1Wed. Mar. 28, 2007 Phy107 Lect 271From Last Time…Quantum system: particle in a box.Quantization: only certain wavelengths allowedDe Broglie: p=h/λ -> quantized momentaKinetic energy E=p2/2m -> quantized energyHW due date extended to Thursday midnite (March 29)Essay topic and paragraph due todayWed. Mar. 28, 2007 Phy107 Lect 272Particle in a boxLowest energystateNext higherenergy state3rd energystateWavefunction ProbabilityLWed. Mar. 28, 2007 Phy107 Lect 273Electron waves in an atom• Electron is a wave.• In the orbital picture, itspropagation direction is aroundthe circumference of the orbit.• Wavelength = h / p(p=momentum, and energydetermined by momentum)• How can we think about waveson a circle?Wed. Mar. 28, 2007 Phy107 Lect 274Waves on a ring• Condition on a ringslightly different.• Integer number ofwavelengths requiredaround circumference.• Otherwise destructiveinterference occurs whenwave travels around ringand interferes with itself.WavelengthWed. Mar. 28, 2007 Phy107 Lect 275Wave representingelectronElectron standing-waves on an atom• Electron wave extendsaround circumference oforbit.• Only integer number ofwavelengths around orbitallowed.Wave representingelectronWed. Mar. 28, 2007 Phy107 Lect 276Hydrogen atom music• These are the five lowest energyorbits for the one electron in thehydrogen atom.• Each orbit is labeled by thequantum number n.• The radius of each is n2ao.• Hydrogen has one electron:the electron must be in one ofthese orbits.• The smallest orbit has the lowestenergy. The energy is larger forlarger orbits.2Wed. Mar. 28, 2007 Phy107 Lect 277Hydrogen atom music• Here the electron is in then=3 orbit.• Three wavelengths fit alongthe circumference of theorbit.• The hydrogen atom is playingits third highest note.• Highest note (shortestwavelength) is n=1.Wed. Mar. 28, 2007 Phy107 Lect 278Hydrogen atom music• Here the electron is in then=4 orbit.• Four wavelengths fit along thecircumference of the orbit.• The hydrogen atom is playingits fourth highest note (lowerpitch than n=3 note).Wed. Mar. 28, 2007 Phy107 Lect 279Hydrogen atom music• Here the electron is in then=5 orbit.• Five wavelengths fit along thecircumference of the orbit.• The hydrogen atom is playingits next lowest note.• The sequence goes on and on,with longer and longerwavelengths, lower and lowernotes.But Remember that these are higher and higher energies!(Coulomb (electrostatic) potential energy dominates).Wed. Mar. 28, 2007 Phy107 Lect 2710Hydrogen atom orbits• Orbit radii are quantized– rn=n2ao– ao = “Bohr radius” = 0.51 Å• Comes from– Quantized momentum -> quantized velocity– Quantized centripetal acceleration– Centripetal acceleration must match thatCoulomb force– Only particular radius will workWed. Mar. 28, 2007 Phy107 Lect 2711Quantized Energy levels• Instead of drawing orbits, we can just indicate the energyan electron would have if it were in that orbit.Zero energyn=1n=2n=3n=4! E1= "13.612 eV! E2= "13.622 eV! E3= "13.632 eVEnergy axisWed. Mar. 28, 2007 Phy107 Lect 2712Hydrogen atom questionHere is Peter Flanary’ssculpture ‘Wave’ outsideChamberlin Hall. Whatquantum state of thehydrogen atom could thisrepresent?A. n=2B. n=3C. n=43Wed. Mar. 28, 2007 Phy107 Lect 2713Another questionHere is Donald Lipski’s sculpture ‘Nail’s Tail’outside Camp Randall Stadium.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)Wed. Mar. 28, 2007 Phy107 Lect 2714Atoms and photons• Experimentally, atoms do emit electromagneticradiation, but not just any radiation!• In fact, each atom has its own ‘fingerprint’ ofdifferent light frequencies that it emits.HydrogenMercuryWavelength (nm)400 nm500 nm600 nm 700 nmWed. Mar. 28, 2007 Phy107 Lect 2715Hydrogen emission spectrum• Hydrogen is simplest atom– One electron orbiting aroundone proton.• The Balmer Series ofemission lines empiricallygiven byn = 3, λ = 656.3 nmHydrogenn = 4, λ = 486.1 nmn=3n=4Wed. Mar. 28, 2007 Phy107 Lect 2716Hydrogen emission• This says hydrogen emits onlyphotons of a particular wavelength, frequency• Photon energy = hf,so this means a particular energy.• Conservation of energy:– Energy carried away by photon is lost by theorbiting electron.Wed. Mar. 28, 2007 Phy107 Lect 2717The Bohr hydrogen atom• Retained ‘planetary’ picture:one electron orbits aroundone proton• Only certain orbits are stable• Radiation emitted only whenelectron jumps from onestable orbit to another.• Here, the emitted photon hasan energy ofEinitial-EfinalStable orbit #2Stable orbit #1EinitialEfinalPhotonWed. Mar. 28, 2007 Phy107 Lect 2718Emitting and absorbing lightPhoton is emitted whenelectron drops from onequantum state to anotherZero energyn=1n=2n=3n=4! E1= "13.612 eV! E2= "13.622 eV! E3= "13.632 eVn=1n=2n=3n=4! E1= "13.612 eV! E2= "13.622 eV! E3= "13.632 eVAbsorbing a photon of correctenergy makes electron jump tohigher quantum state.Photonabsorbed hf=E2-E1Photonemittedhf=E2-E14Wed. Mar. 28, 2007 Phy107 Lect 2719• Each orbit has a specific energyEn=-13.6/n2• Photon emitted when electronjumps from high energy to lowenergy orbit.Ei – Ef = h f• Photon absorption induceselectron jump fromlow to high energy orbit.Ef – Ei = h f• Agrees with experiment!Energy conservation for Bohr atomWed. Mar. 28, 2007 Phy107 Lect 2720Photon emission questionAn electron can jump between the allowed quantum states(energy levels) in a hydrogen atom. The lowest threeenergy levels of an electron in a hydrogen atom are-13.6 eV, -3.4 eV, -1.5 eV.These are part of the sequence En = -13.6/n2 eV.Which of the following photons could be emitted by thehydrogen atom?A. 10.2 eVB. 3.4 eVC. 1.7 eVThe energy carried away by the photon must begiven up by the electron. The electron can giveup energy by dropping to a lower energy state.So possible photon energies correspond todifferences between electron orbital energies.The 10.2 eV photon is emitted when the electronjumps from the -3.4 eV state to the -13.6 eVstate, losing 10.2 eV of energy.Wed. Mar. 28, 2007 Phy107 Lect 2721Example: the Balmer series• All transitions terminateat the n=2 level• Each energy level hasenergy En=-13.6 / n2 eV• E.g. n=3 to n=2 transition– Emitted photon has energy– Emitted wavelength! Ephoton=
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