LECTURE 20 WAVE-PARTICLE DUALITY Instructor: Kazumi Tolich Lecture 20 2 ¨ Reading chapter 30.5 to 30.6 ¤ Matter waves n The de Broglie hypothesis n Electron interference and diffraction ¤ Wave-particle duality ¤ Uncertainty principle de Broglie hypothesis 3 ¨ The wavelength and frequency of matter: ℎ � = � Quiz: 1 4 ¨ An electron and a proton are accelerated through the same voltage. Which has the longer wavelength? A. electron B. proton C. both the same D. neither has a wavelength Quiz: 20-1 answer 5 ¨ electron ¨ Because the electron and the proton travel through the same voltage, the change in potential energy of the electron-electric field is the same as the change in potential energy of the proton-electric field: ∆� = �∆�. ¨ Since mechanical energy is conserved, both particles will get the same kinetic energy: � = − ∆� = −�∆�. 0 + , / ¨ Kinetic energy is given by � = �� = . , ,1 ¨ So the smaller the mass, �, the smaller the momentum, �. 3 ¨ Because � = the particle with the smaller momentum will have the longer wavelength. / Example 1 6 ¨ One of the smallest composite microscopic particles we could imagine using in an experiment would be a particle of smoke or soot. These are about 1 µm in diameter, barely at the resolution limit of most microscopes. A particle of this size with the density of carbon has a mass of about 10-18 kg. What is the de Broglie wavelength for such a particle, if it is moving slowly at 1 mm/s? Quiz: 2 ¨ The electron microscope is a welcome addition to the field of microscopy because electrons have a __________ wavelength than light, thereby increasing the __________ of the microscope. A. longer; resolving power B. shorter; resolving power C. longer; intensity D. shorter; intensity Quiz: 20-2 answer ¨ shorter, resolving power ; ¨ The Rayleigh’s criterion: � = 1.22 567 < ¨ When viewing details of objects with visible light, the details can be resolved only if they are larger than the wavelength of the light. ¨ In electron microscopes, beams of electrons, with small wavelength is used to “see” small objects. Pollen grains Sea urchin sperm Particles of HIV X-ray and matter diffraction 9 ¨ X-rays and matter waves can diffract from crystal planes. The constructive diffraction patterns are given by: 2� sin � = �� � = 1, 2, 3, … Diffraction of matter/Demo: 1 10 ¨ In 1927, C. J. Davisson and L. H. Germer first observed the diffraction of electron waves using electrons scattered from a particular nickel crystal. ¨ G. P. Thomson (son of J. J. Thomson who discovered electrons) showed electron diffraction when the electrons pass through a thin metal foils. ¨ Diffraction has been seen for neutrons, hydrogen atoms, alpha particles, and even more complicated molecules. ¨ In all cases, the measured wavelength matched de Broglie’s prediction. ¨ Demo: electron diffraction X-ray diffraction electron diffraction neutron diffraction Quiz: 3 11 ¨ In an x-ray diffraction experiment with a certain crystal for mono- energetic x-rays, the first order maximum is observed to be at 25º. What higher orders of maxima are possible with this crystal for these x-rays? Choose all that apply. A. None B. 2nd C. 3rd D. 4th E. 5th Quiz: 20-3 answer 12 ¨ 2nd ¨ When x-rays scatter from a crystal the maxima are given by: 2� sin � = ��, � = 1, 2, 3, … or ; sin � = � , � = 1, 2, 3, … ,D ; ¨ If � = 1, sin 25° = 0.42 = ,D ; ¨ For � = 2, sin � = 2 = 2×0.42 = 0.84 ,D ; ¨ But � = 3, sin � = 3 = 3×0.42 = 1.26 which ,D means it’s not possible since sin � ≤ 1! Double-slit interference pattern with matter waves 13 ¨ Even if we have a particle beam so weak that only one particle is present at a time, we still see the final interference pattern after large numbers of particles land. ¨ Quantum weirdness: https://www.youtube.com/watch?v=fwXQjRBLwsQ Wave-particle duality 14 ¨ Light, normally thought of as a wave, exhibits particle properties when it interacts with matter. ¤ photoelectric effect ¨ Electrons, normally thought of as particles, exhibit the wave properties when they pass near the edges of obstacles. ¤ interference and diffraction ¨ All carriers of momentum and energy exhibit both wave and particle characteristics, propagating like a wave and exchanging energy like a particle. Quiz: 4 15 ¨ In a double-slit interference, or in a single-slit diffraction experiment with electrons can we predict where a particular electron will land on the screen? A. Yes B. No Quiz: 20-4 answer 16 ¨ No ¨ It is not possible to predict exactly where that particular electron will land on the screen. ¨ We can give the probability that it will land at different locations. In a single-slit diffraction experiment, an electron is more likely to land at the center of the central maximum than any other position, for example. ¨ However, the fate of each individual electron is uncertain. ¨ This uncertainty is inherent in quantum physics, and is due to the wavelike properties of matter. Quiz: 5 17 ¨ Suppose that a beam of electrons are moving to the right and passing through a single slit. Which of the following statements is/are correct? Choose all that apply. A. The uncertainty in the vertical positions of the electrons as they pass through the slit, ∆�, are larger in Case 1. B. ∆� are larger in Case 2. C. ∆� are the same in both cases. D. The uncertainty in the vertical components of the momentum of the electrons as they pass through the slit, ∆�N, are larger in Case 1. E. ∆�N are larger in Case 2. F. ∆�N are the same in both cases. Case 1 Case 2 Quiz: 20-5 answer 18 ¨ ∆� are larger in Case 2. ¨ The uncertainty in the vertical components of the momentum of the electrons as they pass through the slit, ∆�N, are larger in Case 1. ¨ The larger the vertical components of the momentum of the electrons, the greater the landing angular positions of the electrons. ¨ In Case 1, the width of the diffraction pattern is wider, so the distribution of the vertical components of the momentum of the electrons is larger. Case 1 Case 2 Uncertainty principle 19 ¨ The Heisenberg uncertainty principle for momentum and position is given by ℎ ∆� ∆� ≥ N 2� ¨ The Heisenberg uncertainty principle for energy and time is given by ℎ ∆�∆� ≥ 2� Quiz: 6 20 ¨ An electron in an atom has a minimum uncertainty in position of 0.2 nm. If it is doubled to 0.4 nm by what factor does the minimum uncertainty in momentum change? A. Stays the same B. Factor of 1/4 C. Factor of 1/2 D. Factor of 2 Quiz: 20-6 answer 21 ¨ Factor of 1/2 ¨ The relationship between the two uncertainties is given by the Heisenberg 3 uncertainty principle: ∆� ∆� ≥ N ,S ¨ So, a doubling of the minimum uncertainty in position will change the minimum uncertainty in momentum by 1/2..