結(jié)構(gòu)化學(xué)課件chapter1

StructuralChemistry參考書:<<結(jié)構(gòu)化學(xué)>>廈門大學(xué)化學(xué)系物構(gòu)組編<<PhysicalChemistry>>P.W.Atkins《化學(xué)鍵的本質(zhì)》鮑林上?？茖W(xué)技術(shù)出版社<<結(jié)構(gòu)化學(xué)基礎(chǔ)>>周公度編著,北京大學(xué)出版社《結(jié)構(gòu)化學(xué)習(xí)題解析》周公度等，北京大學(xué)出版社課程教學(xué)組:

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（期中/期末/隨堂考試，期平成績等）課外化學(xué)前沿了解

科技媒體：C&EN,…文獻(xiàn)搜索引擎：webofscience,scifinder,google

文獻(xiàn)閱讀：各大雜志社在線網(wǎng)站，圖書館WhatisChemistryThebranchofnaturalsciencethatdealswithcomposition,structure,propertiesofsubstancesandthechangestheyundergo.TypesofsubstancesNanomaterialsBulkmaterialsGeometricStructureSize

makesthedifferenceElectronicStructureAtomsMoleculesClustersCongeriesStructuredeterminespropertiesPropertiesreflectstructuresStructurevs.PropertiesStructuralChemistryInorganicChemistryOrganicChemistryCatalysisElectrochemistryBio-chemistryetc.MaterialScienceSurfaceScienceLifeScienceEnergyScienceEnvironmentalScienceetc.FunnythingsinStructureChemistryNanoputians:AnthropomorphicMoleculesTour,J.M.etal,J.Org.Chem.2003,68,8750;J.Chem.Edu.2003,80,395."Nanoputian"isaportmanteauofnanoandlilliputian.Lilliputisafictionalislandnationthatappearsinthefirstpartofthe1726novelGulliver'sTravelsbyJonathanSwift.FunnythingsinChemistryNanoputians:AnthropomorphicMoleculesTour,J.M.etal,J.Org.Chem.2003,68,8750;J.Chem.Edu.2003,80,395.NanoBalletdancersNanoToddlerChainofNanoputians

PasdeduexFunnythingsinStructureChemistryNanoputians:AnthropomorphicMoleculesTour,J.M.etal,J.Org.Chem.2003,68,8750;J.Chem.Edu.2003,80,395.Self-assemblyofNanoputiansonGoldSurfaceRoleofStructuralChemistry

inSurfaceScience

fcc(100)fcc(111)fcc(775)fcc(1087)SurfacestructuresofPtsinglecrystalLow-indexsurface：NCSLIS<NCB.

High-indexsurface：Abundantedgesites.NCes<NCSLIS<NCBLowerNC~higherreactivity.NC-CoordinationNumber(111)(775)(100)(1087)Differentsurfacesdodifferentchemistry.Structure-sensitiveCatalysis!418125SurfaceStructurevs.CatalyticActivityN2+3H2

2NH3Fesinglecrystal，20atm/700KAnotherexampleofStructure-sensitiveCatalysisRoleofStructuralChemistry

inMaterialScience

Graphite&DiamondStructuresDiamond:

Insulatororwidebandgapsemiconductor:

Graphite:

Planarstructure:

sp2bonding

2dmetal(inplane)OtherCarbonallotropes“Buckyballs”(C60,C70etc)

“Buckytubes”(nanotubes), otherfullerenes

CCrystalStructuresSameElementvs.DifferentStructuresStructuremakesthedifference!ZhengLS(鄭蘭蓀),etal.Capturingthelabilefullerene[50]asC50Cl10

SCIENCE304(5671):699-699APR302004Thepentagon-pentagonfusionsinpristineC50-D5harestericallystrainedandhighlyreactive.PerchlorinationoftheseactivesitesstabilizesthelabileC50-D5h.NatureMaterials,2008,7,790.

Chlorofullerenesfeaturingtriplesequentiallyfusedpentagons

Xie,S.Y.,Lu,X.,Zheng,L.S.etal

NatureChem.2010,2,269.

#540C54Cl8(a),#864C56Cl12

(b),#4,169C66Cl6(c),#4,169C66Cl10(d).EndohedralMetallofullerene:

Sc4@C82(C3v)vs.Sc4C2@C80(Ih)(Sc2+)4@C828-Shinohara,H.Rep.Prog.Phys.2000,63,843.ProposedQM-predictedSc4@C82DE=28.8kcal/molDE=0.0kcal/molC26-@(Sc3+)4@C806--IhARussian-DollendofullereneX.Lu,J.Phys.Chem.B.2006,110,11098;Lu&Wang,J.Am.Chem.Soc.2009,131,16646.HighlightedbyC&ENandNat.Chem.RoleofStructuralChemistry

inLifeScience

Whatdoproteinsdo?Proteinsarethebasisofhowbiologygetsthingsdone.

Asenzymes,theyarethedrivingforcebehindallofthebiochemicalreactionswhichmakesbiologywork.Asstructuralelements,theyarethemainconstituentsofourbones,muscles,hair,skinandbloodvessels.Asantibodies,theyrecognizeinvadingelementsandallowtheimmunesystemtogetridoftheunwantedinvaders.Whatareproteinsmadeof?Proteinsarenecklacesofaminoacids,i.e.longchainmolecules.DefinitionofStructuralChemistryItisasubjecttostudythemicroscopicstructuresofmattersattheatomic/molecularlevelusingChemicalBondTheory.Chemicalbondsstructuresproperties.

ObjectiveofStructuralChemistryDeterminingthestructureofaknownsubstanceUnderstandingthestructure-propertyrelationshipPredictingasubstancewithspecificstructureandpropertyChapter1basicsofquantummechanics4Chapter2Atomicstructure4Chapter3Symmetry3Chapter4Diatomicmolecules3

MidtermExam!Chapter5/6Polyatomicstructures5Chapter7BasicsofCrystallography4

Chapter8MetalsandAlloys1Chapter9Ioniccompounds3Specialtalksymmetry&chemicalapplicationsOutlineandScheduleChapter1Thebasicknowledgeofquantummechanics

1.1Theoriginofquantummechanics---ThefailuresofclassicalphysicsBlack-bodyradiation,Photoelectriceffect,AtomicandmolecularspectraClassicalphysics:(priorto1900)NewtonianclassicalmechanicsMaxell’stheoryofelectromagneticwavesThermodynamicsandstatisticalphysics1.1.1Black-bodyradiationBlackbody:asubstancethatabsorbsallincidentrayswithnoorveryfewemissions.Devicefortheexperimentationofblack-bodyradiation.Itcannotbeexplainedbyclassicalthermodynamicsandstatisticalmechanics!Black-BodyRadiationWavelength/mAlargenumberofexperimentsrevealedthetemperature-dependenceoflmaxandindependenceonthesubstancemadeoftheblack-bodydevice.ClassicalSolution:Rayleigh-JeansLaw(lowenergy,highT)WienApproximation(highenergy,lowT)QuantizedEnergy--Thedawnofquantummechanics!AtagivenT,theprobabilityratioofatomicvibrationofnhnis1:exp(-hv/kT):…:exp(-nhv/kT)…1.1.2ThephotoelectriceffectThephotoelectriceffectThePhotoelectricEffect1.Thekineticenergyoftheejectedelectronsdependsexclusively

andlinearlyonthefrequencyofthelight.

2.Thereisaparticularthresholdfrequencyforeachmetal.

3.Theincreaseoftheintensityofthelightresultsintheincreaseofthenumberofphotoelectrons(currentintensity).Classicalphysics:Theenergyoflightwaveshouldbedirectlyproportionaltointensityandnotbeaffectedbyfrequency.ExplainingthePhotoelectricEffectAlbertEinstein

Proposedacorpusculartheoryoflight(photons)in1905.wontheNobelprizein19211.Lightisconsistedofastreamofphotons.Theenergyofaphotonisproportionaltoitsfrequency.

=h

h=Planck’sconstant2.Aphotonhasenergyaswellasmass.Mass-energyrelationship:

=mc2

m=h/c23.Aphotonhasadefinitemomentum.p=mc=h/c=h/

4.Theintensityoflightdependsonthephotondensity.Therefore,thephoton’senergyisthesumofthe

photoelectron’skineticenergyandthebindingenergyoftheelectroninmetal.

Ephoton=Ebinding+EKineticenergy

h

=W+Ek

(W=workfunction=Ebinding)ExplainingthePhotoelectricEffectExampleI:CalculationEnergyfromFrequencyProblem:1)WhatistheenergyofaphotonofelectromagneticradiationemittedbyanFMradiostationat97.3x108cycles/sec?2)WhatistheenergyofagammarayemittedbyCs137ifithasafrequencyof1.60x1020/s?Ephoton=hn=(6.626x10-34Js)(9.73x109/s)=6.447098x10-24JEphoton=6.45x10-24JEgammaray=hn=(6.626x10-34Js)(1.60x1020/s)=1.06x10-13JEgammaray=1.06x10-13JSolution:

Plan:Usetherelationshipbetweenenergyandfrequencytoobtaintheenergyoftheelectromagneticradiation(E=hn).ExampleII:CalculationofEnergyfromWavelengthProblem:Whatisthephotonenergyofelectromagneticradiationthatisusedinmicrowaveovensforcooking,ifthewavelengthoftheradiationis122mm?wavelength=122mm=1.22x10-1mfrequency===2.46x1010/s3.00x108m/s1.22x10-1mcwavelengthEnergy=E=hn=(6.626x10-34Js)(2.46x1010/s)=1.63x10-23JPlan:Convertthewavelengthintometers,thenthefrequencycanbecalculatedusingtherelationship;wavelengthxfrequency=c(wherecisthespeedoflight),thenusingE=hntocalculatetheenergy.Solution:ExampleIII:PhotoelectricEffectTheenergytoremoveanelectronfrompotassiummetalis3.7x10-19J.Willphotonsoffrequenciesof4.3x1014/s(redlight)and7.5x1014/s(bluelight)triggerthephotoelectriceffect?Ered=hn=(6.626x10-34Js)(4.3x1014/s)Ered=2.8x10-19JEblue=hn=(6.626x10-34Js)(7.5x1014/s)Eblue=5.0x10-19JThebindingenergyofpotassiumis=3.7x10-19JTheredlightwillnothaveenoughenergytoknockanelectronoutofthepotassium,butthebluelightwillejectanelectron!ETotal=EBindingEnergy+EKineticEnergyofElectronEElectron=ETotal-EBindingEnergyEElectron=5.0x10-19J-3.7x10-19J

=1.3x10-19Joules1.1.3AtomicandmolecularspectraTheLineSpectraofSeveralElementsAnatomcanemitlightsofdiscrete/specificfrequenciesuponelectric/photo-stimulation.FirstproposedbyRutherfordin1911.Theelectronsarelikeplanetsofthesolarsystem---orbitthenucleus(theSun).LightofenergyEgivenoffwhenelectronschangeorbitsofdifferentenergies.Whydotheelectronsnotfallintothenucleus?Whyaretheyindiscreteenergies?Planetarymodel:Basedonclassicalphysics,theelectronswouldbeattractedbythenucleusandeventuallyfallintothenucleusbycontinuouslyemittingenergy/light!!Bohr’satomicmodelNielsBohr,aDanishphysicist,combinedthePlank’squantaidea,Einstein’sphotontheoryandRutherford’sPlanetarymodel,andfirstintroducedtheideaofelectronicenergylevelintoatomicmodel.QuantumTheoryofEnergy.Theenergylevelsinatomscanbepicturedasorbitsinwhichelectronstravelatdefinitedistancesfromthenucleus.Thesehecalled“quantizedenergylevels”,alsoknownasprincipalenergylevels.

n:

principalquantumnumber

TheelectroninHatomcanbepromotedtohigherenergylevelsbyphotonsorelectricity.TheEnergyStatesoftheHydrogenAtomBohrderivedtheenergyforasystemconsistingofanucleusplusasingleelectroneg.Hepredictedasetofquantizedenergylevelsgivenby:-RiscalledtheRydbergconstant(2.18x10-18J)-nisaquantumnumber-ZisthenuclearchargeProblem:Findtheenergychangewhenanelectronchangesfromthen=4leveltothen=2levelinthehydrogenatom?Whatisthewavelengthofthisphoton?Plan:UsetheRydbergequationtocalculatetheenergychange,thencalculatethewavelengthusingtherelationshipofthespeedoflight.Solution:Ephoton=-2.18x10-18J-1n121n22Ephoton=-2.18x10-18J-=

4.09x10-19J142122

==hcE(6.626x10-34Js)(3.00x108m/s)4.09x10-19J=4.87x10-7m=487nm1.1.1Black-BodyRadiationPlanck’squantaideaE=nhnLesson1Summary1.1Thefailuresofclassicalphysics1.1.2ThephotoelectriceffectAcorpusculartheoryoflight(photons)

=h

=mc2

h=Planck’sconstantp=h/

1.1.3Atomicandmolecularspectra

Planetarymodel:orbitsofelectronsaroundthenucleusBohr’satomicmodel:quantizedenergylevelsoforbits1.2Thecharacteristicofthemotionofmicroscopicparticles1.2.1Thewave-particledualityofmicroscopicparticlesInclassicalphysics,wavesandparticlesbehavedifferentlyandcanbedescribedbyratherdifferenttheories.Asmentionedabove,Einstein’sCorpuscularTheoryofLights(photons)toexplainthephotoelectriceffectforthefirsttimeintroducedthewave-particledualityofphoton.In1924deBroglie

suggestedthatmicroscopicparticlessuchaselectronandprotonmightalsohavewavepropertiesinadditiontotheirparticleproperties.DeBroglieconsideredthatthewave-particlerelationshipinlightisalsoapplicabletoparticlesofmatter,i.e.Thelatterequationclearlyreflectsthewave-particledualitybyinvolvingbothaparticleproperty(momentum)andawaveproperty(wavelength)!Thewavelengthofaparticlecouldbedeterminedby

=

h/p=h/mv(v:velocity,m:mass)

E=h

p=h/

h=Planck’sconstant,p=particlemomentum,

=deBrogliewavelengthdeBroglieWavelengthExample:CalculatethedeBrogliewavelengthofanelectronwithspeed3.00x106m/s.electronmass=9.11x10-31kgvelocity=3.00x106m/s

==

h

mv6.626x10-34Js(9.11x10-31kg)(1.00x106m/s)Wavelength

=2.42x10-10m=0.242nmJ=kgm2s2henceThemovingspeedofanelectronisdeterminedbythepotentialdifferenceoftheelectricfield(V)IftheunitofVisvolt,thenthewavelengthis:1eV=1.602x10-19JThedeBroglieWavelengths

ofSeveralparticlesParticlesMass(g)Speed(m/s)

(m)Slowelectron9x10-281.07x10-4Fastelectron9x10-285.9x1061x10-10Alphaparticle6.6x10-241.5x1077x10-15One-grammass1.00.017x10-29Baseball14225.02x10-34Earth6.0x10273.0x1044x10-63DifferentBehaviorsofWavesandParticlesInclassicalphysics,wavesandparticlebehavedifferently!SiCrystalElectronbeam(50eV)ThediffractionofelectronsSTMimageofSi(111)7x7surfacePatternofelectrondiffractionSpatialimageoftheconfinedelectronstatesofaquantumcorral.Thecorralwasbuiltbyarranging48FeatomsontheCu(111)surfacebymeansoftheSTMtip.Rep.Prog.Phys.59(1996)1737ElectronaswavesWave(i.e.,light)-canbewave-like(diffraction)-canbeparticle-like(p=h/

)Particles-canbewave-like(

=h/p)-canbeparticle-like(classical)Thewave-particledualityAwaveofmicroscopicparticlesisaprobabilitywave,neitherlikethemacroscopicmechanicalwavenorlikethenormalelectromagneticwave!Itreflectsthestatisticprobabilityofparticlepresentinginspace!!Forphoton:

p=mc,E=h=h(c/)=pc=mc2ThedifferencesbetweenphotonandmicroscopicparticlesFormicroscopicparticles:

p=mv,E=(1/2)mv2=p2/2m=pv/2

=u/v

…whatisthemeaningofu?p2/2m=(1/2)mv2pvForphoton:v=c,

=h/p=h/(mc)

u=h/m

Forparticles:

=h/p=h/(mv)

u=h/m

E=h

p=h/

Wave-likeParticle-like1.2.2TheuncertaintyprincipleInclassicalPhysics,thepositionandmomentumofamacroscopicparticle(abody)canbecertainlydeterminedatagiventime.Thisisnotthecaseforamicroscopicparticle!Inthediffractionexperimentstorevealthewave-particledualityofelectrons,theobservedwavepatternisjustastatisticdistributionofelectronmotion.Theexactpositionandmomentumofanelectronatagiventimeremainuncertain.CsIfilme-beamImageofelectrondiffractionofCsITheexperimentsofelectronbeamdiffractionrevealedthatthenarrowertheslitis,thelargeristhecentralareaofthediffractionpattern.Whatisbehindthisphenomenon?psinqAOCqqPqAOqqPqAOPqAquantitativeversionElectrondiffractionvs.UncertaintyprincipleyBBIncludinghigherorder,(1st-orderdiffraction)OExampleThespeedofanelectronismeasuredtobe1000m/stoanaccuracyof0.001%.Findtheuncertaintyinthepositionofthiselectron.Themomentumofthiselectronisp=mv=(9.11x10-31kg)(1x103m/s)=9.11x10-28kg.m/sp=px0.001%=9.11x10-33kgm/sx=h/p=6.626x10-34/(9.11x10-33)=7.27x10-2(m)=7.27cmExampleThespeedofabulletofmassof0.01kgismeasuredtobe1000m/stoanaccuracyof0.001%.Findtheuncertaintyinthepositionofthisbullet.Themomentumisp=mv=(0.01kg)(1x103m/s)=10

kg.m/sp=px0.001%=1x10-4kgm/sx=h/p=6.626x10-34/(1x10-4)=6.626x10-30(m)ExampleTheaveragetimethatanelectronexistsinanexcitedstateis10-8s.Whatistheminimumuncertaintyinenergyofthatstate?AnotherformoftheUncertaintyPrinciple!MeasurementClassical:theerrorinthemeasurementdependsontheprecisionoftheapparatus,couldbearbitrarilysmall.Quantum:itisphysicallyimpossibletomeasuresimultaneouslytheexactpositionandtheexactvelocityofaparticle.CLASSICALvsQUANTUMMECHANICSMacroscopicmatter-Matterisparticulate,energyvariescontinuously.

Themotionofagroupofparticlescanbepredictedknowingtheir

positions,theirvelocitiesandtheforcesactingbetweenthem.

Microscopicparticles-microscopicparticlessuchaselectronsexhibit

awave-particle“duality”,showingbothparticle-likeandwave-likecharacteristics.Theenergylevelisdiscrete.…Thedescriptionofthebehaviorofelectronsinatomsrequiresacompletelynew“quantumtheory”.QuantummechanicaldescriptionofElectronQuantummechanicsisbasicallystatisticalinnature.Quantummechanicsdoesnotsaythatanelectronisdistributedoveralargeregionofspaceasawaveisdistributed.Ratheritistheprobabilitypatternsusedtodescribetheelectron’smotionthatbehavelikewaves.1.3Thebasicassumptions(postulates)ofquantummechanicsPostulate1.Thestateofasystemisdescribedbyawavefunctionofthecoordinatesandthetime.

InCM(classicalmechanics),thestateofasystemofNparticlesisspecifiedtotallybygiving3Nspatialcoordinates(Xi,Yi,Zi)and3Nvelocitycoordinates(Vxi,Vyi,Vzi).InQM,thewavefunctiontakestheformy(r,t)

thatdependsonthecoordinatesoftheparticlesandonthetime.Thus,thewavefunction

forasinglemicroscopicparticleof1-Dmotioncanbederivedas:Forexample:Thewavefunctionofplanemonochromaticlight:Accordingtothedualityofmicroscopicparticles,wehaveE=hvandp=h/l

v=E/h

and1/l=p/h.

Awavefunctionmustsatisfy3mathematicalconditions:Single-value;2.Continuous;3.Quadraticallyintegrable.Theproductofwavefunction(r,t)anditscomplexconjugate(r,t)*representstheprobabilitydistributionfunctionofthesystem.(Physicalmeaningofwavefunction!)Thewavefunction(r,t)mustbecontinuousinspace.Otherwiseitssecondderivativewouldnotbeattainable.Thewavefunctionofasystemmustbequadraticallyintegrablesoastoevaluatethestatisticalaveragevaluesofitsproperties.Theprobabilitythattheparticleliesinthevolumeelementdxdydz,locatedatr,attime

t.TheprobabilityTobegenerallynormalizedWavefunctionsofdifferentstatesforagivensystemmustbegenerallyorthogonalPostulate2.EachobservablemechanicalquantityofamicroscopicsystemisassociatedrespectivelywithalinearHermitianoperator.Tofindthisoperator,writedowntheclassical-mechanicalexpressionfortheobservableintermsofCartesiancoordinatesandcorrespondinglinear-momentum,andthenreplaceeachcoordinatexbytheoperatorx,andeachmomentumcomponentpxbytheoperator–i?/x.Intheprevioussubsection,weknowthatthewavefunction

forasinglemicroscopicparticleof1-Dmotioncanbederivedas:ThusAnoperatorisarulethattransformsagivenfunctionintoanotherfunction,e.g.d/dx,sin,logetc.Operatorsobeytheassociativelawofmultiplication:AlinearoperatormeansAHermitianoperatormeansAHermitianoperatorensuresthatitseigenvalueisarealnumber!EigenfunctionsandEigenvaluesSupposethattheeffectofoperatingonafunctionf(x)withtheoperator?issimplytomultiplyf(x)byacertainconstantk.Wethensaythatf(x)isaneigenfunctionof?witheigenvaluek.EigenisaGermanwordmeaningcharacteristic.e2x

isaneigenfunctionoftheoperatord/dx

withaneigenvalue2.ToeveryphysicalobservabletherecorrespondsalinearHermitianoperator.Tofindthisoperator,writedowntheclassical-mechanicalexpressionfortheobservableintermsofCartesiancoordinatesandcorrespondinglinear-momentumcomponents,andthenreplaceeachcoordinatexbytheoperatorxandeachmomentumcomponentpxbytheoperator-i?/x.MechanicalquantitiesandtheirOperatorsPositionx

Momentum(x)pxAngularMomentum(z)Mz=xpy-ypxKineticEnergyT=p2/2mPotentialEnergyVTotalEnergyE=T+V

SomeMechanicalquantitiesandtheirOperators

MechanicalquantitiesMethematicalOperatorHamiltonianIfasystemisinastatedescribedbyanormalizedwavefunctiony,thentheaveragevalueoftheobservableAcorrespondingtooperator?isgivenby–TheaveragevalueofaphysicalobservableThustheonlyvaluewemeasureisthevaluean.Ifthewavefunctionisaneigenfunctionof?,witheigenvaluean,thenameasurementoftheobservablecorrespondingto?willgivethevalueanwithcertainty.Whentwooperatorsarecommutable,theircorrespondingmechanicalquantitiescanbemeasuredsimultaneously.Commutedoperators(對易算符)PoissonbracketPostulate3:Thewave-functionofasystemevolvesintimeaccordingtothetime-dependentSchr?dingerequationToprovethis,weconsiderthesimplestcase,i.e.,aparticleof1-Dmotion.Itswavefunction

isexpressedas:Assumption3:Thewave-functionofasystemevolvesintimeaccordingtothetime-dependentSchr?dingerequation-IngeneraltheHamiltonianHisnotafunctionoft,sowecanapplythemethodofseparationofvariables.Time-independentSchr?dinger’sEquatione.g.HatomEigenvalueequationSphericalpolarcoordinatesFora2Dcirclingparticle,aLaplaceoperatorTheSchr?dinger’sEquationisaneigenequation.

I.TheeigenvalueofaHermitianoperatorisarealnumber.Proof:QuantummechanicaloperatorshavetohaverealeigenvaluesInanymeasurementoftheobservableassociatedwiththeoperatorA,theonlyvaluesthatwilleverbeobservedaretheeigenvalues

a,whichsatisfytheeigenvalueequation.II.TheeigenfunctionsofHermitianoperatorsareorthogonalII.TheeigenfunctionsofHermitianoperatorsareorthogonalConsiderthesetwoeigenequationsMultiplytheleftofthe1steqnbyym*andintegrate,thentakethecomplexconjugateofeqn2,multiplybyynandintegrateThereare2cases,n=m,orn

mSubtractingthesetwoequationsgives-Ifn=m,theintegral=1,bynormalization,soan=an*Ifn

m,andthesystemisnondegenerate(i.e.differenteigenfunctionsdonothavethesameeigenvalues,an

am),thenTheeigenfunctionsofHermitianoperatorsareorthogonal.Example:Postulate4:SuperpositionPrinciple

(態(tài)疊加原理)

If

1,

2,…

narethepossiblestatesofamicroscopicsystem(acompleteset),thenthelinearcombinationofthesestatesisalsoapossiblestateofthesystem.Thecoefficientcireflectsthecontributionofwavefunction

i

to

.Postulate5:Pauli’sprinciple(泡利不相容原理).

Everyatomicormolecularorbitalcanonlycontainamaximumoftwoelectronswithoppositespins.EnergyleveldiagramforHe.Electronconfiguration:1s2Hparamagnetic–one(more)unpairedelectronsHediamagnetic–allpairedelectronsEnergy123012nlms=spinmagneticelectronspin

ms=±?(-?=)(+?=)

Thecompletewavefunctionforthedescriptionofelectronicmotionshouldincludeaspinparameterinadditiontoitsspatialcoordinates.Twoelectronsinthesameorbitalmusthaveoppositespins.Electronspinisapurelyquantummechanicalconcept.Pauliexclusionprinciple:Eachelectronmusthaveauniquesetofquantumnumbers.Thecompletewavefunctionfordescriptionofelectronicmotionshouldincludeaspinparameterinadditiontoitsspatialcoordinates.FermionsParticlesthatdoobeythePauliExclusionPrinciple.BosonsParticlesthatdonotobeythePauliExclusionPrinciple+symmetry(Bosons)-antisymmetry(Fermions)1.4Solutionoffreeparticleinabox–

asimpleapplicationofQuantumMechanics1.4.1Thefreeparticleinaone

dimensionalbox1.The

Schr?dinger’sEquationanditssolutionInareaI,III:

1Dbox(meaningnoparticleinareaIandIII).II:V=0Boundaryconditionandcontinuouscondition:(0)=0,(l)=0Hence,

(0)=Acos0+Bsin0=A+0=0=>A=0

=Bsinx(B0)

(l)=Bsinx=Bsinl=0,Thus,l=n,=n/l(n=1,2,…)Normalizationofwave-function:2.Thepropertiesofthesolutionsa.Theparticlecanexistinmanystatesb.quantizationenergyc.Theexistenceofzero-pointenergy.minimumenergy(h2/8ml2)d.Thereisnotrajectorybutonlyprobabilitydistributione.Thepresenceofnodesn=1n=2n=3EnergyE2E3E4E1n=2Inthegroundstate(n=1),thehighestprobabilityoftheparticleoccursatthelocationl/2.Inthefirstexcitedstate(n=2),thehighestprobabilityoftheparticleoccursatthelocationsl/4and3l/4,thelowestprobabilityatthelocationl/2.nodenodeDiscussion:i.Normalizationandorthogonalityi