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16.107 L01 Mar 11/02 Dilemma • Existence of quanta could no longer be questioned • e/m radiation exhibits diffraction => wave-like • photoelectric & Compton effect => localized packets of energy => particle-like Wave properties Photon detector clicks when a photon is absorbed • “wave-particle duality” Probability Waves • Photon detector might be a photoelectric device • at any point, the clicks will be randomly spaced in time • cannot predict when a photon will be detected at any point on the screen • if we move the detector, the click rate increases near an intensity maximum • the relative probability that a single photon is detected at a particular point in a specified time ∝ Intensity at that point • intensity ∝ Em2 Probability of detection ∝ Em2 Single-photon version (1909) • Light source is so weak that it emits only one photon at a time at random intervals • interference fringes still build up • raises the question: if the photons move through the apparatus one at a time, through which slit does the photon pass? • How does a given photon know that there is another slit? • Can a single photon pass through both slits and interfere with itself? Probability Wave • Probability (per unit time) that a photon is detected in some small volume is proportional to the square of the amplitude of the wave’s electric field in that region • Postulate that light travels not as a stream of photons but as a probability wave • photons only manifest themselves when light interacts with matter • photons originate in the source that produces the light wave (interaction) • photons vanish on the screen (interaction) Electrons and Matter Waves • Light is a wave but can transfer energy and momentum to matter in “photon” sized lumps • can a particle have the same properties? • can it behave as a wave? • “a matter wave” • de Broglie (1924) suggested λ=h/p • “de Broglie wavelength” • a beam of electrons has a wavelength and should diffract if “slit width” comparable to λ=h/p • 1927 Davisson and Germer observed diffraction of electrons from crystals 1 16.107 L01 Mar 11/02 Diffraction • Recall for a single slit, the diffraction minima are at a sinθ = mλ • if λ <<a, then all θ~mλ/a are small and the light essentially travels as a ray and does not spread out • need λ ≈ a for strong diffraction effects • eg. 1 kg billiard ball moving at 5.0 m/s • λ=h/p = h/mv = (6.63x10-34 J.s)/(1kg)(5.0m/s) = 1.3 x 10-34 m << size of atoms • => no diffraction Diffraction Diffraction • eg. Electrons with kinetic energy of 54.0 eV • if K=p2/2me , then p=(2meK)1/2 = (2 x 9.11x10-31 kg x 54eV x 1.6x10-19 J/eV)1/2 =3.97 x 10-24 kg. m/s • λ=h/p = (6.63x10-34 J.s)/(3.97x10-24 kg.m/s) = 1.67 x 10-10 m = .167 nm • smaller mass => larger λ • typical atom has diameter of 10-10 m Double Slit Experiment with electrons (1989) • regular arrays of atoms (crystals) should diffract electrons! • Neutrons can also diffract => study structures of solids and liquids X-ray beam (light wave) Electron beam (matter wave) 2 16.107 L01 Mar 11/02 Problem Solution • Singly charged Na ions are accelerated through a potential difference of 300 V. What is (a) the momentum acquired? (b) what is the de Broglie wavelength? • Solution: K = qV = (1.6x10-19 C)(300V) = 4.80 x10-17 J = 300 eV • m=(22.9898 g/mole)/(6.02x1023 atom/mole) = 3.819 x10-23 g = 3.819 x 10-26 kg • (a) p = (2mK)1/2 = [2(3.819x10-26)(4.8x10-17)]1/2 = 1.91 x10-21 kg.m/s • (b) λ =h/p = (6.63x10-34 J.s)/(1.91x10-21 kg.m/s) = 3.46 x10 -13 m Schrödinger Equation What is Waving? • • • • If particles behave as waves, what is waving? Wave on a string => particles in string execute SHM sound wave in air => air molecules oscillate in SHM light wave => electric and magnetic fields oscillate • e.g. E(x,y,z,t) electric field varies from place to place and with time • intensity ∝ |E|2 • what varies from place to place for a matter wave? • Wave function ψ(x,y,z,t) “psi” d 2 Ψ 8π 2 m + 2 ( E − V )Ψ = 0 dx 2 h • • • • • Schrödinger Equation 1926 Hψ=Eψ ψ(x,t) is a solution of this equation the wave equation for matter waves probability waves probability density is P(x,t)= ψ(x,t) ψ*(x,t) = |ψ(x,t)|2 Example A harmonic wave has a definite value of k but extends to infinity A wave packet has a spread of k- values but is localized • Radar transmitter emits pulses of electromagnetic radiation which last 0.15 µs at a wavelength of λ = 1.2 cm • (a) to what central frequency should the radar receiver be set? • (b) what is the length of the wave packet? • (c ) how much bandwidth should the receiver have? • (a) f0 = c/λ0 = 3 x 108 m/s /1.2 x 10-2 m = 26 GHz • (b) ∆x = c∆t = (3 x 108 m/s)(.15 x 10-6 s) = 45 m • (c ) ∆f = ∆ω/2π = 1/(2π∆t) = 1/(6.28 x .15 x 10-6 s)= 1.1 MHz ∆k.∆x ≈ 1 ∆ω.∆t ≈ 1 3