaryofV.Thisistheconservationlawforprobabilityinquantummechanics.
Inparticular,if
isawavefunctiondescribingasingleparticle,theintegralinthefirsttermoftheprecedingequation(withoutthetimederivative)istheprobabilityofobtainingavaluewithinVwhenthepositionoftheparticleismeasured.ThesecondtermisthentherateatwhichprobabilityisflowingoutofthevolumeV.AltogethertheequationstatesthatthetimederivativeofthechangeoftheprobabilityoftheparticlebeingmeasuredinVisequaltotherateatwhichprobabilityflowsintoV.
Derivationofcontinuityequation
Thecontinuityequationisderivedfromthedefinitionofprobabilitycurrentandthebasicprinciplesofquantummechanics.
Suppose
isthewavefunctionforasingleparticleinthepositionbasis(i.e.
isafunctionofx,y,andz).Then
istheprobabilitythatameasurementoftheparticle'spositionwillyieldavaluewithinV.Thetimederivativeofthisis
wherethelastequalityfollowsfromtheproductruleandthefactthattheshapeofVispresumedtobeindependentoftime(i.e.thetimederivativecanbemovedthroughtheintegral).Inordertosimplifythisfurther,considerthetimedependentSchrödingerequation
anduseittosolveforthetimederivativeof
:
Whensubstitutedbackintotheprecedingequationfor
thisgives
.
Nowfromtheproductruleforthedivergenceoperator
andsincethefirstandthirdtermscancel:
IfwenowrecalltheexpressionforPandnotethattheargumentofthedivergenceoperatorisjust
thisbecomes
whichistheintegralformofthecontinuityequation.
ThedifferentialformfollowsfromthefactthattheprecedingequationholdsforallV,andastheintegrandisacontinuousfunctionofspace,itmustvanisheverywhere:
Forallwholespacewehave
whichmeansthat
mustbecontinuousatanypositioninthewholespace.
Sothewavefunctionanditsderivativemustbecontinuous.(AnexceptiontothisruleoccurswhenVisinfinite.)
Onemore,ifthe
and
isreal,theprobabilitycurrent
overthewhole1Dspacewhichmeans
isalwayscontinuouswhateverthewavefunction
anditsderivative
arecontinuousornot.However,
hastobecontinuousforanacceptablephysicalsolutionforthattheprobabilitydensityisuniquelydefined(唯一确定).Asto
itmaynotbecontinuousespeciallyatthepointwherethepotentialenergyisinfinite.
Itiseasytoprovethat
hastobecontinuousatthepoint
wherethepotentialenergyjusthasalimitedhighstep.
Haveafun!
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