Upload
faizalexandaria
View
212
Download
0
Embed Size (px)
DESCRIPTION
jkm
Citation preview
Chapter41.OneDimensionalQuantumMechanicsMechanics
Quantumeffectsareimportant in nanostructuresimportantinnanostructuressuchasthistinysignbuiltbyscientistsatIBMsresearchlaboratorybymovingxenonatomsaroundonametal
fsurface.
ChapterGoal:Tounderstandand apply the essential ideasandapplytheessentialideasofquantummechanics.
Chapter41.OneDimensionalQuantumMechanics
Topics:
S h di E ti Th L f P i
QuantumMechanics
SchrdingersEquation:TheLawofPsiSolvingtheSchrdingerEquation
A Particle in a Rigid Box: Energies and WaveAParticleinaRigidBox:EnergiesandWaveFunctions
AParticleinaRigidBox:InterpretingtheSolutionTheCorrespondencePrinciple
FinitePotentialWellsWaveFunctionShapes
TheQuantumHarmonicOscillatorMore Quantum ModelsMoreQuantumModels
QuantumMechanicalTunneling
TheSchrdingerEquationTheSchrdingerEquationConsideranatomicparticlewithmassmandmechanicalenergyEinanenvironmentcharacterizedbyapotentialenergyfunctionU(x).TheSchrdingerequationfortheparticleswavefunctionisis
Conditionsthewavefunctionmustobeyare1. (x)and(x)arecontinuousfunctions.2. (x)=0ifxisinaregionwhereitisphysically
impossiblefortheparticletobe.3 (x) 0 as x + and x 3. (x)0asx+andx.4. (x)isanormalizedfunction.
SolvingtheSchrdingerEquationSolvingtheSchrdingerEquation
Ifasecondorderdifferentialequationhastwoindependentsolutions1(x)and2(x),thenageneralp 1( ) 2( ) gsolutionoftheequationcanbewrittenas
whereAandBareconstantswhosevaluesaredeterminedbytheboundaryconditions.
ThereisamoregeneralformoftheSchrodingerequationwhichincludestimedependenceandx,y,zcoordinates;
Wewilllimitdiscussionto1Dsolutions
MustknowU(x),thepotentialenergyfunctiontheparticleexperiencesasitmoves.
Objectiveistosolvefor(x)andthetotalenergyE=KE+Uoftheparticle.In bound state problems where the particle is trapped (localized in space)In boundstate problemswheretheparticleistrapped(localizedinspace),theenergieswillbefoundtobequantizeduponsolvingtheSchrodingerequation.
Inunboundstateswheretheparticleisnottrapped,theparticlewilltravelasatravelingwavewithanamplitudegivenby(x)
E,KE,andPEE,KE,andPE
E,KE,andPEE,KE,andPE
TheSchrdingerEquationwithConstantpotentialTheSchrdingerEquationwithConstantpotential
AParticleinaRigidBoxAParticleinaRigidBox
AParticleinaRigidBoxAParticleinaRigidBox
Consideraparticleofmassmconfinedinarigid,onedimensionalbox.Theboundariesoftheboxareatx =0andx =L.
1. Theparticlecanmovefreelybetween0andLatconstant speed and thus with constant kineticconstantspeedandthuswithconstantkineticenergy.
2. Nomatterhowmuchkineticenergytheparticlehas,itsturningpointsareatx =0andx =L.
3. Theregionsx L areforbidden.Theparticlecannot leave the boxcannotleavethebox.
Apotentialenergyfunctionthatdescribestheparticleinthissituationis
Zeropointenergy:evenatT=0K,aconfinedparticlewillhaveanonzeroenergy of E1; it is movingenergyofE1;itismoving
TheCorrespondencePrincipleTheCorrespondencePrincipleWhenwavelengthbecomessmallcomparedtothesizeofthebox(thatis,wheneitherLbecomeslargeorwhentheenergyoftheparticlebecomeslarge),theparticlemustbehaveclassically.
For particle in a box:Forparticleinabox:
Classically:
TheCorrespondencePrincipleTheCorrespondencePrincipleWhenwavelengthbecomessmallcomparedtothesizeofthebox(thatis,wheneitherLbecomeslargeorwhentheenergyoftheparticlebecomeslarge),theparticlemustbehaveclassically.
For particle in a box: Classically:Forparticleinabox: Classically:
FinitePotentialwell:FinitePotentialwell:1 Solve Schrodingers equation in the1. SolveSchrodinger sequationinthe
threeregions(wealreadydidthis!)2. Connectthethreeregionsbyusingthe
followingboundaryconditions:
1. ThiswillgivequantizedksandEs2. Normalizewavefunction
FinitePotentialwell:FinitePotentialwell:
Finitenumberofboundstates,energyspacingsmallersincewavefunctionmorespreadout(likebiggerL),wavefunctionsextendintoclassicallyforbiddenregion
ClassicallyforbiddenregionClassicallyforbiddenregion penetrationdepthpenetrationdepth
FinitePotentialwellexampleFinitePotentialwellexample QuantumwelllasersQuantumwelllasers
FinitePotentialwellexampleFinitePotentialwellexample 11DmodelofnucleusDmodelofnucleus
QualitativewavefunctionshapesQualitativewavefunctionshapesExponentialdecayifU>E,oscillatoryifE>Ui.e.positiveKE,KE~p2 ~1/2,p y , y p , p / ,Amplitude~1/v~1/Sqrt[KE](particlemovingslowermeansmorelikelytobeinthatplace)
1 Solve Schrodingers equation in the
HarmonicOscillatorHarmonicOscillator1. SolveSchrodinger sequationinthe
threeregions(wealreadydidthis!)2. Connectthethreeregionsbyusingthe
followingboundaryconditions:
3. ThiswillgivequantizedksandEsg q4. Normalizewavefunction
MolecularvibrationsMolecularvibrations HarmonicOscillatorHarmonicOscillator
E=totalenergyofthetwointeractingatoms,NOTofasingleparticleU=potentialenergybetweenthetwotatoms
ThepotentialU(x)isshownfortwoatoms.Thereexistanequilibriumseparation.
Atlowenergies,thisdiplookslikeaparabola Harmonic oscillatorparabola Harmonicoscillatorsolution.
Allowed(total)vibrationalenergies:
ParticleinacapacitorParticleinacapacitor
ParticleinacapacitorParticleinacapacitor
CovalentBond:H2+(singleelectron)CovalentBond:H2+(singleelectron)
CovalentBond:H2+(singleelectron)CovalentBond:H2+(singleelectron)
QuantumTunnelingQuantumTunneling
QuantumTunnelingQuantumTunneling
QuantumTunnelingQuantumTunneling
QuantumTunnelingQuantumTunneling ResonanttunnelingResonanttunneling