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Transcript
About the Resource About the ANimatioNs Welcome to Perimeter Institute’s Revolutions in Science, The three 60-second animations serve to hook your Alice & Bob in Wonderland animations. far more fascinating than they may have realized. a classroom resource based on three serious-but-fun students’ interest—to show them that the everyday world is Challenge and inspire your students with the wonder and Along with the two characters, Alice & Bob, students will Wonderland” world where things are not always what they profound shifts in our understanding of reality. mystery of our universe—we really do live in an “Alice in seem to be: • Gravity is not a force pulling down; it is the ground accelerating up in “curved spacetime.” • Atoms cannot exist in a commonsense universe; they require a strange “quantum” reality. • Energy has inertia; our energy to be alive comes from literally eating the mass of the Sun! Engage your students in the powerful, but surprisingly accessible, creative and critical thinking processes that led to three of the most profound revolutions in science. The resource focuses not only on basic scientific literacy— what these enduring understanding are—but more importantly how they were discovered. Using these discoveries as discover that the simplest questions can lead to the most Alice is a delightfully precocious little girl, brimming with curiosity. Each episode opens with Alice wondering about something that seems so obvious it sounds silly, such as, “What keeps us stuck to the Earth?” Bob is Alice’s older brother who feels it is his duty to ‘educate’ his sister. Without thinking, he blurts out the commonsense answer to her ‘foolish’ questions. Alice gives us reason to question the commonsense answer. Together, our characters use their imaginations and simple reasoning to arrive at amazing insights into the universe. Your students are sure to enjoy their mind-warping adventures with Alice & Bob in Wonderland! exemplars of the power of inquiry, students can experience for themselves how new scientific knowledge is created. About the DVD: The accompanying menu-driven DVD contains the plug-and-play Alice & Bob in Wonderland animations, as well as the following files, which can be accessed by closing the menu software and using your computer’s file browser: this Teachers’ Guide in PDF format and the animations in various file formats. View the animations now! The Student Worksheets and Assessments in editable DOC format can be found at www.perimeterinstitute.ca Curriculum CoNNectioNs 2 What KeepS Us Stuck to the Earth? 3 Introduction Teacher Demonstrations Student Worksheets: SW1: Scientific Models: Gravity SW2: Scientific Revolution: General Relativity Student Assessments: SA1: Scientific Models: Gravity SA2: Scientific Revolution: General Relativity Answers How CaN Atoms Exist? 17 Introduction Student Worksheets: SW1: Scientific Models: The Atom SW2: Scientific Revolution: Quantum Mechanics Student Assessments: SA1: Scientific Models: The Atom SA2: Scientific Revolution: Quantum Mechanics SA3: Applications of Quantum Mechanics Answers Where Does ENergy Come From? Introduction Student Worksheets: Student Assessment: Answers CredIts 4 5-6 7- 8 9-11 12 13 14 -16 18 -19 20 -21 22-23 24 25 26 27- 28 29 SW1: Scientific Models: Time SW2: Scientific Revolution: Special Relativity SA: Scientific Revolution: Special Relativity 30 -31 32 - 34 35-36 37 38 - 40 41 Curriculum coNNectioNs 2 Topic Connection to Resource Module Nature of Science Science involves both creative and critical thinking, leading to new and sometimes revolutionary ways of understanding nature. Educated guesswork and intuitive leaps can lead the scientific imagination to very strange ideas, but as long as these ideas fit the experimental evidence they must be taken seriously. The ultimate judge of a theory is how well it matches the observations, not how well it matches our commonsense. 1, 2, 3 Process of Scientific Modeling We build scientific models to explain complex phenomena. Good models must be logically self-consistent, explain the observations accurately, make testable predictions of new observations, and give new insights into the phenomena. 1, 2, 3 Force and Acceleration Newton s second law of motion dictates that acceleration is the result of a net force. In Newton s model, gravity is a force causing acceleration; in Einstein s model, gravity is not a force so objects in freefall are not accelerating. 1 Weight For Newton, weight is the force of gravity pulling down on you. For Einstein, there is no force of gravity; weight is the magnitude of the force needed to accelerate you up along with the accelerating ground. 1 Gravity Students challenge the underlying assumption of Newton s mysterious “force of gravity,” which has no known cause, and replace it with an alternative explanation for gravity using Einstein s curved spacetime. 1 Frames of Reference All observations and measurements are made relative to a frame of reference. If that frame is moving with constant velocity, there is no experiment that can be done to show that it is moving. If the frame is accelerating, the law of inertia seems to be violated so we invent forces to reconcile our experiences. Bohr-Rutherford Model of the Atom The Bohr-Rutherford model of the atom is an obsolete scientific model. The idea of electrons orbiting around the nucleus is examined and shown to fail due to simple, classical concepts that are within the students grasp. 2 Quantum Mechanical Model of the Atom The quantum mechanical model of the atom uses waves to describe the behaviour of particles. Electrons can behave as if they are in many places at the same time, solving the problems encountered by the classical (and Bohr-Rutherford) models. 2 Wave-Particle Duality The electron is a point-like particle that behaves like a wave. This allows the electron to act as if it is in many places, or traveling in many directions, at the same time. 2 Electromagnetic Fields The electron is charged so it is surrounded by an electric field. Accelerating electrons have changing electric fields so they emit electromagnetic waves. 2 Relative Motion Two observers watching the same event might have very different descriptions of the event if they are moving relative to each other. There is no preferred frame of reference in the universe so all motion is relative. 3 Time Dilation The Newtonian concept of absolute time is wrong. Two observers moving relative to each other will measure the other s time passing at a different rate—moving clocks run slow. . 3 Length Contraction Two observers moving relative to each other will measure the other s space to be contracted in the direction of motion—moving objects occupy less space. 3 Energy Energy is not just “the ability to do work.” Closer inspection of energy leads to the surprising result that all forms of energy have inertia—heating a cup of coffee increases its resistance to acceleration. 3 Inertia Inertia is not just “the ability to resist acceleration.” The inertia of even an object at rest represents the presence of energy, as described by E=mc2. 3 1, 3 Introduction 4 Teacher Demonstrations 5-6 Student Worksheets: SW1: Scientific Models: SW2: Scientific Revolution: 7- 8 Gravity 9-11 General Relativity Student Assessments: SA1: Scientific Models: 12 SA2: Scientific Revolution: 13 Gravity General Relativity Answers 14-16 This module contains two single-period lessons based on the Alice & Bob in Wonderland animation: What Keeps Us Stuck to the Earth? In this episode Alice and Bob ask questions about the nature of gravity and realize that there is a deep connection between gravity and acceleration. Lesson 1 is an introductory level lesson (no prior knowledge of physics is required) that guides students through a critical thinking activity to connect acceleration and gravity. Lesson 2 is a more advanced lesson (prior knowledge of dynamics is an asset) that builds on concepts developed in Lesson 1 to show that the effects of gravity are actually caused by curved spacetime. LessoN 1: LessoN 2: SCIENTIFIC MODELS: GRAVITY SCIENTIFIC REVOLUTION: GENERAL RELATIVITY Use D1: Black Box to engage the students in the Use D3: Toy and Bungee Cord to highlight the Follow with D2: Sagging Rod to explore the force model >> Show the Alice & Bob in Wonderland animation: Distribute SW1: Scientific Models: Gravity after D2. Distribute SW2: Scientific Revolution: General creative process of building and evaluating models. of gravity and introduce the acceleration model. This worksheet walks the students through an exercise in critical thinking about gravity and acceleration. >> Show the Alice & Bob in Wonderland animation: What Keeps Us Stuck to the Earth? SA1: Scientific Models: Gravity. This worksheet includes additional questions to be done in class differences between the two models. What Keeps Us Stuck to the Earth? Relativity. This worksheet guides the students into a discovery of curved spacetime. In Part B, they will use masking tape and beach balls to model curved spacetime. SA2: Scientific Revolution: General Relativity. This worksheet includes additional questions to be done in class or for homework. or for homework. 3 WHAT KEEPS US STUCK TO THE EARTH? Science is a process of building models to explain wrong with Newton’s model—Mercury did not orbit the Sun careful thought and experimentation. Good models explain this discrepancy within the context of Newton’s model, but observations and then refining those models through existing observations and make testable predictions. This Perimeter Institute classroom resource engages students in this process by exploring models of a common real world phenomenon—gravity. Students will exercise their critical and creative thinking skills to demonstrate why Einstein’s model of gravity is better than Newton’s. Our everyday experiences of gravity suggest that the Earth exerts an attractive force on nearby objects. Newton successfully extended this force model of gravity to the Moon, Sun and planets. Nevertheless, the force model of gravity deeply troubled Newton because it did not explain the cause of the force. Moreover, in the 1850’s, a more careful look at existing observations suggested that something might be quite as predicted. Scientists tried various ways to explain all attempts failed. Newton’s model of gravity had reached its limit. Newton’s force model of gravity also troubled Albert Einstein. In his “happiest thought,” Einstein realized that when you are in freefall you do not feel your own weight, like an astronaut floating weightlessly in deep space. However, when an astronaut’s rocket accelerates, she feels as if there is a force pulling her down toward the floor, like weight. In reality, what the astronaut feels is the floor pushing up on her, accelerating her up. Could gravity be like this? Could it be that there is no force pulling us down, but instead the ground is accelerating up? Yes! Einstein showed how curving spacetime can make it possible for the ground to be forever accelerating up without the Earth expanding faster and faster! Students explore this idea through a simple, concrete activity involving just tape and a ball. ur theory is, r how beautiful yo “It doesn’t matte e. If it doesn’t how smart you ar it doesn’t matter ent, it’s wrong.” agree with experim MAN – RICHARD FEYN testable predictions that distinguish it from Newton’s force model. Einstein’s model predicts that time passes more slowly at the surface of a planet compared to farther away. This effect has been precisely measured and is evident daily in the Global Positioning System (GPS). Einstein’s model other, at a may act upon an “…that one body the mediation vacuum, without distance through eir action and by and through th of anything else, another, is to eyed from one to force may be conv ve no man surdity, that I belie me so great an ab a competent sophical matters who has in philo o it.” , can ever fall int faculty of thinking N – ISAAC NEWTO nt office in Bern, chair at the pate “I was sitting in a rred to me: en a thought occu when all of a sudd el his own ely, he will not fe If a person falls fre thought made rtled. This simple sta s wa I t. igh we lled me toward a n on me. It impe a deep impressio tion.” theory of gravita t) ough TEIN (Happiest Th – ALBERT EINS Einstein’s curved spacetime model of gravity makes several also correctly predicts the bending of light as it passes by a massive object, such as a star. Such gravitational lensing has become a powerful tool in astronomy. Einstein’s model also provides a very accurate description of the orbits of all the planets, including Mercury. Einstein’s model of gravity has passed every experimental test to date. These same tests have conclusively ruled out Newton’s model. The old idea of gravity as a force may feel right but it is wrong. The “force of gravity” is an inference, not an observation. We observe the ground compressing under our feet. We infer that gravity is a force pulling us down. In reality, the ground is accelerating up in curved spacetime, pushing up on us, forcing us to accelerate along with it. Dropped objects don’t accelerate down: it is the ground that accelerates up in curved spacetime. These statements may strike us as odd, but they agree with experimental data. Gravity is not a force. Our everyday experiences of gravity are actually the effects of the ground accelerating up through curved spacetime. Gravity is curved spacetime. 4 Teacher DemoNstratioNs D1 - BLACK BOX: (see building instructions below) 1. Pull the top cords back and forth. Invite students to guess how they are connected inside. Now pull one of the bottom cords. Continue pulling different combinations of cords while drawing students into the mystery. 2. Ask students to draw a picture of what they imagine is inside the box. Encourage creative thinking! 3. Have students share their ideas on the board. Engage the class in a discussion about the various models that are on the board. Verify that the models correctly explain the observations. Highlight the following points: • The same set of observations can generate different models. • All models that explain the observations are equally valid. • Models that fail to explain one or more observations are wrong, or need revision. 4. Ask the students if the models on the board predict any new observations that may help distinguish between them. For example, shake the black box to see if it rattles. Return to the models on the board and re-evaluate them, emphasizing the role of testable predictions in the process of developing robust scientific models. Note: Never divulge what is inside the Black Box. In science, we only ever have access to indirect observations—we never “see reality” directly! BUILDING YOUR BLACK BOX Materials: (all dimensions are approximate) • 2 pieces of 8 mm nylon rope, each 70 cm long • 1 harness ring with a 4 cm diameter • 35 cm long piece of drainage pipe (7.5 cm diameter) • 2 drainage pipe end caps (7.5 cm diameter) Tools: • power drill with 3/8” drill bit Procedure: 1. Drill the top holes directly across from one another, each 5 cm from the top. Repeat for the bottom holes, each 5 cm from the bottom (see top Figure). 2. Thread one rope through the top holes and the harness ring (see middle Figure). 3. Tie a knot 15 cm from each end of the rope. 4. Thread the other rope through the bottom holes. Again, ensure that the rope passes through the harness ring as indicated (see bottom Figure). Tie a knot 15 cm from each end of the rope. 5. Secure the end caps. Note: Variations on the design (without a ring for example) will enrich the discussion and work equally well. You may also wish to encourage students to build their own versions of the device with bathroom tissue tubes and string. 5 D2 - Sagging Rod: (a very flexible 2 m long rod with two small masses on each end) 1. Hold the rod horizontally with your hand in the middle so the rod sags. Ask students to explain why the rod is sagging–typically students will say “force of gravity!” 2. Place the rod on a table. Have two students apply horizontal forces on the ends while you hold the middle in place by applying an opposing horizontal force. The class observes the same shape as in step #1. Reinforce the concept that when opposing forces are applied to the rod it will bend. 3. Emphasize the distinction between Observation (when opposing forces are applied to the rod it bends) and Inference (the sagging rod is bent so there must be opposing forces; a “force of gravity” opposes your hand). 4. Have students suggest ways to make the rod bend without using opposing forces. Hold the rod vertically and accelerate it to the side. The ends of the rod will lag behind the middle because of inertia. Emphasize that your hand is applying a force but there is no opposing “force of gravity.” 5. Distribute SW1: Scientific Models. Show the animation: What Keeps Us Stuck to the Earth? after students have worked in small groups to complete the table and discussion sections of the worksheet. D3 - Toy, Bungee Cord and Board: 1. Show the animation: What Keeps Us Stuck to the Earth? 2. Demonstrate Newton’s model of gravity by stretching the bungee cord over the toy (see Figure). “According to Newton gravity is a force, like an invisible bungee cord, that pulls objects to the ground.” Pull the toy away from the board and let it ‘snap’ back down. The bungee cord exerts a force on the toy making it accelerate. 3. Demonstrate Einstein’s model of gravity by removing the bungee cord, holding the toy in the air and accelerating the board up to hit it. “According to Einstein, gravity is not a force. The toy does not accelerate down; rather, the ground accelerates up!” Place the toy on the board and accelerate it up. Ask students to imagine that they are in deep space (no gravity); what would it feel like to stand on an accelerating board? 4. Distribute SW2: Scientific Revolution. Students work in small groups to complete the worksheet. D4 - Curved Spacetime Exemplar: 1. In SW2, the students will use masking tape and a beach ball to model curved spacetime. Read through the activity and make an exemplar on a large exercise ball, if possible (see Figure). 2. The tape describing Alice’s path through spacetime must lie flat. She is experiencing no “force of gravity” and no acceleration so she must follow a straight path. 3. The tape describing Bob’s path must be crinkled. He is experiencing the ground pushing up on him, accelerating him up, and so he must follow a curved path. 4. Time dilation is demonstrated by comparing a length of tape connecting the tops of the ladders with the length of tape connecting the bottoms (Bob’s path). 6 Note: The time dilation demonstrated by this beach ball analogy is actually reversed to the real time dilation– analogies have limits. SW1: Scientific Models: Gravity Scientists use models to try to explain the observations they make. In this activity you are going to use two different models to explain the same observations of an everyday phenomenon—gravity. Force Model: You are standing in a room that is on the Earth; the Earth exerts a downward force on objects inside the room. Explain the following phenomena using this downward force. Follow the sagging rod example. Explain the Sagging Rod Explain Weight Explain Freefall (use words and arrows) (use words and arrows) - the Earth pulls down on the rod and your hand pushes up - the rod bends because your hand is only in the middle - the rod does not accelerate because the two opposing forces are balanced Acceleration Model: You are standing in a room that is inside a rocket; the rocket is accelerating “upwards” in deep space. Explain the following phenomena using this upward acceleration. Follow the sagging rod example. Explain the Sagging Rod Explain Weight Explain Freefall (use words and arrows) (use words and arrows) Force Model Acceleration Model - the room is accelerating up; so are you and the rod - the rod accelerates up because there is now only one force—your hand pushing up - the rod bends because the ends have mass, which resist acceleration (inertia) SUMMARIZE: What is the “big idea” behind each model? How does each explain effects we call “gravity”? 7 Discussion: 1. Examine both of your explanations for freefall. (a) What do you actually observe about an object in freefall? (b) What can you infer about the nature of gravity from your observations of freefall? 2. A flexible rod bends when opposing forces act on it. The same rod bends when suspended horizontally from the middle. Does this prove that gravity is a force? Explain. 3. A friend shows you a video on the Internet of a guy who can make objects “float” in the air. You know this is impossible— how might you explain the video? 4. You wake up in a closed room with no windows, with no idea how you got there. Describe an experiment you could do to determine if the room is on the Earth or inside a rocket accelerating in deep space. >> Watch the animation: What Keeps Us Stuck to the Earth? Thinking Deeper: 1. Both the force model and the acceleration model make claims that are hard to accept. What are they? 2. Both models of gravity explain everyday observations equally well. However, Newton’s force model fails to correctly describe the orbit of Mercury, so it ultimately fails the test for a valid scientific model. Inspired by the acceleration model, Einstein developed an alternative model of gravity. His curved spacetime model made several successful predictions that have conclusively ruled out Newton’s model. Does this mean we should throw out Newton’s model? Does a model have to be correct in order to be useful? 8 SW2: Scientific Revolution: General Relativity Scientific models must make predictions that match our observations, or they must be revised or replaced. New scientific models can be revolutionary. In this activity you are going to examine two models of gravity: Newton’s classical force model, and Einstein’s revolutionary curved spacetime model. Part A: Modeling Gravity Complete this table after watching >> Alice & Bob in Wonderland: What Keeps Us Stuck to the Earth? Force Model Acceleration Model Gravity: How does it work? What’s hard to accept? Alice steps off In the boxes, sketch snapshots of Alice as she falls to the ground and Bob as he stands at the bottom of tall ladder Connect-the-dots of Alice’s position in SPACE the top of a Bob stands at the bottom of the ladder the ladder, showing their progression in time. [Hint: Alice moves faster and faster as she falls.] (height above the ground) as TIME goes on. Is her path through spacetime straight or curved? Connect-the-dots of Bob’s position in SPACE (height above the ground) as TIME goes on. Is Bob’s path through spacetime straight or curved? According to Newton... Alice’s path through spacetime is ______________ because she is accelerating. She is accelerating (straight/curved) because gravity is a force pulling on her. Bob’s path through spacetime is ______________ because he is (straight/curved) not accelerating—the force of gravity is balanced by the ground pushing up. According to Einstein... There is no “force of gravity” pulling down on Alice so she _________ accelerating. Her path through (is/is not) spacetime should be ______________ . The ground pushes up on Bob and since there is no opposing (straight/curved) “force of gravity” to balance this force, he should accelerate up and follow a _______________ path through spacetime. (straight/curved) Discussion: 1. Alice has a video camera in her hands as she falls. If she takes a video of herself as she falls, could she tell that she was accelerating by viewing the video? (Ignore the background.) 2. Alice takes a video of Bob as she falls. Could she tell who was accelerating by viewing the video? (Ignore the background.) 3. Alice closes her eyes as she falls. What does she feel? Can she tell that she is accelerating? 4. Bob closes his eyes. What does he feel? Can he interpret this feeling as accelerating up? 9 Einstein knew that Newton’s model of gravity is wrong. For one thing, it fails to correctly predict the orbit of Mercury; for another, it fails to obey the speed limit of the universe—the speed of light. In his search for a better model, the simple fact that acceleration up mimics force down was too strong of a coincidence to ignore. Einstein needed to find a way to make sense of the ground accelerating up without moving up. How can the ground be accelerating up when the Earth is not expanding? He found the answer in the geometry of spacetime. Part B: Bending Spacetime In Part A, we used the fact that accelerating objects trace out curved paths in spacetime and non-accelerating objects trace out straight paths. We also saw that Newton and Einstein would disagree on who is accelerating and who is not. In this part of the activity you will use tape to transfer the spacetime diagram from Part A onto the surface of a large ball to reveal how curving spacetime resolves the problem of who is accelerating. 1. Use a strip of tape to connect two points on your desk with a straight line. Use another strip of tape to make a curved line. Compare the two pieces of tape. Which strip of tape lies flat on the desk and which is crinkled? 2. Build your spacetime diagram on the surface of a large ball. Start with the space and time axes. • The space axis is a strip of tape that runs vertically along a line of longitude. • The time axis runs horizontally along a circle of latitude (about 15˚ above the equator). 3. Add three identical strips of tape to represent the ladder in three consecutive snapshots. The ladders must follow lines of longitude on the surface, starting about 2 cm above the time axis and ending about 10 cm from the top. 4. Alice’s path is a strip of tape that connects the top of the first ladder with the bottom of the last ladder. Can you make it a straight line? Why would you want to? 5. Bob’s path runs parallel to the time axis along a circle of latitude. It will connect the bottoms of the three ladders. Does the tape lie flat or is it crinkled? What does this indicate? Curved Spacetime: When we transfer the spacetime diagram to the ball we find that the tape for Alice’s path can be ______________ , which means the line is ________________ so Alice is _____________________ (flat/crinkled) (straight/curved) (accelerating/not accelerating) through curved spacetime. The tape describing Bob’s path is ______________________, which (flat/crinkled) means the line is _________________ so Bob is ____________________ through curved spacetime. (straight/curved) (accelerating/not accelerating) Drawing the spacetime diagram on a curved surface reverses who is accelerating and who is not— just what Einstein needed to make the acceleration model make sense. The ground can be forever accelerating up without moving up! Gravity is not a force—it is curved spacetime. 6. The time elapsed for Bob at the bottom of the ladder is the length of his path (i.e. distance in the time direction). If Alice stayed at the top of the ladder, would her elapsed time be the same? Einstein’s model predicts time dilation: time passes at different rates depending on height about the ground, which has been verified by atomic clocks. Newton’s model makes no such claim. Models cannot be proven right—but they can be proven wrong and time dilation proves that Newton’s model of gravity is wrong! Evaluating Models: Newton’s model fails to predict the orbit of Mercury accurately. Einstein’s model does and it also accurately predicts time dilation and the bending of light. We must conclude that the best model of gravity is __________________ ______________________ model. (Newton’s/Einstein’s) 10 (force/curved spacetime) By curving spacetime, Alice’s path changes from curved to straight—she experiences no “force of gravity” and no acceleration. By curving spacetime, Bob’s path changes from straight to curved—he experiences the ground pushing up on him, continually accelerating him up, but without him moving up. Einstein was able to show that gravity is not a mysterious, invisible force—it is the curvature of spacetime. This curved spacetime model asserts that you feel heavy because the surface of the Earth is forever accelerating up without actually moving up. Part C: Accelerating Up without Moving Up Consider the type of motion (accelerating or not) in each of the following scenarios: In Deep Space Rocket 1: Floating in deep space, engines off Rocket 2: Accelerating “up” in deep space, engines on Near the Ground Rocket 3: In freefall near the ground, engines off Rocket 4: Hovering near the ground, engines on 1. In Rocket 1, the astronaut knows she is not accelerating; the rod is straight and she is floating. In which other rocket does she make these observations? 2. In Rocket 2, the astronaut knows he is accelerating; the rod is bent and he feels the force of the floor pushing up on him. In which other rocket does he make these observations? 3. The astronaut in Rocket 3 uncovers the window and looks out. She can see the ground and Rocket 4. (a) What was her type of motion before looking out the window? (Accelerating or not accelerating) (b) How would she describe her motion when she looks out the window? (c) Combine your answers from (a) and (b) into a statement. 4. The astronaut in Rocket 4 uncovers the window and looks out. He can see the ground and Rocket 3. (a) What was his type of motion before looking out the window? (Accelerating or not accelerating) (b) How would he describe his motion when he looks out the window? (c) Combine your answers from (a) and (b) into a statement. We have discovered that astronauts in very different scenarios can experience the same type of motion. This insight is called Einstein’s Equivalence Principle: Freefalling in a uniform gravitational field (Rocket 3) is physically identical to floating in deep space (Rocket 1). Hovering in a uniform gravitational field (Rocket 4) is physically identical to constant acceleration in deep space (Rocket 2). The mass of the Earth curves spacetime so that objects in freefall appear to accelerate down, but there is no force causing this “acceleration”. It is the same kind of “acceleration” you feel when a car accelerates towards you. You are not accelerating—the car is! 11 SA1: Scientific Models: Gravity 1. Observation: using your senses to gather information from your environment. Inference: using logic to interpret the information gathered from your environment. Identify the observations and inferences in the following narrative. Bob wakes up and looks out the window. There are drops of water on the window. “It must have rained last night,” he thinks. He goes downstairs and notices that the ladder is leaning against the house, so he goes outside to help his dad with the roof repair work. “Hey Alice, what are you doing up there?” shouts Bob. Alice is so startled that she loses her grip on the ladder. As she falls to the ground, she sees Bob getting closer and closer. “The force of gravity is making me accelerate down at 9.8 m/s2,” yells Alice. Bob reaches out and catches her just before she hits the ground. “Good thing I was accelerating up at 9.8 m/s2 so I could rescue you,” says Bob. Alice gives Bob a quizzical look and then she tells him about how she was washing the windows when he made her fall. OBSERVATIONS INFERENCES 2. True or False? Rewrite any false statements to make them true. (a) There can only be one model that explains a set of observations. (b) We prove a model is right when we observe the predictions it makes. (c) Models that do not make new predictions are wrong. (d) A model is valid if it can explain the observations. (e) Any model that cannot explain the observations is useless and should be discarded. (f) We design experiments to prove that a given model is correct. 12 SA2: Scientific Revolution: General Relativity 1. Alice and Bob are arguing over whether gravity is a force or curved spacetime. Bob says, “You honestly believe the ground is accelerating up? That’s weird!” Alice replies, “Mysterious invisible force? Who’s weird now, Bob?” Which side of the argument do you hold? How would you convince someone to agree with you? 2. According to Newton, gravity is an invisible, attractive force that acts between massive objects. If his model of gravity is wrong does that mean his equation for universal gravitation is also wrong? 3. According to Einstein, gravity is the curvature of spacetime. If Einstein’s model of gravity is better, why do we still use Newton’s model? When do we have to use Einstein’s model? 4. How is the “force of gravity” similar to centrifugal force? Explain. 5. Newton and Einstein are looking at a book sitting on a table. How would each of them describe the forces acting on that book and how would they justify their description? 13 14 - weight is the sensation of pushing up on an object to force it to accelerate up, along with the accelerating room - the object appears to accelerate down but it is actually the room (and you) accelerating up - the floor continues to accelerate up and meets the object - an object in freefall has no forces acting on it so it does not accelerate - the room is accelerating up - the room is accelerating up; so are you and the object - the object accelerates up because there is now only one force—your hand pushing up Explain Freefall (use words and arrows) Explain Weight (use words and arrows) Acceleration Model - the object accelerates because there is no opposing force - the Earth pulls down on the object - the Earth pulls down on the object and your hand pushes up - the object does not accelerate because the two opposing forces are balanced Explain Freefall (use words and arrows) Explain Weight (use words and arrows) Force Model SW1: Answers effects of this acceleration. 2. Newton’s model is still very useful. It gives a simple intuitive picture of gravity that works for almost all situations. A model does not have to be correct in order to be useful—there are many models that are useful in limited contexts that ultimately fail. We don’t need to use Einstein’s curved spacetime model to calculate the trajectory of a baseball; Newton’s model is adequate for this task. Einstein’s model is necessary only to understand what is really happening to the baseball. Using Newton’s model is analogous to saying that the Sun revolves around the Earth—it is still a convenient way of thinking, even if it is grossly incorrect. 1. The force model claims that there is a mysterious, invisible force that reaches out through space to influence mass but cannot explain the physical nature or cause of this force. The acceleration model claims that the ground is forever accelerating up without moving up. Thinking Deeper: 4. There is no simple experiment that you could do to distinguish between the two scenarios. 3. There are several ways to do this: either a force is applied that you are not able to see (magnets or wires), or the room is in freefall with all objects falling at the same rate as the room. Astronauts train for weightlessness by falling inside a plane that is diving. 2. The fact that a suspended rod will bend exactly like a rod that has opposing forces acting on it does NOT prove that gravity is a force. We can also make the rod bend this way without an opposing force by accelerating it. The “force of gravity” could be a fictitious force we invent to explain the observations made in an accelerating frame of reference. 1. (a) You observe that the distance between the ground and the object decreases at an accelerating rate, regardless of the object’s mass. (b) You can infer either that gravity is a force that causes the object to accelerate or that gravity is our frame of reference accelerating. Discussion: like a mysterious invisible hand. accelerating. Weight and freefall are Our frame of reference is somehow The Earth somehow exerts an attractive force on nearby objects, Acceleration Model Force Model Summarize: 15 Einstein... According to Newton... According to ladder the bottom of the Bob stands at top of a tall ladder Alice steps off the accept? What’s hard to it work? Gravity: How does The ground is accelerating Gravity is a mysterious spacetime. should accelerate up and follow a CURVED path through is no opposing “force of gravity” to balance this force, he STRAIGHT. The ground pushes up on Bob and since there IS NOT accelerating. Her path through spacetime should be There is no “force of gravity” pulling down on Alice so she balanced by the ground pushing up. because he is not accelerating—the force of gravity is pulling on her. Bob’s path through spacetime is STRAIGHT accelerating. She is accelerating because gravity is a force up without moving up Alice’s path through spacetime is CURVED because she is invisible force acceleration. freefall are effects of this that have mass. accelerating. Weight and emanates from objects invisible force that Acceleration Model Our frame of reference is Force Model Gravity is a mysterious Part A: Modeling Gravity SW2: Answers EINSTEIN’S CURVED SPACETIME model. We must conclude that the best model of gravity is ACCELERATING through curved spacetime. which means the line is CURVED so Bob is The tape describing Bob’s path is CRINKLED, NOT ACCELERATING through curved spacetime. which means the line is STRAIGHT so Alice is we find that the tape for Alice’s path can be FLAT, When we transfer the spacetime diagram to the ball 4. (a) The astronaut in Rocket 4 is accelerating. (b) He sees that he is not moving relative to the ground. (c) Objects NOT MOVING relative to the ground are ACCELERATING. 3. (a) The astronaut in Rocket 3 is NOT accelerating. (b) She sees that she is in FREEFALL. (c) Objects in FREEFALL are NOT accelerating. 2. The astronaut also feels a force and the rod is bent in Rocket 4. 1. The astronaut is also floating and the rod is straight in Rocket 3. Part C: Accelerating Up without Moving Up Evaluating Models: Curved Spacetime: 1. The STRAIGHT TAPE is FLAT and the CURVED TAPE is CRINKLED. Part B: Bending Spacetime 4. Bob will feel acceleration. He could just as well be inside a rocket accelerating “up” in deep space. 3. Alice will not feel acceleration. She could just as well be floating weightlessly in space, with a breeze blowing over her face. She cannot tell that she is accelerating until she refers to something in a different frame of reference (e.g. the ground) and even then she can only tell that something is accelerating—not necessarily her. 2. Alice will be able to tell that one of them is accelerating but she can’t tell which one. 1. If Alice ignores the background she cannot tell that she was accelerating. Discussion: 16 - Bob is accelerating up - the force of gravity pulls her down - distance between Alice and Bob decreases at 9.8 m/s2 - dad is fixing the roof - “It must have rained” INFERENCES - notices the ladder - drops of water on window OBSERVATIONS Answers 1. Force is intuitively obvious. A falling object accelerates down, so there must be a force pulling it down. The idea that the ground is accelerating up when the Earth is not expanding just sounds absurd! SA2: FALSE: We design experiments to prove that a given model is wrong. (f) We design experiments to prove that a given model is correct. still be useful in a limited context. discarded. FALSE: Models that do not explain all the observations can (e) Any model that cannot explain the observations is useless and should be (d) A model is valid if it can explain the observations. TRUE explain the existing data to be valid. FALSE: Good models make new predictions, but a model only needs to (c) Models that do not make new predictions are wrong. predictions it makes. FALSE: We prove a model is wrong when we don’t observe the (b) We prove a model is right when we observe the predictions it makes. FALSE: There can be several models that explain a set of observations. (a) There can only be one model that explains a set of observations. 2. True or False? Rewrite any false statements to make them true. 1. SA1: Answers Einstein would agree that the book is “at rest”, but “at rest” relative to what? The ground, which is accelerating up in curved spacetime. The only force acting on the book is the table pushing up and since there is no “force of gravity” opposing the force of the table, the book must accelerate up, along with the accelerating ground. Ultimately, Einstein would justify his description by appeal to experiments—time dilation has been observed using atomic clocks. 5. Newton would say that the book is “at rest” and therefore not accelerating so the forces acting on it are balanced. The compression of the table gives clear evidence that the table is pushing up on the book so a downward force of gravity is needed to balance the forces. 4. Centrifugal force is a fictitious force invoked when objects in a non-inertial frame of reference experience inertia. For example, when a car turns a corner it accelerates but the objects in the car want to keep going straight ahead so they feel a “force” pushing them against the motion of the car. Similarly, the “force of gravity” is a fictitious force created to explain inertial behaviour in a non-inertial frame of reference. The Earth curves spacetime in such a way that the ground is a non-inertial frame of reference. Falling objects seem to accelerate towards the ground, but there is no force causing this “acceleration”; so we invent one— the “force of gravity.” Both forces describe real effects caused by acceleration; neither one describes an actual force. 3. We still use Newton’s model because it is intuitively simple and the math is straightforward. We must use Einstein’s model when accuracy is very important (e.g. space probes and GPS), where Newton’s model breaks down completely (e.g. black holes and neutron stars), or when we are trying to get a clearer picture for how the universe works. 2. Newton’s equation for universal gravitation makes reasonably accurate predictions for the effects of weak gravity (e.g. the effects of the Sun on the orbits of the planets), but gives grossly wrong predictions for the effects of very strong gravity (e.g. near a black hole). The equation is also wrong in the sense that it refers to a force, and gravity is not a force. Acceleration is simpler. Objects “fall” because they have inertia. The frame of reference is accelerating so it looks like objects fall but they don’t. They look like they are accelerating in our frame but in spacetime they are actually not accelerating. Experiments have confirmed the predicted curvature of spacetime, which conclusively rules out the force model. Introduction 18-19 Student Worksheets: SW1: Scientific Models: 20- 21 SW2: Scientific Revolution: 22-23 The Atom Quantum Mechanics Student Assessments: SA1: Scientific Models: 24 SA2: Scientific Revolution: 25 SA3: Applications of 26 The Atom Quantum Mechanics Quantum Mechanics Answers 27- 28 This module contains two single-period lessons based on the Alice & Bob in Wonderland animation: How Can Atoms Exist? In this episode Alice and Bob ask questions about the structure of the atom and discover that the commonly accepted planetary model of the atom (including the Bohr-Rutherford model) cannot possibly exist. Lesson 1 is an introductory level lesson (no prior knowledge of physics is required) that explores why the planetary model fails. Lesson 2 is a more advanced lesson (prior knowledge of waves is an asset) that extends the concepts developed in Lesson 1 to build the quantum mechanical model of the atom—a model that explains how atoms can exist. An additional student activity sheet (SA3) is included that could be combined with either lesson to address applications and implications of scientific discoveries. LessoN 1: LessoN 2: SCIENTIFIC MODELS: THE ATOM SCIENTIFIC REVOLUTION: QUANTUM MECHANICS >> Show the Alice & Bob animation: >> Show the Alice & Bob animation: Distribute SW1: Scientific Models: The Atom. Distribute SW2: Scientific Revolution: Quantum How Can Atoms Exist? This worksheet walks students through a critical examination of atomic models using existing knowledge and a computer simulation to reveal the problems with classical models of the atom. SA1: Scientific Models: The Atom. This worksheet includes additional questions to be done in class or for homework. How Can Atoms Exist? Mechanics. This worksheet engages the students in the creative process of building a quantum model of the atom. Students will use a computer simulation to assist in visualizing the atom. SA2: Scientific Revolution: Quantum Mechanics. This worksheet includes additional questions to be done in class or for homework. 17 How CaN Atoms Exist? Science is a process of building models to explain from? Another possibility would be to invent a new force thought and experimentation. This Perimeter Institute observed. In science, we exhaust all existing possibilities observations and then refining those models through careful classroom resource engages students in this process as they explore models of the atom. Atoms are the building blocks of matter. They are central to our existence; and yet, there is no “commonsense” way to understand how they can exist. The best commonsense atom we can imagine—the one with electrons orbiting the nucleus, like planets orbiting the Sun—would almost instantly self-destruct. Students will exercise their critical and creative thinking skills as they examine how various commonsense models fail and how the very non-commonsensical quantum nature of our universe makes atoms possible. By the early 1900s, experiments had revealed that atoms consist of particles much smaller than the atom itself: one tiny, positively charged nucleus comprising almost all of the atom’s mass, plus a number of even tinier, negatively charged electrons. The challenge was to construct a working model of the atom based on these particles and the forces between them. Electrons are attracted to the nucleus (since opposite that acts inside the atom, but such a force has never been before introducing a new type of matter or force. If the atom cannot exist with static electrons, then the only remaining possibility is a dynamic model where the electrons are moving. In order for a moving electron to stay near the nucleus, its trajectory must bend. The net attractive force towards the nucleus—which defeated the static model—is exactly the sort of force needed to bend an electron’s trajectory into an orbit around the nucleus. But even in the simplest case of a circular orbit, where the electron’s speed is not changing, only its direction is continually changing— the electron is accelerating. This is a problem. When a charged object accelerates (changing its speed or direction), it emits energy in the form of electromagnetic waves. For instance, this is exactly how a cell phone works: electrons in the antenna are accelerated, emitting radio waves. In the atom, the accelerating electrons would emit electromagnetic waves in the form of light. This light would carry energy away from the atom, causing the electron to drop to lower energy orbits, quickly spiraling into the nucleus. charges attract) and repelled from each other (since like So electrons can’t stand still (the static model fails); nor can electrons don’t move is not stable; the attractive force always result in all the atoms in your body collapsing in a blast of charges repel). Any configuration of the atom in which the wins and the electrons collapse into the nucleus. One way to prevent this collapse would be to add “struts” that hold the electrons in place, but we have never seen evidence of any kind of support structure when we strip electrons off an atom. And besides, what type of matter would the struts be made they move (the dynamic model fails). Both models would light energy on par with an atomic bomb. There is no way to escape the catastrophic collapse of any commonsense atom. This raises the question: if the electrons in an atom can’t stand still, and can’t move, what could they possibly be doing? As a first step towards a working model, imagine spreading an orbiting electron into a rotating ring. A perfectly smooth knowledge sm, all scientific “If, in some catacly e sentence oyed, and only on were to be destr of creatures, next generation passed on to the ost uld contain the m what statement wo fewest words? information in the .. that all things omic hypothesis. at e th is it ve lie I be s...” are made of atom MAN – RICHARD FEYN rotating ring is moving but you cannot see any motion—it appears to be static; this is what physicists call a stationary state. A charged rotating ring is stationary so it does not emit electromagnetic waves and would be a simple solution to the energy loss problem. However, such a spreading out of a particle is fraught with severe problems of its own. Each part of the ring would be repelled from all the other parts (since like charges repel), and there would be very strong electrostatic forces tending to make the ring fly apart. We would have to invent a new kind of matter or force to hold it together. Also, whenever we “look” at an electron we always 18 “see” a point-like particle, with the full mass Copernican revolution, with equally vast and far-reaching of a spread out electron in the shape of a ring, or any have allowed us to not only understand how atoms can and full charge of one electron. We never see evidence other shape. Nature’s solution to the unstable atom problem is very strange. An atomic electron does something very much in the spirit of spreading itself out into a rotating ring (avoiding consequences that go well beyond the atom. Quantum ideas exist, and how they work; they also underlie a huge array of technologies from cell phones and computers, to laser surgery and the Internet, representing millions of jobs and trillions of dollars of the world’s economy. the energy loss problem), without literally spreading out its matter (avoiding the other severe problems mentioned above). How can an electron spread out, and not spread out? In an atom, an orbiting electron can be thought of as a particle, like a very tiny baseball, but unlike a baseball, one that doesn’t ly say that nobody “I think I can safe nics.” quantum mecha understands MAN – RICHARD FEYN move along a definite trajectory. It exists in a profoundly weird state in which, at any instant of time, it is not definitely at any location in its orbit. Instead, it is only potentially at each location in its orbit (all at the same time), with an equal potential of being found at any particular location if we were to “look” at the atom (e.g. shine light on it). The very act of light hitting an electron somehow forces the electron to “take a stand”—to assume a definite location. (How this happens is still a mystery today.) This potential, or indefinite, location is described by a fuzzy donut-shaped wave that circulates around the nucleus. It’s not a physical wave, like a sound wave or a water wave; nor is it the electron’s matter physically spread out; instead, it’s a mathematical wave that describes the probability of finding the electron (as a whole point-particle) here or there if we were to “look.” In short, the electron is a particle that behaves like a wave. This weird blending of “particle” and “wave” properties into a single entity is called quantum mechanics. At the foundations of everything we currently know about matter and forces is the discovery of the quantum nature of our universe. The quantum nature of the atom is non-commonsensical. An orbiting electron behaves like a wave, effectively allowing it to be in many places and moving in different directions at the same time! If you wiggle both ends of a Slinky simultaneously, you will create two waves moving along the Slinky in both directions at the same time, resulting in a standing wave. In exactly the same way, we can have two quantum waves circulating in opposite directions around the nucleus. The resulting quantum standing wave describes a single electron behaving as if it is orbiting both clockwise and counter-clockwise at the same time! The mathematics of these waves is well understood. What is not well understood, and still the subject of much debate, is what this mathematics implies about the ultimate nature of reality. The quantum model results in a stable atom and has been experimentally verified to unprecedented precision—it’s decidedly strange, but it works. This breakthrough was a 20th century equivalent to the 19 SW1: Scientific Models: The Atom By the early 1900s, experiments had revealed that atoms consist of particles much smaller than the atom itself: one tiny, positively charged nucleus comprising almost all of the atom’s mass, plus a number of even tinier, negatively charged electrons, such that the total electric charge is zero. In this activity you will build and evaluate possible configurations of these particles to try to produce a stable model of the atom. Part A: Static Model The Law of Static Electricity states that OPPOSITE charges ATTRACT and LIKE charges REPEL. 1. Hydrogen is the simplest atom. It has one negatively charged electron and a positively charged nucleus. What would happen if you put the electron near the nucleus and “let go”? 2. How can Hydrogen exist as a stable atom if its electron and nucleus are attracted to each other? Can you think of a fix for this problem? Part B: The Planetary Model If electrons in the atom cannot be standing still, then they must be moving. Maybe the atom looks like a tiny solar system, with electrons orbiting around the nucleus, like planets around the Sun. As you consider this model, recall that objects that are moving will continue moving on a straight path unless pushed or pulled to the side. 1. What has to happen to a moving electron to change its direction of motion? 2. How might the positively charged nucleus of an atom bend the path of a moving electron? 3. A circular path, or orbit, is the simplest trajectory that an electron could follow. What would happen to the electron’s orbit if we gradually removed energy from the atom? 20 Part C: The Failure of the Planetary Model Any charged object is surrounded by an electric field. It is this field of the nucleus that exerts an attractive force on an electron inside the atom. The electron, too, is surrounded by an electric field. Let’s use the PhET simulation (http://phet.colorado.edu/ en/simulation/radio-waves) to investigate what happens to that field when the electron accelerates (wiggles around). 1. Begin with the following settings: Manual, Full Field, Electric Field, Static Field. What happens to the electric field when you wiggle the electron in the transmitting antenna? 2. Change the settings to: Manual, Full Field, Electric Field, Radiated Field. What happens when you wiggle the electron in the transmitting antenna? 3. Change the settings to: Oscillate, Full Field, Electric Field, Radiated Field. Watch the electron in the receiving antenna. Where does it get the energy to move? 4. An electron orbiting around the nucleus is accelerating just like the electron you wiggled in the antenna. (Imagine looking at the atom from the side. As the electron orbits, it will appear to move up and down.) What would be emitted by the electron as it orbits around the nucleus? 5. Whenever a charged object accelerates (changes its speed or direction of motion), it emits electromagnetic (EM) waves. It takes energy to create these waves, and the waves carry this energy away. Why would this be a problem for the Planetary Model of the atom? Summary: 1. Electrons can’t stand still because: “How wonderful that we have met a paradox. Now we have some hope of making progress.” – NIELS BOHR 2. Electrons can’t move because: There is no way to escape the catastrophic failure of any commonsense atom. This raises the question: if the electrons in an atom can’t stand still, and can’t move, what could they possibly be doing? The answer lies in Quantum Mechanics—a completely new set of laws that describe how nature behaves at a deeper level. 21 SW2: Scientific Revolution: Quantum Mechanics Any commonsense model of the atom is destined to fail. In static models, the atom collapses due to the electrostatic force of attraction the nucleus exerts on the electrons. Dynamic models, like the planetary model, also fail because the atom loses energy as the accelerating electrons emit EM waves, again collapsing the atom. We need a model in which the electron is somehow dynamic (orbiting) but at the same time static (not emitting EM waves)—something physicists call a stationary model. For example, a perfectly smooth spinning top is dynamic (rotating), but appears to be static—you can’t tell that it’s spinning because nothing is changing; it always looks the same. Part A: The Rotating Ring The electron cannot orbit around the nucleus as a point-like particle. What if we spread the mass and charge of the electron out into a rotating ring? 1. A rotating ring of charge behaves like a current-carrying wire. Would the rotating ring emit EM waves? Why or why not? 2. Consider the electrostatic forces acting inside the ring. Would such a structure be stable? Why or why not? Would we be able to observe it? Part B: Standing Waves The rotating ring idea is on the right track, but we have never observed such rings. We always “see” electrons as point-like particles. In preparation for Part C we will need to review some facts about waves: (1) A wave can be in many places at the same time, and (2) Two waves can exist simultaneously in the same place. 1. Stretch a coiled spring (e.g. a Slinky) between two people, on a smooth, horizontal surface (hard floor or table). Wiggle one end of the spring at a constant rate. Where is the wave? What is the direction of the wave? 2. Wiggle both ends of the spring at the same rate. This creates two waves travelling in opposite directions along the spring, existing simultaneously in the same place. Adjust the rate until you get a stable pattern. Notice that the combined wave is not travelling in either direction. It is a wave—it oscillates side to side—but it is not travelling. This is called a standing wave. What happens to the standing wave as you gradually increase the frequency of vibration? Can you create standing waves at higher frequencies of vibration? 22 Part C: The Quantum Model In the quantum model of the atom, the electron is a point-like particle whose behaviour is described by a wave. If the wave is moving, the electron is moving. Wherever the wave exists, the electron can potentially exist. The weird thing is that the electron does not exist at any definite location until its location is measured. Left undisturbed, the electron behaves as if it is spread out like a wave, and stationary states similar to the rotating ring become possible. Use this simulation (http://www.falstad.com/qmatom/) to visualize these waves. Note that these waves are mathematical—the electron’s mass and charge are not physically spread out. 1. Start the simulation. In the top-right drop down menu select “Complex Combos (n=1-4)”. Click on “Clear” then move your mouse over the little circles in the bottom-left panel, noting the yellow text that appears just above the panel. Click on the “n=2, l=1, m= –1” circle, which is the top circle in the second column. Finally, rotate the view by clicking on the z-axis in the top right corner of the main panel and dragging it down until the z disappears at the origin and the y-axis points straight up. This is a “top down” view of a single electron “orbiting” the nucleus of a Hydrogen atom. (The nucleus is at the centre, but not shown.) What do you see? 2. The colours represent the “phase” of a donut-shaped wave circulating around the nucleus, showing that the wave “crests” and “troughs” are moving. Observe that the moving electron is behaving as if it is in two places at once—actually everywhere at once, wherever the wave is non-zero! Select the “View” drop down menu from the top menu bar and deselect “Phase as Color.” You will now see a probability pattern: the probability, at any instant of time, of finding the electron at various locations around the nucleus. In what way is the electron static? In what way is it dynamic? Do you think the electron is emitting EM waves? Draw comparisons with the rotating ring in Part A. 3. Reselect “Phase as Color,” click on “Clear,” and then choose the “n=2, l=1, m=+1” circle. Note the direction of rotation of this wave. Now click on the “n=2, l=1, m=–1” circle. You have just combined two waves circulating in opposite directions around the nucleus to produce a standing wave. This standing wave describes an electron behaving as if it is moving both clockwise and counterclockwise at the same time! Is the electron “moving”? Click on the x-y-z coordinate system and rotate it to view this standing wave from different angles. Deselect “Phase as Color” to reveal the corresponding probability pattern. In what way is the electron static? In what way is it dynamic? Do you think the electron is emitting EM waves? Why or why not? By describing the behaviour of a particle using a wave, anything a wave can do a particle can do. A wave can be in many places at once, or be moving in different directions at once—so can a particle! This leads to very non-commonsensical behaviour of electrons inside atoms, and yet these are the lengths scientists have gone to in order to construct a working model of the atom—one that allows us to understand how atoms can exist in our universe. 23 SA1: Scientific Models: The Atom 1. Why does the Hydrogen atom collapse if the electron isn’t moving? 2. Lithium has 3 electrons and a nucleus with a +3 charge. Show that there is no way to put electrons near the nucleus in a stable, static arrangement. 3. Explain how having the electrons move improves the model. 4. The PhET simulation shows a radio station transmitting EM waves. The energy it takes to create these waves is carried off by the waves. Describe some other examples of EM waves and identify the sources of energy. 5. What is the major problem with the planetary model of the atom? Why do we need new “quantum” rules? Thinking Deeper: 1. If the planetary model doesn’t work, why is it included in almost every introductory Chemistry textbook? 2. What problem does the Law of Electrostatics have for the nucleus of the atom? Suggest a possible solution. 3. All matter is made out of atoms, but there is no way to build a commonsense model of the atom. Summarize the problems and identify properties that a new model must have. 24 SA2: Scientific Revolution: Quantum Mechanics 1. Explain how the rotating ring solves the dilemma of orbiting electrons emitting EM waves. 2. Why does the rotating ring idea fail? 3. A traveling wave is a wave pattern that moves. How does describing the “orbiting” electron by a traveling wave circulating around the nucleus solve the problem of the electron emitting EM waves? 4. A standing wave is composed of two oppositely-directed traveling waves. How is the behaviour of the “orbiting” electron described by a standing wave similar to its behaviour in the static model of the atom? How is it different? Thinking Deeper: 1. Both standing (and traveling) waves can only exist around the nucleus when an integer number of wavelengths fit around the “orbit”. How can this property be used to explain discrete energy levels in an atom? 2. Start the simulation from SW2 (http://www.falstad.com/qmatom/) and use similar settings (and deselect “Phase as Color”). The state produced by selecting “n=2, l=1, m=–1” and “n=2, l=1, m=+1” at the same time is an example of an excited state of the atom. The ground state of the atom (the state of minimum energy) is given by the “n=1, l=0, m=0” circle (top circle in the first column). Both of these are stationary states—they do not emit EM waves. Click on the “n=1, l=0, m=0” circle to put the electron in both states at once—an excited state and the ground state. Increase the Simulation Speed using the slider in the right panel. Do you think the electron is emitting EM waves? Draw comparisons with the antenna simulation in Part C of SW1. What is the quantum atom in the process of doing? 3. Quantum mechanics is often referred to as weird or strange. What is so strange about it? 25 SA3: Applications of Quantum Mechanics Quantum Mechanics is one of the most successful scientific models ever created. Not only has it passed every experimental test to date, but it has become the basis for a huge number of applications, resulting in trillions of dollars of economic activity every year. 1. The quantum mechanical model of the atom says that light is emitted when electrons go from a higher energy state to a lower energy state. A light emitting diode (LED) is a device that uses this property to produce light very efficiently. LEDs do not get hot, do not burn out and do not contain any harmful materials. (a) Where do you find LEDs being used? (b) For a typical home, about 15% of its electricity bill is for lighting. How much money would you be willing to invest in new lighting technology in order to reduce your energy consumption? (c) Research the LED bulb technology that is currently available for residential use. How much would it cost to convert your house over to LED bulbs? How many years would it take for this investment to pay off? (d) What are the factors that you would consider when choosing which technology is the best for you? 2. The quantum mechanical model of the atom says that electrons can only occupy certain energy levels and that the atom will absorb or emit light as the electron changes energy levels. In 1917 Albert Einstein used the laws of quantum mechanics to predict that excited atoms could be stimulated with light to emit their extra energy as more of the same kind of light, thereby amplifying the light. Forty-three years later the first functioning laser was made. (a) Lasers produce very intense, coherent, monochromatic light. List all the applications of lasers that you know of and describe how the properties of laser light are well suited for that application. (b) The scientist who coined the term laser (Light Amplification by Stimulated Emission of Radiation) spent 27 years fighting with the patent office. What would he gain by winning the patent for this technology? 3. The quantum mechanical model of the atom says that electrons are particles that behave like waves. Waves can reflect and produce standing waves. This wave behaviour of electrons is essential for the functioning of transistors, which are the basis for all electronics. (a) Consider your bedroom. List all the devices that contain electronic components. (b) Research the electronics industry. How much money was generated last year by the production of transistors alone? How much money was generated by the production of devices that use transistors? (c) Look up Moore’s Law on the Internet. What does Moore’s Law say and why is it important to the electronics industry? 4. The quantum mechanical model of the atom says that electrons can behave as if they are in more than one place or state of motion at the same time. This strange behaviour of electrons is being explored to design a new type of computer called a quantum computer. Quantum computers will be able to do certain complicated tasks extremely quickly and will allow scientists to make very sophisticated models of quantum systems. As scientists gain the ability to model quantum systems, they will be able to design new and more powerful quantum technologies. Think back over the last century and reflect on how discoveries in basic science and their applications have worked together to produce the world we live in. Where do you think these new applications of quantum mechanics will take us in this next century? Use historical examples to support your insights. 26 27 3. The electron in the receiving antenna gets its energy from the wave emitted by the transmitter. 2. The electric field wiggles, creating a wave pattern in the field that moves away from the electron. 1. The electric field changes as the electric field pattern around the electron moves with the electron. Note for teachers: An electron at rest is surrounded by a static, radial electric field pattern. This field stores energy in the space around the electron (electrostatic energy). When the electron moves with a constant speed and direction, this field pattern (and energy) moves with the electron, like flies buzzing around a moving garbage truck. But when the electron accelerates (changes its speed or direction of motion), some of this energy is “shaken off” (like flies shaken off an accelerating garbage truck) in the form of electromagnetic waves. According to Maxwell’s equations, a changing electric field creates a magnetic field, and vice-versa, setting up a chain reaction that is an electromagnetic wave. (The simulation shows only the electric part of the electromagnetic wave.) In question #1 below, students see just the static part of the field pattern that moves along with the electron. In question #2, students see the radiated part of the electric field—the “flies that are shaken off”. (Note that the energy in the space around the electron is immediately replaced with energy from the “hand” that is wiggling the electron, i.e., it takes more effort to wiggle a charged particle than a neutral particle of equal mass!) When the acceleration is a simple up-and-down oscillation, the EM waves form a simple pattern that radiates outwards from the electron, and carry with them the energy required to make other electrons move. This is what students see in question #3. Part C: The Failure of the Planetary Model 3. The electron would spiral into the nucleus as the energy is gradually removed from the atom. Note for teachers: The electron would actually speed up as it spirals in, increasing its kinetic energy; but the electrostatic potential energy of the electron-nucleus system would decrease by a greater amount, resulting in a net decrease in the atom’s total energy. 2. The nucleus will pull sideways on the moving electron, bending its path. 1. We must exert a sideways force on the moving electron to change its direction of motion. Part B: The Planetary Model 2. The Hydrogen atom cannot be stable if the electron is static. The only other option is to make it dynamic—we need to make the electron move. 1. The electron would be attracted to the nucleus and accelerate towards it, emitting a flash of light. Part A: Static Model SW1: Answers Answers 3. The electron is behaving as if it is moving in two opposite directions at once. In this sense it is not moving (there is no angular momentum), and is like the classical static model of the atom. The difference from the classical static model is that the electron does not get pulled into the nucleus because it is actually moving! As in #2 above, the electron is static in that the probability pattern does not change at all. It is dynamic in that the phase is circulating, albeit in two opposite directions at once! The electron is behaving like two classical, static “blobs” of charge—it will not emit EM waves. 2. The electron is static in that the probability pattern (the “amplitude” of the wave) does not change at all. The electron is dynamic in that the “crests” and “troughs” (the “phase” of the wave) is circulating. The probability pattern tells us that the “potential location” of the electron is spread out into a perfectly smooth ring. The circulating phase tells us that this ring is rotating. So the electron is behaving exactly like a classical, stationary charged rotating ring— it will not emit EM waves. 1. You see a colourful fuzzy ring that slowly rotates in a clockwise direction. The rotation shows the motion of the wave “crests” and “troughs” as the wave circulates around the nucleus. Part C: The Quantum Model 2. As you gradually increase the frequency of vibration, the standing wave will disappear and the Slinky will appear ‘chaotic,’ with random vibrations. Eventually you will reach a frequency which produces another stable standing wave pattern; this pattern will have one more node. 1. The wave is everywhere in the Slinky at the same time. While each part of the Slinky moves side-to-side only, the wave pattern travels in the perpendicular direction, from one end of the Slinky to the other. Part B: Standing Waves 2. The charge inside the ring would repel itself, and the ring would tend to fly apart. Classically, at least, we would be able to observe such a structure by using a microscope with light of sufficiently short wavelength. 1. A rotating ring will not emit EM waves because the electric (and magnetic) fields surrounding the ring are not changing. It is changing electric or magnetic fields that produce EM waves. Part A: The Rotating Ring SW2: 5. The EM wave would remove energy from the atom causing the electron to spiral into the nucleus; the atom would collapse in a flash of light. 4. The electron would emit an EM wave as it orbits around the nucleus because the accelerating electron would create a changing electric field in the reference frame of the atom. 28 1. If an orbiting point-like electron is spread out into a rotating ring, nothing would be “waving” back and forth, or side to side, and so it would not emit EM waves. SA2: Answers 3. The static model collapses due to electrostatic forces. The planetary model collapses due to EM waves draining energy from the atom. We need a model in which electrons somehow “orbit” without emitting EM waves. 2. The protons in the nucleus electrostatically repel one another very strongly. The idea of the strong nuclear force, which holds the nucleus together, could be introduced to students. Here nature does use a new force to solve a stability problem! 1. The planetary model (including the Bohr-Rutherford model) gives a simple, intuitive picture for the atom. It is a good starting point for understanding simple chemical reactions. Models can be useful in a limited context, even if they are wrong. Thinking Deeper 5. An orbiting electron is continually accelerating (toward the nucleus) due to its continually changing direction of motion. This causes the atom to radiate energy in the form of EM waves, and the electron to spiral into the nucleus. Newton’s laws, together with the laws of electromagnetism, predict the collapse of the atom, so atoms cannot exist in a classical universe. Quantum ideas are needed to explain how atoms can exist. 4. Light is emitted when an electron in an atom “drops” to a lower energy level; the energy comes from the excited atom. Cell phones emit EM waves just like a radio station; the energy comes from a battery. X-rays are emitted when electrons are slowed down by a collision; the energy comes from the moving electron. 3. The previous problem points out that there is a net force pulling electrons towards the nucleus. Instead of allowing this force to simply pull static electrons into the nucleus, we use this force to bend the path of the moving electrons, causing them to orbit the nucleus. 2. In any multi-electron atom, each electron will be repelled from the other electrons, and will try to move as far away from the others as possible, in a symmetric way (see Figure for Lithium). But in all cases, each electron experiences a stronger force of attraction towards the nucleus than the net force of repulsion from the other electrons. These unbalanced forces cause the atom to collapse. 1. The nucleus attracts the electron, pulling it into the nucleus. (The nucleus is much heavier, and so hardly moves.) SA1: Answers 3. The weirdness of quantum mechanics may be stated as the wave-particle duality: the idea that all quantum particles (e.g. electrons and photons) exhibit both wave and particle properties. An electron can exhibit the wave properties of being in two locations at once, or moving in two directions at once, which is natural for waves, but not for classical particles. Once we accept this weird wavelike behaviour of particles (and vice-versa), other quintessentially quantum aspects of nature follow naturally. For example, quantization of atomic energy levels is not weird—it is a simple consequence of electrons behaving like waves. 2. The simulation shows a probability pattern that is changing—one that is sloshing back and forth, exactly like an electron moving up and down in an antenna wire. An electron in this non-stationary state is emitting (or absorbing) EM waves, i.e., a photon. In the case of emission, the atom is in the process of “dropping” from the excited state to the ground state; and the reverse in the case of absorption. Note to teachers: Electrons do not mysteriously “jump” between atomic energy levels! There is a very sensible physical process involved. 1. Each energy level corresponds to a different “harmonic,” like the harmonics on a violin string. For the first energy level, one wavelength fits around the “orbit”. For the second energy level, two wavelengths fit around the “orbit”, and so on. The electron is never found between these discrete energy levels because you don’t get stable standing (or traveling) waves there. Thinking Deeper 4. It is the same in that a standing wave does not move, like an electron in the classical static model. It is different in that the electron does not get pulled straight into the nucleus. This is because the electron is actually “moving” (albeit in two directions at the same time!), and so the net force towards the nucleus just bends the path of the electron into two simultaneous, counterrotating “orbits”! 3. The point-like electron behaves as if it is in many places at once (wherever the wave is non-zero), and so it is effectively spread out exactly like a rotating ring. The wave is moving (the electron has angular momentum) but the corresponding probability pattern is not changing—a state called a stationary state. Electrons in such stationary states do not emit EM waves. 2. Different parts of a spread out electron would repel each other, tending to make the ring fly apart. We would have to invent a new type of matter or force to hold the ring together. Also, whenever we “look” at an electron, we always see a point-like particle. If an electron took the form of a ring, it would have to turn into a point-like particle the instant we “look” at it. This would be absurd. It would create static electric and magnetic fields, but EM waves are produced only when these fields change. Introduction 30-31 Student Worksheets: SW1: Scientific Models: 32-34 SW2: Scientific Revolution: 35-36 Time Special Relativity Student Assessment: SA: Scientific Revolution: 37 Special Relativity Answers 38- 40 This module contains two single-period lessons based on the Alice & Bob in Wonderland animation: Where Does Energy Come From? In this episode Alice and Bob ask questions about energy and discover a deep connection between mass and energy. Lesson 1 (introductory level; some prior knowledge of physics is an asset) introduces the concept of relativity, and shows how it led to Einstein’s model of relative time and a universal speed limit. Lesson 2 is a more advanced lesson (prior knowledge is expected) that starts with relative time to show how energy has inertia. LessoN 1: LessoN 2: SCIENTIFIC MODELS: TIME SCIENTIFIC REVOLUTION: SPECIAL RELATIVITY >> Show the Alice & Bob animation: >> Show the Alice & Bob animation: Distribute SW1: Scientific Models: Time. This Distribute SW2: Scientific Revolution: Special Where Does Energy Come From? worksheet presents the students with several thought experiments that will help them develop Einstein’s Special Theory of Relativity. Where Does Energy Come From? Relativity. This worksheet presents the students with several thought experiments that will help guide them to develop E=mc2. SA: Scientific Revolution: Special Relativity is a worksheet that includes additional questions to be done in class or assigned for homework. 29 Where Does ENergy Come From? Science is a process of building models to explain But is relativity universal? Is it really true that you can’t detect careful thought and experimentation. Good models explain inside the room? Ever since Galileo first suggested the observations and then refining those models through existing observations, make testable predictions, and give deeper insights into the phenomena. This Perimeter Institute classroom resource engages students in this process by exploring models of two common real world phenomena— time and energy, with an emphasis on the role of thought experiments in science. Students will exercise critical and creative thinking to discover how Albert Einstein’s intuitive belief that an observer in a closed room cannot tell whether the room is moving leads to a radical new understanding of time and energy, known as the Special Theory of Relativity. We have all experienced relativity. When you are inside a closed room, such as an airplane with the window shades the inertial motion of a closed room by any experiment done concept of relativity in 1632, it has been accepted as true for mechanical experiments (such as drinking coffee, or juggling balls). But in the early 1800s, strong experimental evidence emerged to show that light behaves like a wave, and this presented a problem for relativity. The problem was that, like sound waves, light waves presumably could not travel in empty space. So space must be filled with a light-wave medium they called the “ether”: an immobile substance whose vibrations constitute light, but through which matter could move freely. The wave nature of light created the possibility of detecting motion relative to the medium. Ether might represent a state of absolute rest. drawn, you can’t tell that the room is in motion. You can’t feel Students will explore the speed with no rotation). Everything that happens inside the for relativity using a thought inertial motion, (i.e. motion in a straight line at a constant room (e.g. drinking coffee or juggling balls) happens the same way it does when the room is at rest, no matter how fast the room is moving. From this we learn that “at rest” and “in motion” are relative concepts—they make sense only when compared to objects outside the room, such as the ground moving beneath an airplane. There is no such thing as absolute rest. wed that the very nature of Did you know? Einstein sho tive from detecting motion rela time and space prevents us it and ht mig It ts. exis it if w to the ether. We cannot kno might not. hes us, however, that “More careful reflection teac y does not compel us to the special theory of relativit STEIN deny ether.” – ALBERT EIN only “Speed of Light Principle” Did you know? Einstein’s ion mot the of t t is independen asserts that the speed of ligh e-in wav the is obvious for of the source of light, which of not suggest that the speed s doe It t. ligh ether model of t. ligh of r erve motion of the obs light is independent of the and r, eve ement is true, how This absurd-sounding stat his an obvious consequence of is it how Einstein showed ce. new model of time and spa is, the very concept of time, that “My solution was really for STEIN defined...” – ALBERT EIN that time is not absolutely challenge that waves pose experiment: Alice is inside a spaceship floating in deep space, at rest in the hypothetical ether. She sends a pulse of light (a single wave front) upwards from the floor, which moves at speed c relative to the ether. She measures the time it takes to reflect off the ceiling and return to the floor. Alice then repeats exactly the same experiment in her “closed room”, except now it is drifting through the ether (causing an “ether wind” to blow through the spaceship). She still sees the pulse of light travel vertically up and down, but Bob—floating at rest in the ether, sees the light pulse move diagonally up and down. It is an established fact that waves of any kind move at a fixed speed relative to the medium. Once created, a wave propagates on its own, independent of any motion of the source. Even though Alice’s light source is in motion, the wave front it creates will move with speed c relative to the ether. And since it travels a greater distance in the second experiment, Bob will measure a greater return time. 30 The crucial question is, “What elapsed time will Alice and clock. Bob will see the ball covering less than one metre But, since this time is different from Alice’s “at rest” time, we The additional speed of 1 m/s that Alice has given to the ball measure?” The obvious answer is, “The same as Bob!” would have to admit that it is possible to use an experiment with light inside a closed room to detect the inertial motion of that room. Universal relativity would not hold. For universal relativity to hold, Alice must measure the same time whether she is at rest or moving, but then Isaac Newton’s model of absolute time would be wrong. (Absolute time says when one second elapses for you, one second elapses for everyone in the universe, regardless of their location or state of motion; in this case, Alice must measure the same time as Bob.) Students see that if we adopt a wave-in-ether model of light (as most physicists did until after 1905), then absolute time and universal relativity are incompatible. One is wrong, and must be jettisoned from our thinking. For many years scientists, including Einstein, struggled with the tension between relativity—which seemed so simple and natural that it ought to be universal—and the nature of light. The definitive breakthrough came in 1905 when Einstein realized that the problem had nothing to do with light, but rather the nature of space and time. In particular, he realized that absolute time was just an assumption that had never really been tested beyond everyday experience. He immediately jettisoned Newton’s model of absolute time and worked out the logical consequences of universal relativity. Students will work through the following rational and intuitive progression: (length contraction) in more than one second (time dilation). will be less than 1 m/s for Bob. As Alice’s rocket approaches the speed of light, this effect becomes more pronounced, so that it is never possible for Bob to see the ball reach (or exceed) the speed of light. By universal relativity, no matter how fast Alice is moving relative to Bob, she can consider herself to be “at rest”, and can throw the ball forward as close to the speed of light as she wants, relative to herself. Unified Model of Energy: E=mc2. Using only time dilation, students engage in a thought experiment to conclude that a fast moving ball is more difficult to deflect sideways than one at rest. Its inertia relative to someone at rest increases along with its kinetic energy. By placing two balls connected to the ends of a spring inside a box, and letting the balls oscillate rapidly, students see that the balls’ increased inertia relative to the box increases the mass of the box—kinetic energy has inertia. As the system energy oscillates between kinetic energy and spring potential energy, students realize that potential energy has inertia too. Extending this to the molecules in a hot object, students learn that thermal energy has inertia. Since a block can be heated with light, students also discover that electromagnetic energy has inertia. All forms of energy possess inertia (resistance to changes in motion). Students also explore the converse: how Einstein correctly guessed that the inertia (i.e. mass) of even an object at rest is equivalent to an enormous amount of energy. Time Dilation. According to the second part of our thought This resource introduces students to Einstein’s Special Bob’s. Note that, according to universal relativity, from Alice’s facts, but rather as a logical argument based on the simple experiment, Alice’s moving clock runs slowly relative to perspective she is “at rest”, and it is Bob’s clock that is moving, and running slowly (not faster!). Time dilation is reciprocal. Theory of Relativity not as a sequence of counterintuitive and natural Principle of Universal Relativity, supported by ample experimental evidence. Length Contraction. During the time measured on his clock, Bob sees Alice cover a certain horizontal distance as measured in his frame of reference. But for Alice, it takes less time. For both to be moving at the same relative speed, Alice must measure this distance to be less in her frame of reference. She must see Bob’s space—and everything in it, contracted. And again, by universal relativity, this effect must be reciprocal: Bob must see Alice’s space—and everything in it, equally contracted. Universal Speed Limit. Now suppose Alice throws a ball forward inside her moving spaceship so that it is covering a dilation and length What makes time y Did you know? ect called relativit ocal is a subtle eff t contraction recipr no is b Bo us for hat’s simultaneo , of simultaneity. W rsa! Time dilation ve e vic d an , Alice rk simultaneous for wo y eit an ult sim n, and relativity of lflength contractio y in a logically se vit ati rel l rsa ive un ce for en to er togeth er. consistent mann and mechanics of electrodynamics “The phenomena g to the idea of rties correspondin possess no prope IN ALBERT EINSTE absolute rest.” – distance of one metre every second according to her ruler 31 SW1: Scientific Models: Time In this activity you will conduct several thought experiments and use logic to discover something fascinating about the nature of time and space. You will work in groups of three using chart paper or a large whiteboard. Each group will have a Reader (who keeps the group on task), a Recorder (who writes the answers on the chart paper), and a Reviewer (who asks questions to check for understanding). Part A: Relativity—“Can you tell you are moving?” Commonsense dictates and experience confirms that there is no mechanical experiment that can be done in a closed room to tell whether or not the room is moving at a constant speed in a straight line. 1. Imagine Alice sitting in a parked car, tossing an apple straight up. Later, she repeats this experiment while the car is moving at a constant speed. (a) Sketch the path of the apple, as seen by Alice, both when the car is parked and when it is moving. (b) Can Alice use this experiment to tell that she is moving? (c) Sketch the path of the apple, as seen by Bob standing on the sidewalk, when the car is moving. 2. Imagine Alice on a raft made from two pontoons joined by a couple of poles. She anchors the raft in a still pool of water and sends a wave from one pontoon across to the other. It reflects back and she measures the total time taken. Later, the raft is propelled through the water at a constant speed and she repeats her experiment (see Figure). An important fact to consider is that the speed of a wave produced by a moving source will be the same as the speed of a wave produced by a stationary source. (a) Sketch the path of the wave Alice observes when the boat is anchored. (b) Sketch the path of the wave, as seen by Bob floating at rest in the water, when the boat is moving through the water. (c) Can Alice use the times measured in this experiment to tell that she is moving? 3. Imagine Alice inside a rocket deep in space with a clock that measures the time taken for a pulse of light to go up to a mirror on the ceiling and back again. Alice has learned that light behaves like a wave, and assumes it travels through some medium in space that flows freely through the moving rocket. Suppose she starts at rest in that medium and measures the time taken for the pulse to go up and back again. Later, she repeats the experiment when she is moving at a constant speed. Bob is just floating in space, at rest in the medium. He also times the light pulse as Alice cruises by. (a) Sketch the path of the light pulse Alice observes when she is at rest. (b) Sketch the path of the light pulse Bob observes when Alice is moving. (c) Can Alice use this experiment to tell that she is moving? Part B: Newton vs Einstein Newton believed in absolute time: the rate at which time passes is the same for everyone regardless of their motion. Einstein believed in universal relativity: there is no way to tell, from inside a closed room, that the room is moving. The natural assumption that light waves travel through some medium in space means that the speed of light along both paths in the diagram is the same. Since the path observed by Bob is longer he must record a longer time than Alice did when she was at rest. This puts absolute time and universal relativity into direct conflict—only one of them can be right. 32 1. Absolute time dictates that in the moving experiment Alice should measure the same time as Bob does. What are the implications for universal relativity if this is true? 2. Universal relativity dictates that in the moving experiment Alice should measure the same time as she did when she was at rest. What are the implications for absolute time if this is true? Absolute time and universal relativity are incompatible. One of them must be wrong. The only way to resolve the conflict is through experiment. 3. The Large Hadron Collider (LHC) is the largest scientific experiment in history. It is a 27 km long particle accelerator that smashes protons together with unprecedented energy. Neutral kaons are unstable particles produced during the collisions that decay with a half-life of 8.9x10-11s. Use this half-life as the decay time for the kaons. (a) How far would you expect the kaons to travel before decaying, if they are travelling at 0.995c? (b) Kaons are detected 27 cm from the centre of the collision. How does this data refute Newton? (c) How would Einstein interpret these results? 4. Consider again the kaons produced at the LHC. From the perspective of the kaons, they are at rest and they survive for 8.9x10-11s. The kaons “see” the detector rushing by them at 0.995c. What length of the detector rushes by during this time? How can you reconcile this with the 27 cm mentioned above? A logical consequence of universal relativity is time dilation—“moving clocks run slow.” The flip-side of time dilation is length contraction—“moving objects occupy less space.” Part C: Time Dilation and Length Contraction Experimental evidence supports Einstein’s predictions of time dilation and length contraction. With this as motivation, let’s take a closer look at the diagram above for the rocket experiment. 1. According to universal relativity, Alice cannot tell that she is moving. The time taken by the light pulse must be the same for her, whether she is moving or not. Write the algebraic expression for the time taken, tAlice , when Alice is not moving. 2. Bob sees the light pulse travel up and down in the vertical direction, with a vertical speed of ). Write the algebraic expression for the time Bob measures, tBob. (which simplifies to 3. Compare tAlice and tBob. What is the “time dilation” factor that relates Alice’s time to Bob’s time? 4. In the moving case, Bob sees the vertical speed of the light pulse to be vertical speed of light to be less than c? Explain, using time dilation. . Does this mean that Alice sees the 5. When Alice is moving relative to Bob, Alice and Bob disagree on how much time elapses for the light pulse to return. How will this disagreement affect their understanding of how far Alice has travelled in her rocket? Time and length are both changed by the same amount, the Lorentz factor, and it shows up in so many relativity calculations that it gets its own symbol, γ. Note that γ is always ≥ 1. 33 Part D: Speed Limit One of the misunderstandings about relativity is that Einstein began with this statement that “nothing goes faster than the speed of light” and derived everything from that premise. While this statement is true, it is not fundamental—it is a logical consequence of time dilation and length contraction. 1. Alice is inside her rocket, moving relative to Bob, when she starts running forward at 1 m/s. She covers 1 m in 1 s according to her ruler and clock. (a) How far is Alice’s 1 m as measured by Bob? (Express your answer in terms of γ.) (b) How long is Alice’s 1 s as measured by Bob? (Express your answer in terms of γ.) (c) How fast is Alice running inside the rocket as measured by Bob? (Express your answer in terms of γ.) 2. As Alice begins to run, her speed inside the rocket changes from zero to 1 m/s. How will this speed change appear to Bob? How will this speed change be affected by the speed of the rocket as it gets closer to c? Time dilation and length contraction “enforce” a universal speed limit, and allow it to make sense. 3. Suppose Alice needs to apply a force F, for one second, to get herself running. Now imagine, instead, that Bob “reaches” into her moving rocket and pushes her with the same force, F. (He stays at rest, but his hand moves very fast!) How long does he need to apply the force, according to his clock, to have the same effect on Alice? It’s harder for Bob to accelerate Alice, as if her mass somehow increases. Actually, nothing happens to Alice’s mass. It is time dilation and length contraction that make Alice’s effective inertia greater, relative to Bob. 4. Plot the following historical data for particle accelerators with Energy of the proton (in GeV) on the x-axis and Speed of the proton (as a % of c) on the y-axis. What happens to the speed as more and more energy is given to the particle? Where is the energy going, if it’s not going into increasing the proton’s speed? Proton accelerator Energy (GeV) Speed (%c) CERN Linac 2 0.050 31.4 TRIUMF 0.48 75 CERN PS Booster 1.4 91.6 BNL Cosmotron 3.3 97.5 CERN PS 25 99.93 Suppose we give a particle some energy to accelerate it from rest up to speed v. By universal relativity we can catch up with the particle and see it as “at rest” again. We can then repeat this process—again and again, forever. While its speed is limited (by the nature of time and space), the amount of energy we can give a particle is unlimited. This is just one of the fascinating ideas contained in Einstein’s most famous equation, E=Mc2. Bringing It All Together 1. Review the work that you have done as a group and discuss any points that need clarification. Summarize the concepts in your notebook. Be sure to address the following points: • What is universal relativity and absolute time, and how do they conflict with each other? • What is time dilation and length contraction, and how are they related? • Why is there a universal speed limit? • List some of the experimental evidence for Special Relativity. • What happens to an object’s effective inertia as its speed increases? 34 SW2: Scientific Revolution: Special Relativity Following the logical consequences of universal relativity, we have so far discovered time dilation, length contraction and a universal speed limit. In this activity you will conduct several more thought experiments and use logic to discover something fascinating about energy. You will work in groups of three using chart paper or a large whiteboard. Each group will have a Reader (who keeps the group on task), a Recorder (who writes the answers on the chart paper), and a Reviewer (who asks questions to check for understanding). Part A: A Head-on Collision Imagine that two Super Balls collide head-on. 1. Both balls have speed v going in and speed v going out of the collision. How do their masses compare? 2. Both balls have speed v going in, but one gains speed as a result of the collision. How do their masses compare? 3. One ball has speed v going in and out of the collision, and the other has a greater speed, V, going in and out of the collision. How do their masses compare? Part B: A Glancing Collision Two identical Super Balls undergo a very fast glancing collision that is perfectly symmetrical. Alice is riding on the upper ball. She sees the dashed line whizzing by her with a very large horizontal speed as she drifts toward and then away from it with a very small vertical speed, V. Bob, riding on the lower ball, sees the same thing for himself. 1. Imagine that you are now moving to the left with enough speed that Bob has no horizontal speed relative to you (see lower Figure). Would your new perspective change the vertical speeds that Alice and Bob observe for themselves? 2. Alice is now moving even faster relative to you, and her time is dilated compared to you by a factor of γ. If it takes Alice one second to move the vertical distance to the dashed line according to her clock, will it take more or less time for her to cover the same vertical distance according to your clock? How does time dilation change Alice’s vertical speed, from your perspective? 3. Alice’s vertical speed does not change as a result of the collision, neither does Bob’s. If we ignore Alice’s horizontal motion, this collision is the same as #3 in Part A. How do their masses compare? By time dilation, a moving object has greater “effective inertia” for sideways deflection: M = γ m Part C: All Forms of Energy have Inertia Alice comes across a closed box “at rest” in deep space. 1. (a) Nothing enters or leaves the box. Can the mass of the box suddenly change from M to M’? Why or why not? (b) Bob is drifting by and sees the box moving at a constant speed v. How can Bob use conservation of energy and/or momentum to explain that such a change in mass is impossible? 35 2. Alice opens the box and finds two balls of mass m connected by a spring. The balls are oscillating back and forth very quickly. When they are moving fastest, their time dilation factor is γ. (a) Using the concept of “effective inertia” for sideways deflection, how does the motion of the balls affect the mass of the box? Does kinetic energy have inertia? (b) As the balls move outwards, their kinetic energy changes into elastic potential energy stored in the spring. The total mass of the box cannot change. How does this show that potential energy has inertia? Kinetic Energy 3. A brick is made of atoms connected by spring-like inter-atomic bonds. As the brick is heated, would you expect its mass to increase? Explain. 4. A brick can be heated with the energy in light. The “before” picture shows a box containing a brick of mass M and two pulses of light in midflight heading towards the brick. In the “after” picture, the block has absorbed the light energy and is warmer. Does the mass of the box change when the light is absorbed? What does this say about the inertia of light? All forms of energy have one property in common: inertia. This is a powerful unifying principle in unravelling the mystery of what “energy” is. Part D: E=mc2 We have just discovered that various forms of energy (kinetic, potential, thermal, and electromagnetic) inside a box contribute to the mass (or inertia) of the box. So any change in the energy, ΔE, inside a box must produce a corresponding change in its mass, Δm. The exact relationship is very simple: ΔE = Δmc2. This mass-energy equivalence applies to all physical processes, including chemical and nuclear. All forms of energy have inertia; but do all forms of inertia have energy? Does even mass at rest have energy? 1. The “before” picture shows a box containing two particles at rest, each of mass m; one is matter and the other is antimatter. In the “after” picture, the matter and antimatter have been transformed entirely into light. (a) The box is sitting on a weigh scale. Does its weight change? Explain. (b) We uncover a window on the box and let out all the light. What is the change in mass of the box? How is this related to the amount of energy that left the box? Experiments with elementary particles confirm Einstein’s intuition that even mass at rest has energy: E=mc2. The general form of Einstein’s mass-energy equivalence relation is: E=Mc2, where M= γm is the relativistic mass and m the rest mass. This mass-energy equivalence can be rewritten as a general relation between energy and momentum: E2 = m2c4 + p2c2, where p = γmv is the relativistic momentum of the system. When the system has no rest mass (e.g. a photon) the general relation reduces to E=pc, a result that agrees with both a wave model of light and a particle model of light. When the system is at rest (p=0), the general relation reduces to E=mc2. It is a remarkable fact of nature that matter can transform into light, and vice versa, but notice that in such processes both the total mass and the total energy stay the same. Mass is not “converted” into energy, or vice versa. The energy in light has inertia, and the inertia in matter has energy—as described by E=mc2. Putting It All Together: 1. Review the work that you have done as a group and discuss any points that need clarification. Summarize the concepts in your notebook. Be sure to address the following points: • How does time dilation result in moving objects having extra “effective inertia”? • How is an increase in the “effective inertia” different from an increase in the mass of the object itself? • How would you explain to your friend that kinetic and potential energy have inertia? • Does a cup of coffee weigh more when it is hot? Explain. • Why is it incorrect to say that mass is converted into energy, and vice-versa? • As the Sun emits light energy, what must happen to its mass? Where has the mass gone? 36 SA: Scientific Revolution: Special Relativity 1. The scientists at CERN accelerate protons to 99.9999991% c as the protons travel around a 27 km long ring. How does this extreme speed create challenges for them? 2. One of the most famous equations in science is E=mc2. There are several ways to understand where this equation comes from. Here is one published by Albert Einstein: (a) Start with a block of mass M at rest. Let it absorb two pulses of light, each with energy E/2, so the block heats up (see Figure). (b) Consider this same experiment from the point of view of an observer drifting by with speed v. To this observer, the block has momentum Mv, and the pulses of light approach the block at an angle. (c) Draw a diagram to show how this angle is related to the relative speed of the block and the speed of the light pulses. Remember that light moves at speed c. (d) Express the sine of the angle as a ratio of these two speeds. (e) The momentum of light is given by Maxwell’s equations as: p= . What is the momentum of each pulse of light approaching the block? What is the vertical component of this momentum? (f) What is the change in momentum of the block when it absorbs the two pulses of light? (g) The block does not speed up when it absorbs the light. Why not? (h) If momentum changes but speed does not, then the mass of the block must increase from M to M’. Using conservation of momentum, find an expression for E in terms of the change in mass. 3. In the animation Where Does Energy Come From? Alice and Bob discover that we are literally eating the Sun! The Sun provides the energy that is the basis for virtually every food chain. We measure the energy output of the Sun by its luminosity and find that the Sun emits 3.8x1026 W. (a) Use E=mc2 to determine how much mass the Sun is losing every second. (b) The Sun has a total mass of 2x1030 kg. How long will it take to use up the mass of the Sun? (c) The Sun is expected to survive for another 5 billion years. Will the mass loss be significant by then? 4. The general relation between energy and momentum in Special Relativity is E2 = m2c4 + p2c2 , where m is the rest mass for the system and p is the relativistic momentum, p = γmv. (a) Start with the general energy-momentum relation and use common factors to derive E = γmc2. (b) Start with E = γmc2 and use the binomial expansion of to show that E ≈ mc2 + ½mv2. (c) What does this last equation contribute to our understanding of energy? 37 38 Answers (b) 2. (a) (b) 3. (a) 2. If Alice measures the same time as when she was at rest, then she would not know that she is moving. But since this time is different from Bob’s, time is not absolute. Two observers moving relative to each other will disagree on how much time has elapsed. 1. If Alice measures the same time as Bob, which is different from when she was at rest, then she would know that she is moving. If she can tell that she is moving then relativity is not universal. Relativity would not apply to experiments done with light in a closed room. Part B: (c) This problem is in principle the same as #2, and students might expect that the return time will be different—Alice can use this experiment to tell that she is moving. In fact she cannot, and students challenge the reason for their expectation (absolute time) in Part B. 3. (a) (c) Alice can tell that she is moving relative to the water because the time taken for the wave to return is longer when she is moving than when she is at rest. The time is longer because the path in longer, but the speed of the wave relative to the still water is the same in both cases. (c) (b) Alice cannot tell that she is moving. Both paths will be identical. 1. (a) Part A: SW1: The detector rushes by at 0.995c for 8.9x10-11 s. During this time the kaons will “see” 2.7 cm of the detector pass by. But actually they have travelled through 27 cm of the detector. From the kaons’ perspective, the detector must be contracted in the direction of motion by a factor of 10, so the 27 cm of detector occupies a space of only 2.7 cm in the kaons’ frame of reference. 4. Bob sees the light moving up and down more slowly than speed c, and so it might seem that Alice would see a vertical speed of less than c for the light pulse in her frame. But Alice’s time is also passing more slowly relative to Bob, and so she sees the normal speed for light—c. It is the nature of time and space, and not the nature of light, that makes all observers measure the same speed for light (or anything else moving at the universal speed limit). Alice cannot use the return time or the speed of the light pulse to detect her motion! 3. 2. 1. If d is the length of the path (up and down), then Part C: 4. (c) Einstein would interpret these results by saying that when 8.9x10-11 s elapses in the detector frame, less time elapses for the kaons. While the kaons don’t feel it, time passes more slowly for them relative to the detector. The kaons still have time to travel further (in fact, 10 times as far) before they decay. (b) The kaons are travelling 10 times further than they “should”—27 cm versus 2.7 cm. According to Newton, when 8.9x10-11 s elapses in the detector frame, the same amount of time should have elapsed for the kaons. They “should” decay after travelling only 2.7 cm. 39 b) (c) 4. Giving more energy to a very fast moving particle has little effect on its speed. If we catch up with the particle, giving it energy will dramatically increase its speed in this new reference frame, but this speed increase is small as seen in the frame of the accelerator. (See #2.) As in #3, time dilation causes the effective inertia of the particle to increase as its speed approaches c. 0 25 50 75 100 0 16.7 Energy (GeV) 8.3 25.0 3. One second for Alice is γ times one second for Bob. He will need to apply the force for a longer time, according to his clock, to have the same effect on Alice. Alternatively, if he applies a force F for some time according to his clock, Alice will claim that he applied that force for less time. Time dilation makes his effort to accelerate Alice less effective than he thinks! 2. Bob will measure Alice’s change in speed to be less than 1 m/s—her speed change will be reduced by a factor of γ2. As the speed of Alice’s rocket approaches c, γ becomes larger and larger so her changes in speed will appear to Bob to be less and less. That “nothing goes faster than the speed of light” is a consequence of the very nature of time and space. 1. (a) Part D: will not have gone as far as Bob thinks. Alice’s perspective, Bob’s space—and everything in it is contracted so she Bob moving at the same speed at which that Bob sees Alice moving). From tBob), she must see this distance as less: LAlice < LBob (in order that Alice see will see Alice travel a distance LBob. Since Alice’s travel time is less (tAlice < 5. Alice and Bob will disagree on how far she travels. During the time tBob, Bob Speed (%c) Answers 3. As the brick is heated the atoms will vibrate, much like the balls in #2, so the inertia of the system will increase and the hot brick will have a greater mass. 2. (a) The moving balls have more effective inertia for sideways deflection, which means the box will present a greater resistance to upward acceleration. (It can be shown that this effect is the same for all directions of acceleration.) The kinetic energy of the balls increases the inertia (or mass) of the box. (b) The mass (or inertia) of the box cannot change so the potential energy stored in the spring must have an equal amount of inertia as the kinetic energy that was previously stored in the moving balls. 1. (a) The mass of the box cannot change without gaining or losing something to the surroundings. (b) The mass of the box cannot change because that would violate both energy and momentum conservation laws. For example, the kinetic energy of the moving box is ½ Mv2. The kinetic energy can’t change because there is no outside force acting on the box. So if v doesn’t change, M can’t change. Part C: 3. If Alice’s vertical speed in and out is lower than Bob’s, then she must have more mass (or inertia). Note to teachers: Ignoring the horizontal motion of the ball may be challenging for students. This difficulty is alleviated in Part C by effectively putting the ball in a box and letting it move side-to-side very rapidly. 2. It will take Alice longer from your perspective to cover the same vertical distance, so you see her moving more slowly than she does. 1. A change in your perspective will not affect the speeds observed by Alice and Bob. Part B: 3. If two balls of differing speeds collide and maintain their original speeds then their masses must be different by the same ratio. The slower ball must have more mass than the faster moving ball. 2. One ball gains speed as a result of the collision so it must have less mass than the other. 1. The speeds going in and out are equal so the masses must also be equal. Part A: SW2: 40 Answers 2. The speed of the block does not change in its rest frame. The speed of the block in the moving frame is due to the motion of the observer, and so obviously it will not change. 1. The protons are travelling so fast that there is a huge increase in their effective inertia for sideways deflection (γ=7500). They need a magnetic field of over 8 Tesla to bend their trajectories into a circle—this requires superconducting magnets. SA: (b) When all the light has escaped, nothing remains, and the change in the mass of the box is Δm = 2m, where 2m is the mass of the original particle/ anti-particle pair. Using ΔE = Δmc2, the change in the energy of the box is ΔE = (2m)c2. But since energy is conserved, we must have ΔE = E, where E is the energy in the escaped light. So E = (2m)c2 in this case. In general, an object with mass m at rest has energy E=mc2. Even mass at rest has energy! 1. (a) The mass of the box will not change. The inertia of the light must be the same as the mass of the particle/anti-particle pair. Mass is conserved. Note to teachers: While students might not make the connection, the fact that the weight of the box also does not change follows from the nature of gravity: the weigh scale is accelerating up in curved spacetime, and is measuring the resistance of the box to acceleration! Part D: 4. The box contains the brick and the light (the light is not being added from outside the box) so the total mass of the box cannot change when the brick absorbs the light. According to #3 the warmer brick has more mass, which means that light must have inertia! A box with light bouncing around inside will resist acceleration more than the same box when it’s empty. Even after 5 billion years the amount of mass lost by the Sun is negligible. (c) Einstein showed that even when an object is at rest and has no kinetic energy (½mv2=0), it still has energy—a new kind of energy called “rest energy”. (b) Binomial Expression: 4. (a) (c) (b) 3. (a) This thought experiment shows that adding an amount of energy E to a system will increase the mass of the system by E/c2. As a general formula we write ΔE = Δmc2. A change in the energy of a system is always accompanied by a corresponding change in its mass. Energy has inertia. CREDITS Authors Executive Producer Scientific Advisor Dr. Richard Epp Manager of Educational Outreach Perimeter Institute for Theoretical Physics Greg Dick Director of Educational Outreach Perimeter Institute for Theoretical Physics Dr. Niayesh Afshordi Associate Faculty Perimeter Institute for Theoretical Physics Dave Fish Senior Physics Teacher Sir John A. Macdonald Secondary School Educational Advisors Joan Crawford Glenview Park Secondary School Cambridge, Ontario Rob Crawford Turner Fenton Secondary School Brampton, Ontario Philip Freeman Richmond Secondary School Richmond, British Columbia Olga Michalopoulos Georgetown District High School Georgetown, Ontario Dennis Mercier Turner Fenton Secondary School Brampton, Ontario Dr. Damian Pope Manager of Educational Outreach Perimeter Institute for Theoretical Physics Melissa Reist Elizabeth Ziegler Public School Waterloo, Ontario David Vrolyk Sir John A. Macdonald Secondary School Waterloo, Ontario Brandon Lloyd PIP Animation Services Director & Design Justin Aresta PIP Animation Services Design Cynthia DeWit Graphic Design Sara Leblanc Illustrations Kevin Donkers Preston High School Cambridge, Ontario Elisa Gatz Sterling High School Sterling, Illinois Animation Production Frank Taylor Title Entertainment Executive Producer Document Production Susan Fish Editor Corporate Sponsor Steve Kelly DVD Authoring