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Lord Of The Rings Optics challenge
A great physics problem!
In J. R. R. Tolkien’s The Lord of the Rings (volume 2, p. 32), Legolas the Elf claims to be able to accurately count horsemen and discern their hair color (yellow) 5 leagues away on a bright, sunny day.
“Riders!” cried Aragorn, springing to his feet. “Many riders on swift steeds are coming towards us!”
“Yes,” said Legolas,”there are one hundred and five. Yellow is their hair, and bright are their spears.
Their leader is very tall.”
Aragorn smiled. “Keen are the eyes of the Elves,” he said.
“Nay! The riders are little more than five leagues distant,” said Legolas.”
Make appropriate estimates and argue that Legolas must have very strange-looking eyes, have some means of non-visual perception, or have made a lucky guess. (1 league ~ 3.0 mi.)
On land, the league is most commonly defined as three miles, though the length of a mile could vary from place to place and depending on the era. At sea, a league is three nautical miles (3.452 miles; 5.556 kilometres).
Several solutions are possible, depending on the estimating assumptions
When parallel light waves strike a concave lens the waves striking the lens surface at a right angle goes straight through but light waves striking the surface at other angles diverge. In contrast, light waves striking a convex lens converge at a single point called a focal point. The distance from the long axis of the lens to the focal point is the focal length. Both the cornea and the lens of the eye have convex surfaces and help to focus light rays onto the retina. The cornea provides for most of the refraction but the curvature of the lens can be adjusted to adjust for near and far vision.
I. Here is one solution
By Chad Orzel is an Associate Professor in the Department of Physics and Astronomy at Union College in Schenectady, NY
The limiting factor here is the wave nature of light– light passing through any aperture will interfere with itself, and produce a pattern of bright and dark spots.
So even an infinitesimally small point source of light will appear slightly spread out, and two closely spaced point sources will begin to run into one another.
The usual standard for determining whether two nearby sources can be distinguished from one another is the Rayleigh criterion:
The sine of the angular separation between two objects = 1.22 x ratio of the light wavelength to the diameter of the (circular) aperture, through which the light passes.
To get better resolution, you need either a smaller wavelength or a larger aperture.
Legolas says that the riders are “little more than five leagues distant.”
A league is something like three miles, which would be around 5000 meters, so let’s call it 25,000 meters from Legolas to the Riders.
Visible light has an average wavelength of around 500 nm, which is a little more green than the blond hair of the Riders, but close enough for our purposes.
The sine of a small angle can be approximated by the angle itself.
The angle = the size of the separation between objects divided by the distance from the objects to the viewer.
Putting it all together, Legolas’s pupils would need to be 0.015 m in diameter.
That’s a centimeter and a half, which is reasonable, provided he’s an anime character. I don’t think Tolkien’s Elves are described as having eyes the size of teacups, though.
We made some simplifying assumptions to get that answer, but relaxing them only makes things worse. Putting the Riders farther away, and using yellower light would require Legolas’s eyes to be even bigger. And the details he claims to see are almost certainly on scales smaller than one meter, which would bump things up even more.
Any mathematical objections to these assumptions? Sean Barrett writes:
“The sine of a small angle can be approximated by the angle itself, which in turn is given, for this case, by the size of the separation between objects divided by the distance from the objects to the viewer.”
Technically this is not quite right; the separation divided by the distance is not the angle itself, but rather the tangent of the angle. (SOHCAHTOA: sin = opposite/hypotenuse; tangent = opposite/adjacent.)
Because the cos of a very small angle is very nearly 1, however, the tangent is just as nearly equal the angle as is the sine. But that doesn’t mean you can just skip that step.
And there’s really not much need to even mention the angle; with such a very tiny angle, clearly the hypotenuse and the adjacent side have essentially the same length (the distance to either separated point is also essentially 25K meters), and so you can correctly say that the sine itself is in this case approximated by the separation divided by the distance, and never mention the angle at all.
(You could break out a calculator to be on the safe side, but if you’re going to do that you need to know the actual formulation to compute the angle, not compute it as opposite/adjacent! But, yes, both angle (in radians) and the sine are also 1/25000 to about 10 sig figs.)
II. Another solution
Using the Rayleigh Criterion. In order for two things, x distance apart, to be discernible as separate, at an angular distance θ, to an instrument with a circular aperture with diameter a:
θ > arcsin(1.22 λ/a)
5 leagues is approximately 24000 m.
We can sssume that each horse is ~2 m apart from each other
So arctan (1/12000) ≅ θ.
We can use the small-angle approximation (sin(θ) ≅ tan(θ) ≅ θ when θ is small)
So we get 1/12000 ≅ 1.22 λ/a
Yellow light has wavelengths between 570 and 590 nm, so we’ll use 580.
a ≅ 1.22 * (580E-9 m)* 12000 ≅ .0085 m.
8 mm is about as far as a human pupil will dilate, so for Legolas to have pupils this big in broad daylight must be pretty odd-looking.
Edit: The book is Six Ideas that Shaped Physics: Unit Q, by Thomas Moore
III. Great discussion on the Physics StackExchange
Could Legolas actually see that far? Physics StackExchange discussion
Here, Kyle Oman writes:
The best seeing, achieved from mountaintops in perfect conditions, is about 1 arcsec,
Let’s first substitute the numbers to see what is the required diameter of the pupil according to the simple formula:
θ = 1.220.4 μmD = 2m24 km θ=1.220.4μm D= 2m24km
I’ve substituted the minimal (violet…) wavelength because that color allowed me a better resolution i.e. smaller θθ. The height of the knights is two meters.
Unless I made a mistake, the diameter DD is required to be 0.58 centimeters. That’s completely sensible because the maximally opened human pupil is 4-9 millimeter in diameter.
Just like the video says, the diffraction formula therefore marginally allows to observe not only the presence of the knights – to count them – but marginally their first “internal detailed” properties, perhaps that the pants are darker than the shirt. However, to see whether the leader is 160 cm or 180 cm is clearly impossible because it would require the resolution to be better by another order of magnitude. Just like the video says, it isn’t possible with the visible light and human eyes. One would either need a 10 times greater eye and pupil; or some ultraviolet light with 10 times higher frequency.
It doesn’t help one to make the pupils narrower because the resolution allowed by the diffraction formula would get worse. The significantly more blurrier images are no helpful as additions to the sharpest image. We know that in the real world of humans, too. If someone’s vision is much sharper than the vision of someone else, the second person is pretty much useless in refining the information about some hard-to-see objects.
The atmospheric effects are likely to worsen the resolution relatively to the simple expectation above. Even if we have the cleanest air – it’s not just about the clean air; we need the uniform air with a constant temperature, and so on, and it is never so uniform and static – it still distorts the propagation of light and implies some additional deterioration. All these considerations are of course completely academic for me who could reasonably ponder whether I see people sharply enough from 24 meters to count them. 😉
Even if the atmosphere worsens the resolution by a factor of 5 or so, the knights may still induce the minimal “blurry dots” at the retina, and as long as the distance between knights is greater than the distance from the (worsened) resolution, like 10 meters, one will be able to count them.
In general, the photoreceptor cells are indeed dense enough so that they don’t really worsen the estimated resolution. They’re dense enough so that the eye fully exploits the limits imposed by the diffraction formula, I think. Evolution has probably worked up to the limit because it’s not so hard for Nature to make the retinas dense and Nature would be wasting an opportunity not to give the mammals the sharpest vision they can get.
Concerning the tricks to improve the resolution or to circumvent the diffraction limit, there aren’t almost any. The long-term observations don’t help unless one could observe the location of the dots with the precision better than the distance of the photoreceptor cells. Mammals’ organs just can’t be this static. Image processing using many unavoidably blurry images at fluctuating locations just cannot produce a sharp image.
The trick from the Very Large Array doesn’t work, either. It’s because the Very Large Array only helps for radio (i.e. long) waves so that the individual elements in the array measure the phase of the wave and the information about the relative phase is used to sharpen the information about the source.
The phase of the visible light – unless it’s coming from lasers, and even in that case, it is questionable – is completely uncorrelated in the two eyes because the light is not monochromatic and the distance between the two eyes is vastly greater than the average wavelength.
So the two eyes only have the virtue of doubling the overall intensity; and to give us the 3D stereo vision. The latter is clearly irrelevant at the distance of 24 kilometers, too. The angle at which the two eyes are looking to see the 24 km distant object are measurably different from the parallel directions. But once the muscles adapt into this slightly non-parallel angles, what the two eyes see from the 24 km distance is indistinguishable.
V. This is also analyzed in “How Far Can Legolas See?” by minutephysics (Henry Reich)
MCAS Open Response questions
Content Objectives: SWBAT construct answers to open-response questions on the physics MCAS.
2015, High School Intro Physics: sample open response question
2011 sample open response questions
2012 sample open response questions
2013 sample open response questions
2014 sample open response questions
2015 sample open response questions
2016 sample open response questions
- Learning Standards:
- For answering open-response questions – ELA Core Curriculum
- CCRA.R.1 – Read closely to determine what the text says explicitly and to make logical inferences from it; cite
- specific textual evidence when writing or speaking to support conclusions drawn from the text.
- For answering problems involving equations: Massachusetts Curriculum Framework for Mathematics
- Functions: Connections to Expressions, Equations, Modeling, and Coordinates.
- Determining an output value for a particular input involves evaluating an expression; finding inputs that yield a
- given output involves solving an equation.
Emmy Noether
Amalie Emmy Noether (1882 – 1935) was a German mathematician known for her landmark contributions to abstract algebra and theoretical physics. She was described by Pavel Alexandrov, Albert Einstein, Hermann Weyl, and Norbert Wiener as the most important woman in the history of mathematics. In physics, Noether’s theorem explains the connection between symmetry and conservation laws.
Our related articles
https://kaiserscience.wordpress.com/physics/mathematics/symmetry/
External articles
In her short life, mathematician Emmy Noether changed the face of physics. ScienceNews.Org
http://www.thephysicsmill.com/2014/03/09/international-womens-day-spotlight-emmy-noether/
Data needs an interpretation to have meaning
Lesson: “Data has no meaning without a physical interpretation”
Content objectives:
1. SWBAT to identify trends in data (apparent linear plots; apparent linear data plus noise; and simple harmonic motion.)
Thesis: raw data doesn’t tells us anything physical phenomenon. We always first need to know what physical phenomenon we are analyzing, before we can interpret it.
Tier III vocabulary: Simple harmonic motion
Launch: Students are given graph paper, and data. Plot the given data points, and connect the dots in a way that they think is logical.
Question: Justify why you connected the dots in that way. Why not in some other way?
Direct Instruction/guided practice
Teacher instructions
Create a sine wave. I do so here using Desmos – desmos.com/calculator/xxmkiptej7
I modified this function to be Y = 4•sin(1.5x)
Don’t tell the students yet. This sine wave is a position versus time graph of any object in the real world undergoing simple harmonic motion.
Y-axis can be interpreted as height; X-axis is time.
Let’s get some data points from this function. Draw a straight line across it, from upper right to lower left.
The line will intersect the sine wave at many points.
Overlay some semi-transparent graph paper on top of this, and plot these points. Or, as I have done here, do it on an app. In this example we have seven data points.
Give the students the coordinates for these points but do not show them the graph! Just give them the data. Ask them to interpret it, plot it, and hypothesize about what the data could mean.
Tag six more points from the sine wave, that are not on the original straight line.
Here I chose some data points that we could sample from actual motion, if we happened to be sampling at just the right time interval.
Again, give students these coordinates without showing them the graph. Ask them to interpret it, plot it, and hypothesize about what the data could mean.
If one were to plot only these points then they would appear as a straight line.
A naïve reading of the raw data would lead one (mistakenly) to believe that we are studying some kind of linear phenomenon.
If one were to plot only these points then they would appear as a straight line.
A naïve reading of the raw data would lead one (mistakenly) to believe that we are studying some kind of linear phenomenon.
Very few students will quickly see that these points fit a sine curve. They will have all sorts of answers
When we are done with all of these examples, then we can show them the original sine curve; show them each of these graphs, and how all the different data came from the same data set/phenomenon.
Part A: Justify your choice: What real world motion would produce such a function? Think-Pair-Share
After the discussion, the teacher reveals what produces such data: SHM, Simple Harmonic Motion:
Summative question, tying this all together:
Why couldn’t most students plot the data correctly, even after the final data points were added?
Answer: Unless you know what kind of phenomenon you are studying, you have no idea whether the data is supposed to be linear, harmonic, exponential, etc. Data – by itself – has no meaning without a physical interpretation.
Closure: Query multiple students: Where do you experience SHM in your own life?
Possible answers: Moving back-and-forth on a swing, pendulum of a clock, automobile suspension system
Something more to think about:
Image from https://m.xkcd.com/2048/
Learning standards
A Framework for K-12 Science Education: Practices, Crosscutting Concepts, and Core Ideas (2012)
Dimension 1: Scientific and Engineering Practices: Practice 4: Analyzing and Interpreting Data.
“Once collected, data must be presented in a form that can reveal any patterns and relationships and that allows results to be communicated to others. Because raw data as such have little meaning, a major practice of scientists is to organize and interpret data through tabulating, graphing, or statistical analysis. Such analysis can bring out the meaning of data—and their relevance—so that they may be used as evidence.”
Emergent phenomenon
Thomas T. Thomas writes:
From our perspective at the human scale, a tabletop is a flat plane.
but at the atomic level, the flat surface disappears into a lumpy swarm of molecules.
Aficionados of fractal imagery will understand this perfectly: any natural feature like the slope of a hill or shore of a coast can be broken down into smaller and smaller curves and angles, endlessly subject to refinement. In fractal geometry, which is driven by simple equations, the large curves mirror the small curves ad infinitum.
The emergent property is not an illusion… The flatness of the tabletop is just as real—and more useful for setting out silverware and plates—than the churning atoms that actually compose it. The hill and its slope are just as real—and more useful for climbing—than the myriad tiny angles and curves, the surfaces of the grains of sand and bits of rock, that underlie the slope.
Emergent property works on greater scales, too. From space the Earth presents as a nearly perfect sphere, a blue-white marble decorated with flashes of green and brown, but still quite smooth. That spherical shape only becomes apparent from a great distance. Viewed from the surface, it’s easy enough for the eye to see a flat plane bounded by the horizon and to focus on hills and valleys as objects of great stature which, from a distance of millions of miles, do not even register as wrinkles.
Emergent properties come into play only when the action of thousands, millions, or billions of separate and distinct elements are perceived and treated as a single entity. “Forest” is an emergent property of thousands of individual trees. The concept of emergent properties can be extremely useful to describe some of the situations and events that we wrestle with daily.
The Human Condition: Emergent Properties, Thomas T. Thomas, 8/11/2013
also
NOVA ScienceNow Emergence, PBS
Examples
Conway’s game of life
https://en.wikipedia.org/wiki/Conway%27s_Game_of_Life
http://emergentuniverse.wikia.com/wiki/Conway%27s_Game_of_Life
http://www.scholarpedia.org/article/Game_of_Life
http://www.conwaylife.com/
BOIDS: Birds flocking
Boids Background and Update by Craig Reynolds
http://www.red3d.com/cwr/behave.html
http://www.emergentmind.com/boids
Coding: 3 Simple Rules of Flocking Behaviors: Alignment, Cohesion, and Separation
https://en.wikipedia.org/wiki/Flocking_(behavior)
Classical physics
Classical physics is an emergent property of quantum mechanics
TBA
External links
Online Interactive Science Museum about Emergence
How Complex Wholes Emerge From Simple Parts Quanta magazine
Learning Standards
2016 Massachusetts Science and Technology/Engineering Curriculum Framework
Appendix VIII Value of Crosscutting Concepts and Nature of Science in Curricula
In grades 9–12, students can observe patterns in systems at different scales and cite patterns as empirical evidence for causality in supporting their explanations of phenomena. They recognize that classifications or explanations used at one scale may not be useful or need revision using a different scale, thus requiring improved investigations and experiments. They use mathematical representations to identify certain patterns and analyze patterns of performance in order to re-engineer and improve a designed system.
Next Gen Science Standards HS-PS2 Motion and Stability
Crosscutting Concepts: Different patterns may be observed at each of the scales at which a system is studied and can provide evidence for causality in explanations of phenomena. (HS-PS2-4)
A Framework for K-12 Science Education
Scale, proportion, and quantity. In considering phenomena, it is critical to recognize what is relevant at different measures of size, time, and energy and to recognize how changes in scale, proportion, or quantity affect a system’s structure or performance…. The understanding of relative magnitude is only a starting point. As noted in Benchmarks for Science Literacy, “The large idea is that the way in which things work may change with scale. Different aspects of nature change at different rates with changes in scale, and so the relationships among them change, too.” Appropriate understanding of scale relationships is critical as well to engineering—no structure could be conceived, much less constructed, without the engineer’s precise sense of scale.
Dimension 2, Crosscutting Concepts, A Framework for K-12 Science Education: Practices, Crosscutting Concepts, and Core Ideas (2012)
http://necsi.edu/guide/concepts/emergence.html
Ray Tracing
This lesson is from Rick Matthews, Professor of Physics, Wake Forest University.
Lesson 1, convex lens
The object is far from the lens.
Lesson 2, convex lens
The object is near the lens.
The rules for concave lenses, are similar:
A horizontal ray is refracted outward, as if emanating from the near focal point.
A ray that strikes the middle of the lens continues in a straight line.
A ray coming from the object, far from the far focal point, will leave the lens horizontal.
Lesson 3, concave lens.
Note that object placement has little effect on the nature of the image.
The rays diverge.
__________________________
In every case:
if the rays leaving the lens actually intersect then the image is real.
If the rays leaving the lens diverge then someone looking back through the lens
would see a virtual image:
Your mind would extrapolate where you think the image should be,
even though one isn’t really there, as shown below with the dotted lines.
image from Giancoli Physics, 6th edition
http://users.wfu.edu/matthews/courses/tutorials/RayTrace/RayTracing.html
Sonar and ultrasound
Sonar (SOund Navigation And Ranging)
The use of sound to navigate, communicate with, or detect objects – on or under the surface of the water – such as another vessel.
Active sonar uses a sound transmitter and a receiver.
Active sonar creates a pulse of sound, often called a “ping”, and then listens for reflections (echo) of the pulse.
Several animals developed sonar through evolution by natural selection.
Example: whales
Example: dolphins
Example: bats
Aquaman uses sonar! (Superfriends, 1970s, ABC)
How do we know what the ocean floor looks like?
Figure 6.8: A ship sends out sound waves to create a picture of the seafloor below it.
The echo sounder has many beams of sound. It creates a three dimensional map of the seafloor beneath the ship.
Early echo sounders had only a single beam and only created a line of depth measurements.
Boston Harbor
Data from USGS Construction of Digital Bathymetry for the Gulf of Maine
What would it look like if we could use sonar to map out the entire Atlantic ocean?
National Oceanic and Atmospheric Administration (NOAA), ETOPO1 Global Relief Model, http://www.virginiaplaces.org/geology/rocksdui4.html
Ultrasound
Medical ultrasound – a diagnostic imaging technique using ultrasound.
Used to see internal body structures such as tendons, muscles, joints, vessels and internal organs.
The practice of examining pregnant women using ultrasound is called obstetric ultrasound.
Ultrasound is sound waves with frequencies which are higher than those audible to humans (>20,000 Hz).
Ultrasonic images also known as sonograms are made by sending pulses of ultrasound into tissue using a probe.
The sound echoes off the tissue; with different tissues reflecting varying degrees of sound. These echoes are recorded and displayed as an image to the operator.
Medical ultrasound (Wikipedia)
“Amniocentesis is a prenatal test in which a small amount of amniotic fluid is removed from the sac surrounding the fetus for testing. The sample of amniotic fluid (less than one ounce) is removed through a fine needle inserted into the uterus through the abdomen, under ultrasound guidance.”
“The fluid is then sent to a laboratory for analysis. Different tests can be performed on a sample of amniotic fluid, depending on the genetic risk and indication for the test.”
.
Blue sky
This is the outline for a future lesson on Rayleigh Scattering: Why the sky is blue
– Rayleigh scattering occurs when light is scattered off many very small particles.
– Mie scattering occurs when light is scattered off of many larger particles.
text
Addressing misconceptions
Question: Particles in the air cause shorter wavelengths (blue-ish0 to scatter more than the longer wavelengths (reddish.) This causes us to see the sky as being blue. So why does the sunrise (or sunset) and sun look red/orange?
Answer: “When you look at the sky and see blue you’re seeing blue light being scattered towards your eye.”
“When you look at the sun and it looks red or orange that’s because the blue light is being scattered away from your eye – leaving the remaining light to enter your eye.”
“The blue light is being scattered in all directions by Raleigh scattering. The colors you see depend on what direction you’re looking.”
Reference Physicsforums.com How-does-rayleigh-scattering-work
External resources
Why the sky is blue, by Chuck Weidman, Atmo 170A1 Sect. 3 Fall 2013
http://math.ucr.edu/home/baez/physics/General/BlueSky/blue_sky.html
https://www.itp.uni-hannover.de/~zawischa/ITP/scattering.html
http://hyperphysics.phy-astr.gsu.edu/hbase/atmos/blusky.html
http://www.thephysicsmill.com/2014/03/23/sky-blue-lord-rayleigh-sir-raman-scattering/
Brownian motion app galileoandeinstein Brownian motion app
Lesson EarthRef.org Digital Archive ematm.lesson3.scattering.pptx
EM in the Atmosphere: Reflection, Absorption, and Scattering Lesson Plan
Powerpoint for the lesson plan
Learning standards
SAT subject test in Physics: Waves and optics
• General wave properties, such as wave speed, frequency, wavelength, superposition, standing wave diffraction, and Doppler effect
• Reflection and refraction, such as Snell’s law and changes in wavelength and speed
• Ray optics, such as image formation using pinholes, mirrors, and lenses
• Physical optics, such as single-slit diffraction, double-slit interference, polarization, and color
AP Learning Objectives
IV.A.2.b: Students should understand the inverse-square law, so they can calculate the intensity of waves at a given distance from a source of specified power and compare the intensities at different distances from the source.
IV.B.2.b: Know the names associated with electromagnetic radiation and be able to arrange in order of increasing wavelength the following: visible light of various colors, ultraviolet light, infrared light, radio waves, x-rays, and gamma rays.
L.2: Observe and measure real phenomena: Students should be able to make relevant observations, and be able to take measurements with a variety of instruments (cannot be assessed via paper-and-pencil examinations).
L.3: Analyze data: Students should understand how to analyze data, so they can:
– a) Display data in graphical or tabular form.
– b) Fit lines and curves to data points in graphs.
L.5: Communicate results: Students should understand how to summarize and communicate results, so they can:
– a) Draw inferences and conclusions from experimental data.
– b) Suggest ways to improve experiment.
– c) Propose questions for further study
Rainbows
Rainbows are produced by electromagnetic radiation – visible light – reflecting in marvelous ways from the dispersion of light.
Let’s start with the basics:
A prism separates white light into many colors
How? Each wavelength of light refracts by a different amount
The result is dispersion – each wavelength is bent by a different amount
The physics of rainbow formation
Rainbows: At Atmospheric optics
http://hyperphysics.phy-astr.gsu.edu/hbase/atmos/rbowpri.html
Rebecca McDowell How rainbows form
The shape of a rainbow
A discussion of this comic is here Explain XKCD. 1944: The End of the Rainbow
If one considers the path that light takes to form a rainbow, then it forms a two-cone structure, where the Sun (the vertex of the outer cone) emits light rays that move towards the Earth (forming the faces of the outer cone),
Then the rays reflect off water droplets located at just the right angle (the circular base) to reach our eyes (the vertex of the inner cone).
Thus, such a rainbow structure can be said to have “ends”, represented by the vertices of the two cones: one at the eye of the viewer, and another at the light source (usually the sun).
from the webcomic XKCD.
Do rainbows have reflections?
It certainly seems like rainbows can have reflections.
Consider this great photo by Terje O. Nordvik, September ’04 near Sandessjøen, Norway.
http://www.atoptics.co.uk/rainbows/bowim6.htm
But rainbows aren’t real objects – and so they literally can’t have reflections!
So what are we seeing here? See Rainbow reflections: Rainbows are not Vampires
Learning Standards
2016 Massachusetts Science and Technology/Engineering Curriculum Framework
HS-PS4-3. Evaluate the claims, evidence, and reasoning behind the idea that electromagnetic radiation can be described by either a wave model or a particle model, and that for some situations involving resonance, interference, diffraction, refraction, or the photoelectric effect, one model is more useful than the other.
SAT subject test in Physics: Waves and optics
• General wave properties, such as wave speed, frequency, wavelength, superposition, standing wave diffraction, and Doppler effect
• Reflection and refraction, such as Snell’s law and changes in wavelength and speed
• Ray optics, such as image formation using pinholes, mirrors, and lenses
• Physical optics, such as single-slit diffraction, double-slit interference, polarization, and color.
Schrödinger’s cat
Schrödinger’s cat is a thought experiment, sometimes described as a paradox, devised by Austrian physicist Erwin Schrödinger in 1935.
It illustrates what he saw as the problem of the Copenhagen interpretation of quantum mechanics when applied to everyday objects.
Here is how the Schrödinger’s cat thought experiment works:
Acat, a flask of poison, and a radioactive source are placed in a sealed box.
If an internal monitor detects radioactivity (i.e., a single atom decaying), the flask is shattered, releasing the poison, which kills the cat.
The Copenhagen interpretation of quantum mechanics implies that after a while, the cat is simultaneously alive and dead.
Yet, when one looks in the box, one sees the cat either alive or dead, not both alive and dead.
This poses the question of when exactly quantum superposition ends and reality collapses into one possibility or the other.
The Copenhagen interpretation implies that the cat remains both alive and dead – until the state is observed.
Schrödinger did not wish to promote the idea of dead-and-alive cats as a serious possibility.
On the contrary, he intended the example to illustrate the absurdity of the existing view of quantum mechanics
Since Schrödinger’s time, other interpretations of quantum mechanics have been proposed that give different answers to the questions posed by Schrödinger’s cat of how long superpositions last and when (or whether) they collapse.
This introduction has been adapted from “Schrödinger’s cat.” Wikipedia, The Free Encyclopedia, 5 Feb. 2017.
Many-worlds interpretation and consistent histories
In 1957, Hugh Everett formulated the many-worlds interpretation of quantum mechanics, which does not single out observation as a special process.
In the many-worlds interpretation, both alive and dead states of the cat persist after the box is opened, but are decoherent from each other.
In other words, when the box is opened, the observer and the possibly-dead cat split into an observer looking at a box with a dead cat, and an observer looking at a box with a live cat.
But since the dead and alive states are decoherent, there is no effective communication or interaction between them. We have created parallel universes!
Decoherence interpretation
When opening the box, the observer becomes entangled with the cat.
Therefore “observer states” corresponding to the cat’s being alive and dead are formed; each observer state is entangled or linked with the cat so that the “observation of the cat’s state” and the “cat’s state” correspond with each other.
Quantum decoherence ensures that the different outcomes have no interaction with each other. The same mechanism of quantum decoherence is also important for the interpretation in terms of consistent histories.
Only the “dead cat” or the “alive cat” can be a part of a consistent history in this interpretation.
External resources
https://www.newscientist.com/article/2097199-seven-ways-to-skin-schrodingers-cat/
Learning Standards
SAT Subject Test: Physics
Quantum phenomena, such as photons and photoelectric effect
Atomic, such as the Rutherford and Bohr models, atomic energy levels, and atomic spectra
Nuclear and particle physics, such as radioactivity, nuclear reactions, and fundamental particles
Relativity, such as time dilation, length contraction, and mass-energy equivalence
AP Physics Curriculum Framework
Essential Knowledge 1.D.1: Objects classically thought of as particles can exhibit properties of waves.
a. This wavelike behavior of particles has been observed, e.g., in a double-slit experiment using elementary particles.
b. The classical models of objects do not describe their wave nature. These models break down when observing objects in small dimensions.
Learning Objective 1.D.1.1:
The student is able to explain why classical mechanics cannot describe all properties of objects by articulating the reasons that classical mechanics must be refined and an alternative explanation developed when classical particles display wave properties.
Essential Knowledge 1.D.2: Certain phenomena classically thought of as waves can exhibit properties of particles.
a. The classical models of waves do not describe the nature of a photon.
b. Momentum and energy of a photon can be related to its frequency and wavelength.
Content Connection: This essential knowledge does not produce a specific learning objective but serves as a foundation for other learning objectives in the course.
A Framework for K-12 Science Education: Practices, Crosscutting Concepts, and Core Ideas (2012)
Electromagnetic radiation can be modeled as a wave of changing electric and magnetic fields or as particles called photons. The wave model is useful for explaining many features of electromagnetic radiation, and the particle model explains other features. Quantum theory relates the two models…. Knowledge of quantum physics enabled the development of semiconductors, computer chips, and lasers, all of which are now essential components of modern imaging, communications, and information technologies
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