Last Time…Hydrogen atom energiesQuantum ‘Particle in a box’Classical vs QuantumQuantum versionDifferent quantum statesParticle in box questionParticle in box energy levelsQuestionQuantum dot: particle in 3D boxInterpreting the wavefunctionHigher energy wave functionsProbability of finding electronQuantum CorralScanning Tunneling MicroscopyParticle in a box, againQuantum mechanics says something different!Two neighboring boxesSlide 19Example: Ammonia moleculeAtomic clock questionTunneling between conductorsSlide 23Surface steps on SiManipulation of atomsSlide 26The Stadium CorralSome fun!Particle in box again: 2 dimensionsQuantum Wave Functions2D excited statesParticle in a boxNext higher energy stateThree dimensionsParticle in 3D boxPowerPoint PresentationSlide 37The ‘principal’ quantum numberOther quantum numbers?Sommerfeld: modified Bohr modelAngular momentum questionThe orbital quantum number ℓOrbital mag. momentOrbital magnetic dipole momentOrbital mag. quantum number mℓSlide 46Slide 47Interaction with applied B-fieldSlide 49Summary of quantum numbersAdditional electron propertiesElectron magnetic momentElectron spin orientationsSlide 54Spin: another quantum numberInclude spinSlide 57Slide 58Slide 59Putting electrons on atomOther elements: Li has 3 electronsSlide 62The periodic tableWavefunctions and probabilityHydrogen atom: Lowest energy (ground) staten=2: next highest energyn=3: two s-states, six p-states and……ten d-statesSlide 69Emitting and absorbing lightThe wavefunctionSiliconParticle in box wavefunctionMaking a measurementLast Time… Bohr model of Hydrogen atomWave properties of matterEnergy levels from wave propertiesThu. Nov. 29 2007 Physics 208, Lecture 25 2Hydrogen atom energiesZero energyn=1n=2n=3n=4€ E1= −13.612 eV€ E2= −13.622 eV€ E3= −13.632 eVEnergy€ En= −13.6n2 eVQuantized energy levels:Each corresponds to differentOrbit radiusVelocityParticle wavefunctionEnergyEach described by a quantum number nThu. Nov. 29 2007 Physics 208, Lecture 25 3Quantum ‘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.This is a ‘classical state’ of the ball. Identify each state by speed, momentum=(mass)x(speed), or kinetic energy.Classical: any momentum, energy is possible.Quantum: momenta, energy are quantizedLThu. Nov. 29 2007 Physics 208, Lecture 25 4Classical 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 wave function: superposition of both at same timeThu. Nov. 29 2007 Physics 208, Lecture 25 5Quantum versionQuantum state is both velocities at the same timeGround state is a 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 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟Thu. Nov. 29 2007 Physics 208, Lecture 25 6Different 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( )=2Lsin2πλx ⎛ ⎝ ⎜ ⎞ ⎠ ⎟Wavefunction:Thu. Nov. 29 2007 Physics 208, Lecture 25 7Particle 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.Thu. Nov. 29 2007 Physics 208, Lecture 25 8Particle 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 numberThu. Nov. 29 2007 Physics 208, Lecture 25 9QuestionA 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. unchangedThu. Nov. 29 2007 Physics 208, Lecture 25 10Quantum 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−n2h28mL2Thu. Nov. 29 2007 Physics 208, Lecture 25 11Interpreting the wavefunctionProbabilistic interpretationThe square magnitude of the wavefunction ||2 gives the probability of finding the particle at a particular spatial locationWavefunction Probability = (Wavefunction)2Thu. Nov. 29 2007 Physics 208, Lecture 25 12Higher energy wave functionsn=1n=2n=3Wavefunction ProbabilityL€ h2L€ 2h2L€ 3h2L€ h28mL2€ 22h28mL2€ 32h28mL2n p EThu. Nov. 29 2007 Physics 208, Lecture 25 13Probability of finding electronClassically, equally likely to find particle anywhereQM - true on average for high nZeroes in the probability!Purely quantum, interference effectThu. Nov. 29 2007 Physics 208, Lecture 25 14Quantum Corral48 Iron atoms assembled into a circular ring.The ripples inside the ring reflect the electron quantum states of a circular ring (interference effects).D. Eigler (IBM)Thu. Nov. 29 2007 Physics 208, Lecture 25 15Scanning Tunneling MicroscopyOver the last 20 yrs, technology developed to controllably position tip and sample 1-2 nm apart.Is a very useful microscope! TipSampleThu. Nov. 29 2007 Physics 208, Lecture 25 16Particle in a box, againLWavefunctionProbability = (Wavefunction)2Particle contained entirely within closed tube.Open top: particle can escape if we shake hard enough.But at low energies, particle stays entirely within box.Like an electron in metal (remember photoelectric effect)Thu. Nov. 29 2007 Physics 208, Lecture 25 17Quantum mechanics says something different!Quantum Mechanics:some probability of the particle penetrating walls of box!Low energy Classical stateLow energy Quantum stateNonzero
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