1Friday Honors lectureProf. Clint Sprotttakes us on a tourof fractals.Thu. 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 eV Quantized energy levels: Each corresponds todifferent Orbit radius Velocity Particle wavefunction Energy Each described by aquantum 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 L Classically, 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 Quantum Classical: particle bounces back and forth. Sometimes velocity is to left, sometimes to rightL Quantum mechanics: Particle represented by wave: p = mv = h / λ Different motions: waves traveling left and right Quantum wave function: superposition of both at same timeThu. Nov. 29 2007 Physics 208, Lecture 25 5Quantum version Quantum state is both velocities at the same time Ground state is a standing wave, made equally of Wave traveling right ( p = +h/λ ) Wave traveling left ( p = - h/λ ) Determined by standing wave condition L=n(λ/2) :! "= 2LOne half-wavelength! p =h"=h2LmomentumLQuantum wave function:superposition of both motions.! "x( )=2Lsin2#$x% & ' ( ) * Thu. Nov. 29 2007 Physics 208, Lecture 25 6Different quantum states p = mv = h / λ Different speeds correspond to different λ subject to standing wave conditioninteger number of half-wavelengths fit in the tube.! "= LTwo half-wavelengths! p =h"=hL= 2pomomentum! "= 2LOne half-wavelength! p =h"=h2L# pomomentumn=1n=2! "x( )=2Lsin2#$x% & ' ( ) * Wavefunction:2Thu. 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 levelsA. 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 levels Quantized momentum Energy = kinetic Or Quantized Energy! E =p22m=npo( )22m= n2Eo! En= n2Eo! p =h"= 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 box Energy level spacing increases as particle size decreases. i.eCdSe quantum dotsdispersed in hexane(Bawendi group, MIT)Color from photonabsorptionDetermined by energy-level spacingDecreasing particle size! En +1" En=n + 1( )2h28mL2"n2h28mL2Thu. Nov. 29 2007 Physics 208, Lecture 25 11Interpreting the wavefunction Probabilistic interpretationThe square magnitude of the wavefunction |Ψ|2 gives theprobability of finding the particle at a particular spatiallocationWavefunction 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 E3Thu. Nov. 29 2007 Physics 208, Lecture 25 13Probability of finding electron Classically, equally likely to find particle anywhere QM - true on average for high nZeroes in the probability!Purely quantum, interference effectThu. Nov. 29 2007 Physics 208, Lecture 25 14Quantum Corral 48 Iron atoms assembled into a circular ring. The ripples inside the ring reflect the electron quantum states of acircular ring (interference effects).D. Eigler (IBM)Thu. Nov. 29 2007 Physics 208, Lecture 25 15Scanning Tunneling Microscopy Over the last 20 yrs, technology developed to controllablyposition 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 entirelywithin closed tube.Open top: particle can escape ifwe shake hard enough.But at low energies, particlestays entirely within box.Like an electron in metal(remember photoelectric effect)Thu. Nov. 29 2007 Physics 208, Lecture 25 17Quantum mechanics sayssomething different!Quantum Mechanics:some probability of theparticle penetratingwalls of box!Low energyClassical stateLow energyQuantum stateNonzero probability of being outside the box.Thu. Nov. 29 2007 Physics 208, Lecture 25 18Two neighboring boxes When another box is brought nearby, theelectron may disappear from one well, andappear in the other! The reverse then happens, and the electronoscillates back an forth, without ‘traversing’ theintervening distance.4Thu. Nov. 29 2007 Physics 208, Lecture 25 19Question‘low’ probability‘high’ probabilitySuppose separation between boxes increases by a factor of two.The tunneling probabilityA. Increases by 2B. Decreases by 2C. Decreases by <2D. Decreases by >2E. Stays sameThu. Nov. 29 2007 Physics 208, Lecture 25 20Example:Ammonia molecule Ammonia molecule: NH3 Nitrogen (N) has two equivalent‘stable’ positions. Quantum-mechanically tunnels2.4x1011 times per second (24 GHz) Known as ‘inversion line’ Basis of first ‘atomic’ clock (1949)NHHHThu. Nov. 29 2007 Physics 208, Lecture 25 21Atomic clock questionSuppose we changed the ammonia molecule sothat the distance between the two stable positionsof the nitrogen atom INCREASED.The clock wouldA. slow down.B. speed up.C. stay the same.NHHHThu. Nov. 29 2007 Physics 208, Lecture 25 22Tunneling between conductors Make one well deeper: particle tunnels, then stays in other well. Well made deeper by applying electric field. This is the principle of scanning tunneling microscope.Thu. Nov. 29 2007 Physics 208, Lecture 25 23Scanning Tunneling Microscopy Over the last 20 yrs, technology developed to controllablyposition tip and sample 1-2 nm apart. Is a very useful microscope!TipSampleTip, sample are quantum‘boxes’Potential difference inducestunnelingTunneling extremely sensitiveto tip-sample
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