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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

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

CovalentBond:H2+(singleelectron)CovalentBond:H2+(singleelectron)

QuantumTunnelingQuantumTunneling

QuantumTunnelingQuantumTunneling ResonanttunnelingResonanttunneling