Physics_and_Reality_Week_3

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Name _________________ I attended this week’s lecture 0 Homework Questions for Physics and Reality Week 3 Due the week of February 26 Metric System Prefixes Table T “terra” 10 12 G “giga” 10 9 M “mega” 10 6 k “kilo” 10 3 c “centi” 10 -2 m “milli” 10 -3 μ “micro” 10 -6 n “nano” 10 -9 p “pico” 10 -12 Useful Values: Planck’s constant: h = 6.63 × 10 -34 J s Mass of an electron: m e = 9.11 × 10 -31 kg Speed of light in a vacuum: c = 3.00 × 10 8 m/s 1 J = 1 kg m 2 /s 2 1 mile = 1.61 km Useful Equations De Broglie wavelength ࠵? = ! "# λ = wavelength, h = Planck’s constant, m = mass, v = speed Heisenberg’s uncertainty principle ∆࠵?∆࠵? ≥ ! "#$ Δ x = quantum uncertainty in position, Δ v = quantum uncertainty in speed, h = Planck’s constant, m = mass 1. Double Slit Experiment, Wave Particle Duality, and Uncertainty Principle In this week’s lecture, you learned about the double slit experiment. As a reminder, the experiment is set up such that you send various objects through an impenetrable wall containing two slits and those objects hit a detecting screen, as shown in Figure 1 (next page):

2 Figure 1. Sketch of the setup for the double slit experiment. A. Suppose that you send millions of electrons through the setup shown in Figure 1. In the sentences below circle the option that best describes what we can say about the electrons: a. We CAN/ CANNOT predict the location where an individual electron will definitely land. b. We CAN/CANNOT predict locations where an individual electron will definitely not land. c. We CAN/ CANNOT predict locations where an individual electron is most likely to land. d. We CAN/ CANNOT precisely determine both the position and the speed of the electron right before it enters the slits. B. Imagine you now run three different versions of the double slit experiment, each utilizing the same basic experimental setup, as described in Figure 1 above: Experiment I: Electrons pass from the source through the double slit. Experiment II: Marbles pass from the source through the double slit. Experiment III: Electrons are fired from the source; one of the slits (chosen at random) is blocked as each electron is fired, such that each slit is only open half of the time. Which of the patterns shown in Figure 2 (next page) BEST represents the outcome of each of the three experiments (I-III) described above? Choose the correct answer from the list provided and explain each of the matches you chose in 1-2 sentences.

3 Figure 2. Three possible patterns that might be detected on the screen of the double slit experiment. Patterns combinations: a. I-B, II-B, III-A b. I-C, II-B, III-A c. I-C, II-B, III-C d. I-C, II-B, III-B As Prof. Greene indicated in his lecture, the double slit experiment teaches us that microscopic objects, like electrons, are both particles and waves. In fact, as Prof. Greene also described, Louis de Broglie suggested in 1923 that all material objects display wave-like properties and have wavelengths associated with them. According to de Broglie, these “matter waves” have a wavelength given by the equation ࠵? = ! $% , where λ is the wavelength of the object, h is Planck’s constant, m is the mass of the object, and v is its speed. Let’s consider what this formula and its application tell us about the electrons in our double slit experiment. C. (i) Suppose that in your double slit experiment, your electrons are moving at a speed of 900 m/s. What is the wavelength of these electrons in units of nm? (ii) Do the electrons in the double slit experiment exhibit any particle-like behavior, and if so, when? Give your answer as YES or NO and explain in 1-2 sentences.

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4 D. An object's "wave-like" properties are apparent on size scales comparable to (or smaller than) their de Broglie wavelength. At scales much larger than its de Broglie wavelength, an object behaves more like a particle. In 2012 researchers at the University of Nebraska–Lincoln performed the double-slit experiment, exactly as it was described in the lecture. Electrons were fired by an electron gun and passed through a wall with two slits, which were 272 nm apart from each other . (i) Compare the wavelength that you calculated in part C (i) with the slit separation used in the University of Nebraska-Lincoln experiment. Based on this comparison, what kind of behavior do you think the electrons will manifest as they approach the slits? Explain your choice in one sentence. a. wave-like b. particle-like c. neither wave-like, nor particle-like (ii) Given your answer to part D (i) above, which slit does each electron pass through? Explain your choice in one sentence. a. one of the slits only, which we could determine if we had sufficiently precise measuring apparatus b. one slit only, but we can never determine which one c. both slits at the same time In the lecture, you also learned about the Heisenberg Uncertainty Principle (∆࠵?∆࠵? ≥ ! "#$ ) . According to this principle, there is a limit on the product between the uncertainty in position (a particle-like quality of the object) and the uncertainty in speed (a wave-like quality of the object, since, as de Broglie showed, speed and wavelength of the object are related). Let’s consider what we can learn from the Heisenberg Uncertainty Principle both qualitatively and quantitatively. E. Suppose that the scientists conducting the experiment at the University of Nebraska-Lincoln, did a version of the experiment in which the electrons were sent out towards the slits one by one. Furthermore, suppose that they wanted to determine exactly which slit each electron went through. In order to do so, one would need to determine the position of the electron to within the width of each slit. In other words, the uncertainty in the position of the electrons would equal the width of the slits. In their setup, the width of each slit was 62 nm.

5 (i) What would be the minimum uncertainty in the speed of the electrons as they passed through the slit, given this uncertainty in position? (ii) Is the uncertainty in the electron’s speed small or large? To answer this, compare the uncertainty you calculated in part E (i) above to the electron's speed (900 m/s). Report your answer as a percentage. (iii) Recall that a particle has a very well-defined position, whereas a wave has a very well- defined speed. In this experiment, in which experimenters determine which slit the electrons pass through, do the electrons behave in a wave-like or particle-like fashion? Justify your answer. F. Suppose that instead of electrons we use marbles in the same setup as above. Note that the size of the slits would have to change in order for the marbles to fit through, but we would still like to determine the position of the marbles to within 62 nm. The mass of a marble is ~10 g. (i) What would be the minimum uncertainty in the speed of the marbles? (ii) Is the uncertainty in the marble’s speed small or large? To answer this, compare the minimum uncertainty in the speed of an electron from part E (i) to the minimum uncertainty in the speed of a marble from part F (i). Which is larger and by how many orders of magnitude? G. Consider your answers to questions E and F above. What do they suggest about quantum mechanics more generally, as it applies to microscopic vs. macroscopic objects?

6 2. Probability Wave The famous Danish physicist Niels Bohr devised one of the first successful models of the hydrogen atom. According to the Bohr model, the hydrogen atom consists of an electron orbiting the proton in well-defined orbits. The first orbit in the Bohr model is located at a distance of 0.0529 nm, which is also known as the Bohr radius (a Bohr ). According to quantum mechanics, the electron is located somewhere inside the hydrogen atom, with a probability that varies according to distance from the proton, as shown in Figure 3. Figure 3. Probability wave of the location of an electron in a hydrogen atom (for the lowest energy state). The x-axis shows the distance from the center of the hydrogen atom (where the proton is located) as a multiple of the Bohr radius. Use Figure 5 to answer the following questions: (i) Where is the electron most likely to be found? Express your answer in units of a Bohr . (ii) Where will the electron never be found? 3. Entanglement B. Twin brothers, Fred and George Weasley, are each preparing for separate trips to opposite corners of England. Between them they own just one pair of dragonhide shoes and always argue who gets to bring the shoes along, so they ask their brother Ron to decide for them. Ron, however, chooses to play a prank. Without telling his brothers, he packs a shoebox for each, but r/a Bohr Probability wave

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7 he puts only the left shoe in one and only the right shoe in the other. Ron then scrambles the shoeboxes and randomly gives one to Fred and one to George. (i) Suppose that according to Ron, Fred reaches his destination and begins to unpack first. Which of the following is true when Fred is about to open the shoebox he brought along? Pick ALL the correct statements. a. The shoe in Fred’s box is either the left shoe or the right shoe, but not both. b. The shoe in George’s box is either the left shoe or the right shoe, but not both c. The shoe in Fred’s box is both the left shoe and the right shoe at the same time. d. The shoe in George’s box is both the left shoe and the right shoe at the same time. e. If Fred is told that his box contains only one shoe, he cannot definitively predict whether he will get the right shoe or the left shoe. f. If Fred is told that his box contains only one shoe, he can definitively predict whether he will get the right shoe or the left shoe. (ii) Suppose that Fred found the left shoe inside his box, but has not told anyone yet. Which of the following is true when George is about to open the shoebox he brought along? Pick ALL the correct statements. a. The shoe in George’s box is definitely the right shoe. b. The shoe in George’s box is both the left shoe and the right shoe at the same time. c. If George is told that his box contains only one shoe, he can definitively predict whether he will get the right shoe or the left shoe. d. George can determine whether he has the left shoe or the right shoe, only if he is told what Fred found in his box. C. Let’s imagine that instead of shoes, Ron prepares for his brothers a pair of entangled electrons and gives a box containing a single entangled electron to each twin. The entangled electrons are in a quantum superposition of the following two states: [Fred’s electron has spin UP, George’s electron has spin DOWN] AND [Fred’s electron has spin DOWN, George’s electron has spin UP]. (i) Suppose that according to Ron, Fred reaches his destination and begins to unpack first. Which of the following is true when Fred is about to open the electron box he brought along? Pick ALL the correct statements. a. The electron in Fred’s box has either UP spin or DOWN spin, but not both. b. The electron in George’s box has either UP spin or DOWN spin, but not both. c. The electron in Fred’s box has both UP spin and DOWN spin at the same time. d. The electron in George’s box has both UP spin and DOWN spin at the same time. e. If Fred is told that his box contains only one electron, he cannot definitively predict whether his electron will have UP spin or DOWN spin. f. If Fred is told that his box contains only one electron, he can definitively predict whether his electron will have UP spin or DOWN spin.

8 (ii) Suppose that Fred opened his electron box and found his electron to have UP spin, but has not told anyone yet. Which of the following is true when George is about to open the electron box he brought along? Pick ALL the correct statements. a. The electron in George’s box definitely has DOWN spin b. The electron in George’s box has both UP spin and DOWN spin at the same time c. If George is told that his box contains only one electron, he can definitively predict whether his electron will have UP spin or DOWN spin. d. George can determine whether his electron will have UP spin or DOWN spin only if he is told what Fred found in his box.

Physics_and_Reality_Week_3

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