1Wed. Mar. 22, 2006 Phy107 Lect 231From Last Time…• Essay topic and paragraph due Friday, Mar. 24• Light waves are particles and matter particles are waves!• Electromagnetic radiation (e.g. light)made up of photon PARTICLESwith energy E=hf and momentum p=E/c=h/ λ• Matter particles are waveswith wavelength λ=h/p.HW8: M Chap 15: Question B, Exercises 2, 6M Chap 16: Question B, Exercises 1M Chap 17: Questions C, D Wed. Mar. 22, 2006 Phy107 Lect 232Wavelength of 1 eV electron• For an electron,! "=1240 eV # nm2 $ 0.511 MeV1Ekinetic=1.23 eV1/ 2# nmEkinetic• 1 eV electron, λ=1.23 nm• 10 eV electron λ=0.39 nm• 100 eV electron λ=0.12 nmconstantkinetic energyrest energyWed. Mar. 22, 2006 Phy107 Lect 233QuestionA 10 eV electron has a wavelength of ~ 0.4 nm.What is the wavelength of a 40 eV electron?A. 0.2 nmB. 0.4 nmC. 0.8 nm ! Wavelength = constantrest energy Kinetic energyWed. Mar. 22, 2006 Phy107 Lect 234Can this be correct?• If electrons are waves, they shoulddemonstrate wave-like effects– e.g. Interference, diffraction• A 25 eV electron has wavelength 0.25 nm,similar to atomic spacings in crystalsWed. Mar. 22, 2006 Phy107 Lect 235Wave reflection from crystal• Interference of waves reflecting from differentatomic layers in the crystal.• Difference in path length ~ spacing between atomsside viewReflection fromtop planeReflectionfrom nextplaneWed. Mar. 22, 2006 Phy107 Lect 236Constructive & DestructiveInterference• Interference arises when waves change their‘phase relationship’.• Can vary phase relationship of two waves bychanging physical location of speaker.ConstructiveDestructive‘in-phase’‘1/2 λ phase diff’2Wed. Mar. 22, 2006 Phy107 Lect 237X-raydiffractionpatternX-ray diffraction• Diffraction spot arrangement indicatesatomic arrangement• Used to determine atomicarrangements of complex molecules.– e.g. DNAMolecularstructureWed. Mar. 22, 2006 Phy107 Lect 238Davisson-Germerexperiment• Diffraction ofelectrons from anickel single crystal.• Established thatelectrons are waves54 eV electrons (λλ=0.17nm) Bright spot:constructiveinterferenceDavisson:NobelPrize 1937Wed. Mar. 22, 2006 Phy107 Lect 239Suppose an electron is a wave…• Here is a wave:…where is the electron?– Wave extends infinitely far in +x and -x directionλx! "=hpWed. Mar. 22, 2006 Phy107 Lect 2310Analogy with sound• Sound wave also has the same characteristics• But we can often locate sound waves– E.g. echoes bounce from walls. Can make a sound pulse• Example:– Hand clap: duration ~ 0.01 seconds– Speed of sound = 340 m/s– Spatial extent of sound pulse = 3.4 meters.– 3.4 meter long hand clap travels past you at 340 m/sWed. Mar. 22, 2006 Phy107 Lect 2311Beat frequency: spatial localization• What does a sound ‘particle’ look like?– One example is a ‘beat frequency’ between two notes– Two sound waves of almost same wavelength added.ConstructiveinterferenceLargeamplitudeConstructiveinterferenceLargeamplitudeDestructiveinterferenceSmallamplitudeWed. Mar. 22, 2006 Phy107 Lect 2312Making a particle out of waves440 Hz +439 Hz440 Hz +439 Hz +438 Hz440 Hz +439 Hz +438 Hz +437 Hz +436 Hz3Wed. Mar. 22, 2006 Phy107 Lect 2313Spatial extentof localized sound wave• Δx = spatial spread of ‘wave packet’• Spatial extent decreases as the spread inincluded wavelengths increases.-8-4048-15 -10 -5 0 5 10 15JΔxWed. Mar. 22, 2006 Phy107 Lect 2314Same occurs for a matter wave• Construct a localized particle by adding togetherwaves with slightly different wavelengths.• Since de Broglie says λ = h /p, each of thesecomponents has slightly different momentum.– We say that there is some ‘uncertainty’ in the momentum• And still don’t know exact location of the particle!– Wave still is spread over Δx (‘uncertainty’ in position)– Can reduce Δx, but at the cost of increasing the spread inwavelength (giving a spread in momentum).Wed. Mar. 22, 2006 Phy107 Lect 2315Interpreting• For sound, we would just say that the sound pulse iscentered at some position, but has a spread.• Can’t do that for a quantum-mechanical particle.• Many measurements indicate that the electron isindeed a point particle.• Interpretation is that the magnitude of electron ‘wave-pulse’ at some point in space determines theprobability of finding the electron at that point.-8-4048-15 -10 -5 0 5 10 15JWed. Mar. 22, 2006 Phy107 Lect 2316Heisenberg Uncertainty Principle• Using– Δx = position uncertainty– Δp = momentum uncertainty• Heisenberg showed that the product ( Δx ) • ( Δp ) is always greater than ( h / 4π )Often write this aswhere is pronounced ‘h-bar’Planck’sconstant ! "x( )"p( )~ h /2 ! h =h2"Wed. Mar. 22, 2006 Phy107 Lect 2317Thinking about uncertaintyFor a classical particle, p=mv, so anuncertainty in momentum corresponds to anuncertainty in velocity. ! "x( )"p( )~ h /2 ! "x( )"v( )~ h /2mThis says that the uncertainty is small for massive objects,but becomes important for very light objects, such aselectrons.Large, massive objects don’t show effects of quantummechanics.Wed. Mar. 22, 2006 Phy107 Lect 2318Uncertainty principle questionSuppose an electron is inside a box 1 nm in width.There is some uncertainty in the momentum ofthe electron. We then squeeze the box to makeit 0.5 nm. What happens to the momentum?A. Momentum becomes more uncertainB. Momentum becomes less uncertainC. Momentum uncertainty unchanged4Wed. Mar. 22, 2006 Phy107 Lect 2319Using quantum mechanics• Quantum mechanics makes astonishinglyaccurate predictions of the physical world• Can apply to atoms, molecules, solids.• An early success was in understanding– Structure of atoms– Interaction of electromagnetic radiation with atomsWed. Mar. 22, 2006 Phy107 Lect 2320Planetary model of atom• Positive charge is concentrated inthe center of the atom ( nucleus )• Atom has zero net charge:– Positive charge in nucleus cancelsnegative electron charges.• Electrons orbit the nucleus likeplanets orbit the sun• (Attractive) Coulomb force playsrole of gravitynucleuselectronsWed. Mar. 22, 2006 Phy107 Lect 2321Difference between atoms• No net charge to atom– number of orbiting negative electrons same asnumber of positive protons in nucleus– Different elements have different number oforbiting electrons• Hydrogen: 1 electron• Helium: 2 electrons• Copper: 29 electrons• Uranium: 92 electrons!• Organized into periodic
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