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Lecture 28: TUE 04 MAY 2010 Physics 2102 Jonathan Dowling Ch. 37 Einstein’s Theory of Relativity Ch. 38: Ch. 38: Photons and Matter Waves

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Chapter 37 Relativity Relativity is an important subject that looks at the measurement of where and when events take place, and how these events are measured in reference frames that are moving relative to one another. In this chapter we will explore the special theory of relativity (which we will refer to simply as "relativity"), which only deals with inertial reference frames (where Newton's laws are valid). The general theory of relativity looks at the more challenging situation where reference frames undergo gravitational acceleration. In 1905, Albert Einstein stunned the scientific world by introducing two "simple" postulates with which he showed that the old, commonsense ideas about relativity are wrong. Although Einstein's ideas seem strange and counterintuitive, e.g., rate at which time passes depends on the speed of reference frame, these ideas have not only been validated by experiment, they are also being used in modern technology, e.g., global positioning satellites. (37-1)

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The Postulates 1. The Relativity Postulate: The laws of physics are the same for observers in all inertial reference frames. No frame is preferred over any other. 2. The Speed of Light Postulate: The speed of light in vacuum has the same value c in all directions and in all inertial reference frames. Both postulates tested exhaustively, no exceptions found! (37-2)

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The Relativity of Time Fig The time interval between two events depends on how far apart they occur in both space and time; that is, their spatial and temporal separations are entangled. (37-8)

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The Relativity of Time, cont'd When two events occur at the same location in an inertial reference frame, the time interval between them, measured in that frame, is called the proper time interval or the proper time. Measurements of the same time interval from any other inertial reference frame are always greater. Lorentz factor: Speed Parameter: (37-9)

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Lorentz factor as a function of the speed parameter The Relativity of Time, cont'd Fig (37-10)

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2. Macroscopic Clocks. Super precision atomic clocks (large systems) flown in airplanes ~7x10 -7 (Hafele and Keating in 1977 within 10%, and U. Maryland a few years later within 1% of predictions) repeated the muon lifetime experiment on a macroscopic scale If the clock on the U. Maryland flight registered hours as the flight duration, how much would a clock that stayed on earth (lab frame) have measured for the duration? More or less? Does it matter whether airplane returns to same place? Two Tests of Time Dilation, cont'd (37-12)

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Twin Paradox

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The Relativity of Length The length L 0 of an object in the rest frame of the object is its proper length or rest length. Measurement of the length from any other reference frame that is in motion parallel to the length are always less than the proper length. (37-13)

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Does a moving object really shrink? Fig You must measure front and back of moving penguin simultaneously to get its length in your frame. Let's do this by having two lights flash simultaneously in the rest frame when the front and back of the penguin align with them. In penguin's frame, your measurements did not occur simultaneously. You first measured the front end (light from front flash reaches moving observer first as in slide 37-7) and then the back (after the back has moved forward), so the length that you measure will appear to be shorter than in the penguin's rest frame. (37-14)

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A New Look at Energy Mass energy or rest energy ObjectMass (kg)Energy Equivalent Electron≈ 9.11x ≈ 8.19x J(≈ 511 keV) Proton ≈ 1.67x ≈ 1.50x J(≈ 938 MeV) Uranium atom ≈ 3.95x ≈ 3.55x10 -8 J(≈ 225 GeV) Dust particle ≈ 1x ≈ 1x10 4 J(≈ 2 kcal.) U.S. penny ≈ 3.1x10 -3 ≈ 2.8x10 14 J(≈ 78 GWh) Table 37-3 The Energy Equivalents of a Few Objects (37-25)

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A New Look at Energy, cont'd Total energy The total energy E of an isolated system cannot change (37-26)

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Fig Experiment by Bertozzi in 1964 accelerated electrons and measured their speed and kinetic energy independently. Kinetic energy →∞ as speed → c The Ultimate Speed Ultimate Speed→Speed of Light: (37-3)

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Chapter 38 Photons and Matter Waves The subatomic world behaves very differently from the world of our ordinary experiences. Quantum physics deals with this strange world and has successfully answered many questions in the subatomic world, such as: Why do stars shine? Why do elements order into a periodic table? How do we manipulate charges in semiconductors and metals to make transistors and other microelectronic devices? Why does copper conduct electricity but glass does not? In this chapter we explore the strange reality of quantum mechanics. Although many topics in quantum mechanics conflict with our commonsense world view, the theory provides a well-tested framework to describe the subatomic world. (38-1)

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Quantum physics: Study of the microscopic world Many physical quantities found only in certain minimum (elementary) amounts, or integer multiples of those elementary amounts These quantities are "quantized" Elementary amount associated with this quantity is called a "quantum" (quanta plural) Analogy example: 1 cent or $0.01 is the quantum of U.S. currency. Electromagnetic radiation (light) is also quantized, with quanta called photons. This means that light is divided into integer number of elementary packets (photons). The Photon, the Quantum of Light (38-2)

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The energy of light with frequency f must be an integer multiple of hf. In the previous chapters we dealt with such large quantities of light that individual photons were not distinguishable. Modern experiments can be performed with single photons. So what aspect of light is quantized? Frequency and wavelength still can be any value and are continuously variable, not quantized: The Photon, the Quantum of Light, cont'd However, given light of a particular frequency, the total energy of that radiation is quantized with an elementary amount (quantum) of energy E given by: where the Planck constant h has a value: where c is the speed of light 3x10 8 m/s (38-3)

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When short-wavelength light illuminates a clean metal surface, electrons are ejected from the metal. These photoelectrons produce a photocurrent. First Photoelectric Experiment: Photoelectrons stopped by stopping voltage, V stop. The kinetic energy of the most energetic photoelectrons is The Photoelectric Effect Fig K max does not depend on the intensity of the light! → single photon ejects each electron (38-4)

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The Photoelectric Effect, Einstein’s Analysis! Fig Second Photoelectric Experiment: Photoelectric effect does not occur if the frequency is below the cutoff frequency f 0, no matter how bright the light! → single photon with energy greater than work function ejects each electron (38-5)

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The Photoelectric Effect, Einstein’s Analysis Photoelectric Equation The previous two experiments can be summarized by the following equation, which also expresses energy conservation: Using equation for a straight line with slope h/e and intercept – /e Multiplying this result by e: (38-6)

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Photons Have Momentum Fig Fig (38-7)

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Photons Have Momentum, Compton shift Fig Conservation of energy Since electrons may recoil at speeds approaching c we must use the relativistic expression for K: where is the Lorentz factor Substituting K in the energy conservation equation Conservation of momentum along x: Conservation of momentum along y: (38-8)

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Loose end: Compton effect can be due to scattering from electrons bound loosely to atoms ( m = m e → peak at ≠ 0) or electrons bound tightly to atoms ( m ≈ m atom >> m e → peak at ≈ 0). Photons Have Momentum, Compton shift cont’d Want to find wavelength shift: Conservation of energy and momentum provide 3 equations for 5 unknowns (, ’, v, , and ), which allows us to eliminate 2 unknowns, v and ., ’, and can be readily measured in the Compton experiment. is the Compton wavelength and depends on 1/ m of the scattering particle. (38-9)

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How can light act both as a wave and as a particle (photon)? Light as a Probability Wave Fig Standard Version: Photons sent through double slit. Photons detected (1 click at a time) more often where the classical intensity: is maximum. The probability per unit time interval that a photon will be detected in any small volume centered on a given point is proportional to E 2 at that point. Light is not only an electromagnetic wave but also a probability wave for detecting photons. (38-10)

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Light as a Probability Wave, cont'd Single Photon Version: Photons sent through double slit one at a time. First experiment by Taylor in We cannot predict where the photon will arrive on the screen. 2. Unless we place detectors at the slits, which changes the experiment (and the results), we cannot say which slit(s) the photon went through. 3. We can predict the probability of the photon hitting different parts of the screen. This probability pattern is just the two-slit interference pattern that we discussed in Ch. 35. The wave traveling from the source to the screen is a probability wave, which produces a pattern of "probability fringes" at the screen. (38-11)

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Light as a Probability Wave, cont'd Conclusions from the previous three versions/experiments: 1. Light is generated at source as photons. 2. Light is absorbed at detector as photons. 3. Light travels between source and detector as a probability wave. (38-13)

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If electromagnetic waves (light) can behave like particles (photons), can particles behave like waves? Electrons and Matter Waves Fig where p is the momentum of the particle Electrons (38-14)

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In the previous example, the momentum ( p or k ) in the x -direction was exactly defined, but the particle’s position along the x -direction was completely unknown. This is an example of an important principle formulated by Heisenberg: Measured values cannot be assigned to the position r and the momentum p of a particle simultaneously with unlimited precision. Heisenberg’s Uncertainty Principle (38-21)

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As a puck slides uphill, kinetic energy K is converted to gravitational potential energy U. If the puck reaches the top its potential energy is U b. The puck can only pass over the top if its initial mechanical energy E> U b. Otherwise the puck eventually stops its climb up the left side of the hill and slides back. For example, if U b = 20 J and E = 10 J, the puck will not pass over the hill, which acts as a potential barrier. Barrier Tunneling Fig (38-22)

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Fig Fig What about an electron approaching an electrostatic potential barrier? Fig Due to the nature of quantum mechanics, even if E< U b there is a nonzero transmission probability (transmission coefficient T ) that the electron will get through (tunnel) to the other side of the electrostatic potential barrier! Barrier Tunneling, cont’d (38-23)

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