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Exam 3 covers Lecture, Readings, Discussion, HW, Lab Exam 3 is Tue. Nov. 25, 5:30-7 pm, 2103 Chamberlin (here) Biot-Savart Law - currents produce magnetic fields Ampere’s law - shortcut to determining mag. fields from currents. Magnetic flux, Faraday effect, Lenz’ law, inductance, inductors Electromagnetic waves: Wavelength, freq, speed E&B fields, intensity, power, rad. pressure, Poynting vec Polarization Modern Physics (quantum mechanics) Photons & photoelectric effect Bohr atom: Energy levels, absorbing & emitting photons Tue. Nov. 25, 2008 Physics 208, Lecture 25 1 Last Time… Photons as particles Photon absorption and emission Bohr atom Photon properties of light Photon of frequency f has energy hf E photon hf hc / h 6.626 1034 J s 4.14 1015 eV s hc 1240eV nm Red light made of ONLY red photons The intensity of the beam can be increased by increasing the number of photons/second. (#Photons/second)(Energy/photon) = energy/second = power Tue. Nov. 25, 2008 Physics 208, Lecture 25 3 Photon interference? Only one photon present here Do an interference experiment again. But turn down the intensity until only ONE photon at a time is between slits and screen ? Is there still interference? A. Yes Tue. Nov. 25, 2008 B. No C. I’m confused Physics 208, Lecture 25 4 Single-photon interference 1/30 sec exposure Tue. Nov. 25, 2008 1 sec exposure Physics 208, Lecture 25 100 sec exposure 5 Probabilities Quantum mechanic says: Cannot predict where on camera photon will arrive. Individual photon hits determined probabilistically. Photon has a probability amplitude through space. Square of this quantity gives probability that photon will hit particular position on detector. The photon is a probability wave. Tue. Nov. 25, 2008 Physics 208, Lecture 25 6 Matter waves If light waves have particle-like properties, maybe matter has wave properties? de Broglie postulated that the wavelength of matter is related to momentum as h p This is called the de Broglie wavelength. Tue. Nov. 25, 2008 Physics 208, Lecture 25 Nobel prize, 1929 7 Why h / p ? Works for photons Wave interpretation of light: wavelength = (Speed of Light) / Frequency =c/f Particle interpretation of light (photons): Energy = (Planck’s constant) x Frequency E = hf, so f = E / h c c h Wavelength = = f E /h E /c But photon momentum = p = E / c… h for a photon p Tue. Nov. 25, 2008 Physics 208, Lecture 25 8 h We argue that applies to everything p Photons and footballs both follow the same relation. Everything has both wave-like and particle-like properties Tue. Nov. 25, 2008 Physics 208, Lecture 25 9 Wavelengths of massive objects h deBroglie wavelength = p p=mv h mv Tue. Nov. 25, 2008 Physics 208, Lecture 25 10 Matter Waves deBroglie postulated that matter has wavelike properties. deBroglie wavelength h / p Example: Wavelength of electron with 10 eV of energy: Kinetic energy p2 E KE p 2mE KE 2m h hc 1240eV nm 0.39nm 2 6 2mE KE 2mc E KE 20.51110 eV 10eV Tue. Nov. 25, 2008 Physics 208, Lecture 25 11 Wavelength of a football Need m, v to find Momentum: Make the Right Call: The NFL's Own interpretations and guidelines plus 100s of official rulings on game situations. National FootBall League, Chicago. 1999: "... short circumference, 21 to 21 1/4 inches; weight, 14 to 15 ounces.” (0.43 - 0.40 kg) “Sometimes I don’t know how they catch that ball, because Brett Aaron wings that thing 60, 70 mph,” Flanagan said. (27 - 32 m/s) Wells mv 0.4 kg30 m /s 12 kg m /s h 6.6 1034 J s 35 26 5.5 10 m 5.5 10 nm p 12 kg m /s Tue. Nov. 25, 2008 Physics 208, Lecture 25 12 This is very small 1 nm = 10-9 m Wavelength of red light = 700 nm Spacing between atoms in solid ~ 0.25 nm Wavelength of football = 10-26 nm • What makes football wavelength so small? h h p mv Tue. Nov. 25, 2008 Large mass, large momentum short wavelength Physics 208, Lecture 25 13 Suppose an electron is a wave… Here is a wave: h p x …where is the electron? Wave extends infinitely far in +x and -x direction Tue. Nov. 25, 2008 Physics 208, Lecture 25 14 Analogy 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/s Tue. Nov. 25, 2008 Physics 208, Lecture 25 15 Beat frequency: spatial localization What does a sound ‘particle’ look like? Example:‘beat frequency’ between two notes Two waves of almost same wavelength added. Qui ckTi me™ a nd a TIF F (L ZW ) d e co mp re ssor a re ne e d ed to se e thi s p i cture . Qui ckTi me™ a nd a TI F F (L Z W ) d e co mp re ssor a re ne e d ed to se e thi s p i cture . Constructive interference Large amplitude Tue. Nov. 25, 2008 Destructive interference Small amplitude Physics 208, Lecture 25 Constructive interference Large amplitude 16 Making a particle out of waves 440 Hz + 439 Hz 440 Hz + 439 Hz + 438 Hz 440 Hz + 439 Hz + 438 Hz + 437 Hz + 436 Hz Tue. Nov. 25, 2008 Physics 208, Lecture 25 17 Adding many sound waves Six sound waves with different wavelength added together 1= 4= /1.15 2= /1.05 5= /1.20 3= /1.10 6= /1.25 •Wave now resembles a particle, but what is the wavelength? – Sound pulse is comprised of several wavelength – The exact wavelength is indeterminate 8 4 0 -4 x -8 -15 -10 -5 0 5 10 15 J Tue. Nov. 25, 2008 Physics 208, Lecture 25 18 Spatial extent of ‘wave packet’ 8 4 0 -4 x -8 -15 -10 -5 0 5 10 15 J x = spatial spread of ‘wave packet’ Spatial extent decreases as the spread in included wavelengths increases. Tue. Nov. 25, 2008 Physics 208, Lecture 25 19 Same occurs for a matter wave Localized particle: sum of waves with slightly different wavelengths. = h /p, each wave has different momentum. There is some ‘uncertainty’ in the momentum Still don’t know exact location of the particle! Wave still is spread over x (‘uncertainty’ in position) Can reduce x, but at the cost of increasing the spread in wavelength (giving a spread in momentum). Tue. Nov. 25, 2008 Physics 208, Lecture 25 20 Heisenberg Uncertainty Principle Using x = position uncertainty p = momentum uncertainty Planck’s constant Heisenberg showed that the product ( x ) ( p ) is always greater than ( h / 4 ) Often write this as x p ~ /2 h is pronounced ‘h-bar’ 2 where Tue. Nov. 25, 2008 Physics 208, Lecture 25 21 Uncertainty principle question Suppose an electron is inside a box 1 nm in width. There is some uncertainty in the momentum of the electron. We then squeeze the box to make it 0.5 nm. What happens to the momentum uncertainty? A. Momentum becomes more uncertain B. Momentum becomes less uncertain C. Momentum uncertainty unchanged Tue. Nov. 25, 2008 Physics 208, Lecture 25 22 The wavefunction Quantify this by giving a physical meaning to the wave that describing the particle. This wave is called the wavefunction. Cannot be experimentally measured! But the square of the wavefunction is a physical quantity. It’s value at some point in space is the probability of finding the particle there! Tue. Nov. 25, 2008 Physics 208, Lecture 25 23 Electron waves in an atom Electron is a wave. Its ‘propagation direction’ is around circumference of orbit. Wavelength = h / p Waves on a circle? Tue. Nov. 25, 2008 Physics 208, Lecture 25 24 Waves on a circle Wavelength My ‘ToneNut’. Produces particular pitch. Sound wave inside has wavelength =v/f (red line). Integer number of wavelengths required around circumference Otherwise destructive interference Blow in here Tue. Nov. 25, 2008 wave travels around ring and interferes with itself Physics 208, Lecture 25 25 Electron Standing Waves Electron in circular orbit works same way Integer number of deBroglie wavelengths must fit on circumference of the orbit. Circumference = (2)x(orbit radius) = 2r h h So condition is 2r n n n p mv This says mvr n This is quantization angular momentum (L=mvr) Ln Tue. Nov. 25, 2008 Physics 208, Lecture 25 26 Electron standing-waves on an atom Wave representing electron Electron wave extends around circumference of orbit. Only integer number of wavelengths around orbit allowed. Tue. Nov. 25, 2008 Wave representing electron Physics 208, Lecture 25 27 Hydrogen atom energies Wavelength gets longer in higher n states, (electron moving slower) so kinetic energy goes down. But energy of Coulomb interaction between electron (-) and nucleus (+) goes up faster with bigger n. Zero energy n=4 n=3 E3 13.6 eV 32 n=2 E2 13.6 eV 22 E1 13.6 eV 12 Energy End result is 13.6 E n 2 eV n n=1 Tue. Nov. 25, 2008 Physics 208, Lecture 25 28 Hydrogen atom question Here is Peter Flanary’s sculpture ‘Wave’ outside Chamberlin Hall. What quantum state of the hydrogen atom could this represent? A. n=2 B. n=3 C. n=4 Tue. Nov. 25, 2008 Physics 208, Lecture 25 29 Another question Here is Donald Lipski’s sculpture ‘Nail’s Tail’ outside Camp Randall Stadium. What could it represent? QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture. A. A pile of footballs B. “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) Tue. Nov. 25, 2008 Physics 208, Lecture 25 30