Quantum Physics-Photoelectric Effect,X-ray production

8.04 Quantum Physics Lecture VIPhotoelectric effect Observed 1888; explanation, Einstein 1905. A negatively charged metal plate loses charge slowly if illuminated with light, while a positive charge remains. Light of Figure I: Photoelectric effect. Electrons are released from a metal plate illuminated by monochromatic light. The energy of the released electrons is measured. sufficiently short wavelength releases electrons, the electron current is proportional to the intensity of the light. Light below a certain cutoff frequency ν0 does not release any electrons, no matter how high the intensity. Energy of released electrons can be measured by determining voltage V that prevents them from reaching cathode. A linear relation between the electron’s kinetic energy Wkin and the frequency ν of the incident light is observed. The kinetic energy of the electrons does not depend on the intensity of the incident light. Classical prediction Incident power per unit area is given by intensity of light, and independent of frequeency thus electron’s kinetic energy should be proportional to intensity (independeen of ν), and there should be no cutoff frequency. Einstein’s explanation, 1905 Nobel prize, 1921. Einstein assigned physical reality to Planck’s mathematical constrruct (quanta of light): Massachusetts Institute of Technology VI-1 8.04 Quantum Physics Lecture VIFigure II: The value of the cutoff frequency ν0 depends on the metal. The slope does not. 1. Light consists of smallest energy units (quanta, photons) of energy E = hν. 2. An electron is bound to the metal with a binding energy W0. 3. Each electron is ejected by a single photon. Figure III: Work function These three assumptions can explain all of the observed features: -Energy conservation in the process requires that hν = W0 + WKIN -Linear dependence of light frequency ν on electron’s kinetic energy WKIN with intercept −W0. -The slope of WKIN is given by Planck’s constant h. Massachusetts Institute of Technology VI-2 8.04 Quantum Physics Lecture VI-No photoelectrons can be released below cutoff frequency ν0 = Wh 0 , the value of which is metal-dependent. W0 is typically a few eV. -Photoelectric current is proportional to the photon arrival rate, i.e. to the light intensity. Einstein’s explanation, in combination with experiments, also implied the wave-particle duality and the stochastic nature of the quantization hypothesis: When Figure IV: Photoelectric effect and wave-particle duality. we lower the incident power to a level P , corresponding to a photon arrival rate R = ¯P , the first photoelectron is observed on average after a time t0 = R 1 . This is hω very surprising since the electron in an atom only occupies a small area σ � A. One might think that σ = πr2, where r ≈ 1˚A is the atom radius. However, this is wrong because the oscillating electron represents an (optimally matched) dipole antenna, so σ ∼ λ2 ∼ (1 µm)2 A)2 (by some 8 orders of magnitude). Nevertheless, � π(1 ˚σ/A � 1(σ = 10−8 cm2, A=1 cm2, so it is quite impossible for enough light energy to have accumulated on average within the area σ after a time t0. We must conclude that the full photon energy is located (randomly) somewhere within the beam area A to a spot size less or equal to σ: stochastic location of photons within beam, particle-like nature of photons. Nevertheless, the same light still behaves like a wave, e.g. exhibits diffraction, interference etc.: wave-particle duality. In the photoelectric effect photons produce electrons whose kinetic energy depends on the photon frequency due to energy conservation. We nw consider the reverse proceess namely, the production of electromagnetic waves (photons) by electron impact. Energy conservation now determines the photon spectrum: x-rays. Massachusetts Institute of Technology VI-3 8.04 Quantum Physics Lecture VIX-ray productionFigure V: X-ray production. Electrons are accelerated in an electric field and gain a kinetic energy WKIN = qV0 that can be converted into electromagnetic radiation (photons) as they hit (decelerate in) the target. The observed spectrum of the emitted radiation is: λmin is independent Figure VI: Spectrum of emitted X-rays. of target material and limited by energy conservation: electron’s total KE is converted into a single x-ray photon. c qeV0 = hVmax = h (6-1) λmin Short-wavelength cutoff of x-rays: hc 1.24 ˚A λmin = = (6-2) qeV0 V0/10 kV X-ray emission is due to the deceleration of electrons in the field of the nucleus. Classically, since the electron can turn around very sharply within a very short time Δt, we expect a continuous spectrum extending to very high frequencies Δ = Δ1 t . Instead, we observe a cutoff due to energy conservation: all of the electron’s KE is converted into a single photon. The superimposed line spectrum is due to electronic transitions in atoms (strongly bound inner-shell electrons). X-ray spectra can be measured using crystals as diffraction gratings. Massachusetts Institute of Technology VI-4 8.04 Quantum Physics Lecture VIFigure VII: German: Bremsstrahlung “braking emission” of an electron in the field of a nucleus. Double-slit experiment, electrons as waves To observe wave behavior for particles, we need a situation where a single classical path can no longer be ascribed to the particle. One example is single-slit diffraction, another even more striking example is double-slit interference: The interference pat-Figure VIII: Interference pattern for double slit. The interference pattern persists even if there is only one electron in the apparatus at any given time. tern is observed even if the beam is sufficiently attenuated so that there is only one electron in the apparatus at any given time: This indivisible electron passes through slit 1 and slit 2. Any measurement that allows one to determine which slit the electrro actually passed through destroys the interference pattern: Blocking one of the slits obviously destroys the interference, but what about more subtle methods, such as optical observation (light scattering)? Which-path measurements and destruction of interference To optically resolve which slit the electron passed through we need short-wavelength light with λp d, i.e. when we can no longer resolve which slit the electron passed through. What happened if instead we lower the intensity of the light until, on average, only half of the electrons scatter a photon? The contrast of the interference pattern will be reduced to 21 . However, there is an event-by-event correlation between interference and scattering: Post-selection of only those electron arrivals where no photon was observed produce full contrast, post selection of those trials where a photon was scattered by the electron produces no Massachusetts Institute of Technology VI-6 8.04 Quantum Physics Lecture VIcontrast at all. We say that if the electron scatters a photon, the direction of the scatteere photon becomes correlated (entangled) with the electron’s path. Correlations (“entanglement”) between different parts of a system constitute an essential feature of QM and of the measurement process. The latter involves averaging over (“tracing over”) those parts and degrees of freedom of the measurement apparatus that were not observed. If measurement (“which-path information”) destroys the interference pattern, when does the measurement actually occur? -at the moment when electron scatters photon -at the moment when photon hits the photodetector -at the moment when we learn about the result (hear click) Massachusetts Institute of Technology VI-7