From Last time…Exam 3 resultsThe wavefunctionProbabilityDiscrete vs continuousExample wavefunctionWavefunctionsQuantum ‘Particle in a box’Classical vs QuantumQuantum versionDifferent quantum statesParticle in box questionParticle in box energy levelsZero-point motionQuestionQuantum dot: particle in 3D boxInterpreting the wavefunctionHigher energy wave functionsProbability of finding electronQuantum CorralParticle in box again: 2 dimensionsQuantum Wave Functions2D excited statesParticle in a boxNext higher energy stateThree dimensionsParticle in 3D boxPowerPoint PresentationSlide 293-D particle in box: summaryFrom Last time…De Broglie wavelength Uncertainty principleWavefunction of a particleTue. Dec. 1 2008 Physics 208, Lecture 25 2Exam 3 resultsExam average ~ 70%Scores posted on learn@uwCourse evaluations:Tuesday, Dec. 9QuickTime™ and aTIFF (LZW) decompressorare needed to see this picture.AABBBCCDTue. Dec. 1 2008 Physics 208, Lecture 25 3The wavefunctionParticle has a wavefunction (x)x€ Ψ2x( )dx= probability to find particle in infinitesimal range dx about x-0.0500.050.10.150.2-4 -3 -2 -1 0 1 2 3 4x2dx2(x)Very small x-rangeLarger x-range€ Ψ2x( )dxx1x2∫= probability to find particle between x1 and x2Entire x-range€ Ψ2x( )dx−∞∞∫=1particle must be somewhereTue. Dec. 1 2008 Physics 208, Lecture 25 4ProbabilityHeadsTailsProbability0.50.01Probability0.50.02 3 4 5 61/6P(heads)=0.5P(tails)=0.5P(1)=1/6P(2)=1/6etc€ P 1( )+ P 2( )+ P 3( )+ P 4( )+ P 5( )+ P 6( )=1€ P heads( )+ P tails( )=1Tue. Dec. 1 2008 Physics 208, Lecture 25 5Discrete vs continuousInfinite-sided die, all numbers between 1 and 6“Continuous” probability distribution 1Probability0.50.02 3 4 5 61/6Loaded die6-sided die, unequal prob1P(x)0.50.02 3 4 5 61/60x€ P(x)dx =116∫Tue. Dec. 1 2008 Physics 208, Lecture 25 6Example wavefunctionWhat is P(-2<x<-1)?-3 -2 -1 0 1 2 3x2-3 -2 -1 0 1 2 3x€ = Ψ2x( )dx−2−1∫= fractional area under curve -2 < x < -1= 1/8 total area -> P = 0.125What is (0)?€ Ψ2x( )dx−∞−∞∫= entire area under 2 curve = 1= (1/2)(base)(height)=22(0)=1€ → Ψ 0( )=1/ 2Tue. Dec. 1 2008 Physics 208, Lecture 25 7WavefunctionsEach quantum state has different wavefunctionWavefunction shape determined by physical characteristics of system.Different quantum mechanical systemsPendulum (harmonic oscillator)Hydrogen atomParticle in a boxEach has differently shaped wavefunctionsTue. Dec. 1 2008 Physics 208, Lecture 25 8Quantum ‘Particle in a box’Particle confined to a fixed region of spacee.g. ball in a tube- ball moves only along length LClassically, ball bounces back and forth in tube.A classical ‘state’ of the ball. State indexed by speed, momentum=(mass)x(speed), or kinetic energy.Classical: any momentum, energy is possible.Quantum: momenta, energy are quantizedLTue. Dec. 1 2008 Physics 208, Lecture 25 9Classical vs QuantumClassical: particle bounces back and forth. Sometimes velocity is to left, sometimes to rightLQuantum mechanics: Particle represented by wave: p = mv = h / Different motions: waves traveling left and rightQuantum wavefunction: superposition of both at same timeTue. Dec. 1 2008 Physics 208, Lecture 25 10Quantum versionQuantum state is both velocities at the same timeSuperposition waves is standing wave, made equally of Wave traveling right ( p = +h/ )Wave traveling left ( p = - h/ )Determined by standing wave condition L=n(/2) : € λ =2LOne half-wavelength€ p =hλ=h2LmomentumLQuantum wave function: superposition of both motions.€ ψ x( )=2Lsin2πλx ⎛ ⎝ ⎜ ⎞ ⎠ ⎟Tue. Dec. 1 2008 Physics 208, Lecture 25 11Different quantum states p = mv = h / Different speeds correspond to different subject to standing wave condition integer number of half-wavelengths fit in the tube.€ λ =LTwo half-wavelengths€ p =hλ=hL= 2 pomomentum€ λ =2LOne half-wavelength€ p =hλ=h2L≡ pomomentumn=1n=2€ ψ x( )=2Lsin n2πλx ⎛ ⎝ ⎜ ⎞ ⎠ ⎟Wavefunction:€ λ =2L /nn=1n=2Tue. Dec. 1 2008 Physics 208, Lecture 25 12Particle in box questionA particle in a box has a mass m. Its energy is all kinetic = p2/2m. Just saw that momentum in state n is npo. It’s energy levels A. are equally spaced everywhereB. get farther apart at higher energyC. get closer together at higher energy.Tue. Dec. 1 2008 Physics 208, Lecture 25 13Particle in box energy levelsQuantized momentumEnergy = kinetic Or Quantized Energy€ E =p22m=npo( )22m= n2Eo€ En= n2Eo€ p =hλ= nh2L= npoEnergyn=1n=2n=3n=4n=5n=quantum numberTue. Dec. 1 2008 Physics 208, Lecture 25 14Zero-point motionLowest energy is not zeroConfined quantum particle cannot be at restAlways some motionConsequency of uncertainty principlep cannot be zerop not exactly knownp cannot be exactly zero € ΔxΔp > h /2Tue. Dec. 1 2008 Physics 208, Lecture 25 15QuestionA particle is in a particular quantum state in a box of length L. The box is now squeezed to a shorter length, L/2.The particle remains in the same quantum state.The energy of the particle is nowA. 2 times biggerB. 2 times smallerC. 4 times biggerD. 4 times smallerE. unchangedTue. Dec. 1 2008 Physics 208, Lecture 25 16Quantum dot: particle in 3D boxEnergy level spacing increases as particle size decreases.i.e CdSe quantum dots dispersed in hexane(Bawendi group, MIT)Color from photon absorptionDetermined by energy-level spacingDecreasing particle size€ En +1− En=n +1( )2h28mL2−n2h28mL2Tue. Dec. 1 2008 Physics 208, Lecture 25 17Interpreting the wavefunctionProbability interpretationThe square magnitude of the wavefunction ||2 gives the probability of finding the particle at a particular spatial locationWavefunction Probability = (Wavefunction)2Tue. Dec. 1 2008 Physics 208, Lecture 25 18Higher energy wave functionsn=1n=2n=3Wavefunction ProbabilityL€ h2L€ 2h2L€ 3h2L€ h28mL2€ 22h28mL2€ 32h28mL2n p ETue. Dec. 1 2008 Physics 208, Lecture 25 19Probability of finding electronClassically, equally likely to find particle anywhereQM - true on average for high nZeroes in the probability!Purely quantum, interference effectTue. Dec. 1 2008 Physics 208, Lecture 25 20Quantum Corral48 Iron atoms assembled into a circular ring.The ripples inside the ring reflect the electron quantum states of a circular ring (interference
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