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UW-Madison PHYSICS 107 - Lecture 25 Notes

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1Phy107 Fall 20061• Observation of atoms indicated quantized energy states.• Atom only emitted certain wavelengths of light• Structure of the allowed wavelengths indicated thewhat the energy structure was• Quantum mechanics and the wave nature of the electronallowed us to understand these energy levels.From Last Time…Today• The quantum wave function• The atom in 3 dimensions• Uncertainty principle againPhy107 Fall 20062Hydrogen atom energies• Electrons orbit the atom inquantized energy states• Energy states are resonantstates where the electronwave constructivelyinterferes with itself. nwhole wavelengths around• Wavelength gets longer inhigher n states and thekinetic energy goes downproportions to 1/n2• Potential energy goes up aswith gravity also as 1/n2Zero energyn=1n=2n=3n=4E1= 13.612 eVE2= 13.622 eVE3= 13.632 eVEnergyEn= 13.6n2 eVPhy107 Fall 20063Hydrogen atom questionHere is Peter Flanary’ssculpture ‘Wave’ outsideChamberlin Hall. Whatquantum state of thehydrogen atom could thisrepresent?A. n=2B. n=3C. n=4Phy107 Fall 20064Another questionHere is Donald Lipski’s sculpture‘Nail’s Tail’ outside Camp RandallStadium.What could it represent?A. A pile of footballsB. “I hear its made of plastic. For 200 grand,I’d think we’d get granite”- Tim Stapleton (Stadium Barbers)C. “I’m just glad it’s not my money”- Ken Kopp (New Orlean’s Take-Out)D. Amazingly physicists make bettersculptures!Phy107 Fall 20065Compton scattering andPhotoelectric effect• Collision of photon and electron• Photon loses energy and momentum, transfers it to electron•Either:– Loses enough energy/momentum to bump it up one level– Electron later decays back to ground state releasing a photon– See reflected and emitted photons when looking at an object– Or has enough energy to completely knock the electron out of thesystem. Photoelectric effect!Before collisionAfter collisionPhy107 Fall 20066Simple Example: ‘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.– No friction, so ball continues to bounce back and forth,retaining its initial speed.– This is a ‘classical state’ of the ball. A different classical state wouldbe ball bouncing back and forth with different speed.– Could label each state with a speed,momentum=(mass)x(speed), or kinetic energy.– Any momentum, energy is possible.Can increase momentum in arbitrarily small increments.L2Phy107 Fall 20067Quantum Particle in a Box• In Quantum Mechanics, ball represented by wave– Wave reflects back and forth from the walls.– Reflections cancel unless wavelength meets thestanding wave condition:integer number of half-wavelengths fit in the tube.= LTwo half-wavelengthsp =h=hL= 2 pomomentum= 2LOne half-wavelengthp =h=h2L pomomentumn=1n=2Phy107 Fall 20068Particle in a boxLowest energystateNext higherenergy state3rd energystateWave functionProbability: Square ofthe wave functionLPhy107 Fall 20069Particle in box questionA particle in a box has a mass m.It’s energy is all energy of motion = p2/2m.We just saw that it’s 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.Phy107 Fall 200610General aspects of Quantum Systems• System has set of quantum states, labeled by an integer(n=1, n=2, n=3, etc)• Each quantum state has a particular frequency and energyassociated with it.• These are the only energies that the system can have:the energy is quantized• Analogy with classical system:– System has set of vibrational modes, labeled by integerfundamental (n=1), 1st harmonic (n=2), 2nd harmonic (n=3), etc– Each vibrational mode has a particular frequency and energy.– These are the only frequencies at which the system resonates.Phy107 Fall 200611Wavefunction of pendulumn=2n=3n=1ground stateHere are quantumwavefunctions of apendulum. Which hasthe lowest energy?Phy107 Fall 200612Probability density of oscillatorMoves fast here,low prob of finding in a‘blind’ measurementMoves slow here,high prob offindingClassical prob3Phy107 Fall 200613Wavefunctions in two dimensions• Physical objects often can move in more thanone direction (not just one-dimensional)• Could be moving at one speed in x-direction,another speed in y-direction.• From deBroglie relation, wavelength related tomomentum in that direction• So wavefunction could have differentwavelengths in different directions.=hpPhy107 Fall 200614Two-dimensional (2D) particle in boxGround state: samewavelength (longest) inboth x and yNeed two quantum #’s,one for x-motionone for y-motionUse a pair (nx, ny)Ground state: (1,1)Probability(2D)Wavefunction Probability = (Wavefunction)2One-dimensional (1D) casePhy107 Fall 2006152D excited states(nx, ny) = (2,1)(nx, ny) = (1,2)These have exactly the same energy, but theprobabilities look different.The different states correspond to ball bouncingin x or in y direction.Phy107 Fall 200616Particle in a boxWhat quantum state could this be?A. nx=2, ny=2B. nx=3, ny=2C. nx=1, ny=2Phy107 Fall 200617Three dimensions• Object can have different velocity (hencewavelength) in x, y, or z directions.– Need three quantum numbers to label state•(nx, ny , nz) labels each quantum state(a triplet of integers)• Each point in three-dimensional space has aprobability associated with it.• Not enough dimensions to plot probability• But can plot a surface of constant probability.Phy107 Fall 2006183D particle in box• Ground statesurface of constantprobability•(nx, ny, nz)=(1,1,1)• Like the 2D case -highest probabilityin the center andless further out2D case4Phy107 Fall 200619(211)(121)(112)All these states have the sameenergy, but different probabilitiesPhy107 Fall 200620(221)(222)Phy107 Fall 200621Hydrogen atom• Hydrogen a little different, in that it hasspherical symmetry• Not square like particle in a box.• Still need three quantum numbers, but theyrepresent ‘spherical’ things like– Radial distance from nucleus– Azimuthal angle around nucleus– Polar angle around nucleus• Quantum numbers are integers (n, l, ml)Phy107 Fall 200622Hydrogen atom:Lowest energy (ground) state• Lowest energy state issame in all directions.• Surface of constantprobability is surface ofa sphere. n = 1, l = 0, ml= 0Phy107 Fall 200623n=2:


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UW-Madison PHYSICS 107 - Lecture 25 Notes

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