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Posts Tagged ‘science’

One day in the summer of 2011, while mowing the lawn, I saw a strange creature flying through the air.

Actually, “flying” is too generous a term.  The creature was lilting through the air.  Lurching.  It appeared to have ten legs, and was about the size of a silver dollar.  I was puzzled, to say the least, but that lawn wasn’t going to mow itself so I went back to work.

dip_phantom_crane_fly01

The mystery bug…

I live in a rural area in the mountains of North Carolina, only 30 minutes away from the entrance to the Great Smoky Mountains National Park.  There’s a lot of wildlife here: I’ve seen bear, elk, deer, raccoons, opossums, groundhogs, voles, and squirrels; our bird feeders are always full of cardinals, chickadees, towhees, finches, and titmice; and I once came home to a 4-foot black snake inside my house.  As for arthropods, I’m very familiar with flies, no-see-ums (family Ceratopogonidae), moths, wasps, honeybees, crickets, ants, beetles, and spiders of all kinds.  I’ve had close encounters with black widow spiders no less than 3 times in my life.

But this thing?  With 10 legs?  Lurching through the air like a drunken hang glider?  Incomprehensible.

Over the course of that summer, I saw such creatures on numerous occasions.  I gradually came to realize that they were insects, since subsequent sightings showed 6 legs, not 10.  My working hypothesis was that what I saw that first day was a mating pair: two of these things stuck together.  But I still had no idea what the confounded creatures were.

In appearance, the insects were bizarre to say the least.  They were striped, like zebras, and their legs appeared  to have at least 3 joints each, so that the legs took on a zigzag character.  They didn’t appear to use their wings, which I guessed were vestigial; rather, picture a 6-legged starfish up on one end, clawing and grasping its way forward.  As a physicist, it looked very much like the creatures were literally swimming through the air.  And so I resolved, with the help of the internet, to positively identify them.

Rutherford said that “all science is either physics or stamp collecting.”  A lot of people take this to be a disparaging comment about sciences other than physics, but I don’t.  I kind of like stamp collecting.  I like being meticulous, and being detailed.  That’s why I like pastimes such as putting together 1000-piece puzzles.

But my search for the identity of the “mystery bug” took stamp collecting to a whole new level.  It literally took me a month of sleuthing to identify the things.  I tried the obvious first: I googled things like “strange zebra striped bug” and “bug that swims in the air” but had no luck.  I posted a question on an entomology bulletin board.  I looked at websites dedicated to “insects of the Appalachians.”

Finally, I had a breakthrough: I saw one of the bugs hitting up against a window in our house.  For the first time, I could see the creature close up and for more than just a second or two.  I verified that the creature did have six legs; I verified that it did have wings, although they seemed useless.  I realized that my mystery bug was a crane fly.  Here’s a more typical, run-of-the-mill crane fly:

800px-Crane_fly_halteres

A typical crane fly

Regular crane flies are common where I live; kids often mistake them for gigantic mosquitoes (which they are not).

Once I realized that the mystery bug was a type of crane fly, my task was eased enormously.  And eventually I found this assortment of photographs.  Eureka!  I had done it!  They were phantom crane flies, of the family Ptychopteridae.  Specifically, they were the species Bittacomorpha clavipeswhich, according to this Wikipedia article, are “known for the odd habit of spreading out [their] legs while flying, using expanded, trachea-rich tarsi to waft along on air currents.”

It turns out that the phantom crane fly is one of the very, very few creatures on Earth that fly without using their wings.  They are literally swimming, somehow taking advantage of a high Reynolds number (let’s say, 265?) to sludge through the atmosphere without those wings that evolution gave them.  Consequently they look more like seed pods drifting on the wind than they do insects.

What is my point?  I don’t have one.  I just think these bugs are cool, and you should try to find them if you ever visit the Eastern United States.  They hang out in marshy areas in late summer.  Oh, and if you’re a physicist or an entomologist, think about studying these little guys.  The field’s wide open as far as I can tell.  Somebody needs to video the flight of the phantom crane fly, so get on it!  [Note added later: I did find this video which shows the weird flight, are there more?]

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R60118GREEN

I don’t want this shirt for my birthday.

What do the colors pink, gray, and beige have in common?

For one thing, they’re all annoying.  I mean, come on…this isn’t rocket science.

But why are they annoying?  Why is lilac (RGB = [220, 208, 255]) so insipid?  Why does jasmine (RGB = [248, 222, 126]) make one vaguely nauseated?  Why is Crayola fuchsia (RGB = [193, 84, 193]) worse than a bout of the common cold?  (Use this applet to investigate these combinations.)

My thesis is this: that these colors are so annoying because they’re extra spectral colors.  And on some primal, instinctual level, humans don’t like extra spectral colors very much.

In a previous post, I talked about how humans have 3 kinds of cones in their retinas.  Roughly speaking, these cones react most strongly with light in the red, green, and blue parts of the visible spectrum.  Now, as I mentioned, “color” is a word we give to the sensations that we perceive.  Light that has a wavelength of 570 nm, for example, stimulates “red” and “green” cones about equally, and we “see” yellow.  That’s why we say that R+G=Y.  That’s why we also say that 570 nm light is “yellow” light.

Extra spectral colors are colors that don’t correspond to any one single wavelength of light.  They are “real” colors, in the sense that retinal cones get stimulated and our brains perceive something.  However, extra spectral colors don’t appear in any rainbow.  To make an extra spectral color, more than one wavelength of light must hit our retinas.  Our brains then take this data and “create” the color we perceive.

In terms of the RGB color code, extra spectral colors are those in which both R and B (corresponding to the cones at either end of the visible spectrum) are non-zero.  And I don’t know about you, but I have a very heavy preference against extra spectral colors.

Now, admittedly, white (RGB = [255,255,255]) is about as extra spectral as you can get.  Does white annoy me?  Not really; but as a color, it’s also pretty dull.  Does anyone paint their bedroom pure white on purpose?  Does anyone really want an entirely white car?

But the other extra spectral colors I mentioned earlier are a who’s who of mediocrity.  Does anyone older than 16 actually like pink?  Has anyone in the history of the world every uttered the sentence, “Gray is my favorite color”?  And beige—ugh.  Just, ugh.

Standard pink has an RGB code of [255, 192, 203].  Surprisingly, there are combinations that are much, much worse.  Hot pink [255, 105, 180] disturbs me.  Champagne pink [241, 221, 207] bothers me.  Congo pink [248, 131, 121] doesn’t actually make your eyes bleed, but I had to check a mirror to verify this for myself.

Beiges are less offensive, but that’s like saying cauliflower tastes better than broccoli.   Of particular note are “mode beige” [150, 113, 23] which used to be called “drab” but was re-branded in Orwellian fashion, and feldgrau [77, 93, 83] which was used in World War II by the German army, in an apparent attempt to win the war by losing the fashion battle.

This is speculation, but I’ve often wondered if these colors bother me because they are stimulating all three kinds of cones in my retina.  Maybe in some deep part of the reptilian complex portion of my brain, I know (on an intuitive level) that these colors don’t correspond to any particular wavelength.  These colors don’t appear in the rainbow.  You can’t make a laser pointer with one of these colors.  You can’t have a magenta, or a beige, or a gray photon.  And somehow, my aesthetic sense knows this.  So when I see the color “dust storm” [229, 204, 201] my limbic system tells me to wince, and I’m saved from even having to know why.

Anyway, I’d be interested in seeing which color(s) bother you the most.  I’m going to guess the color(s) are extra spectral.

[Note: my book Why Is There Anything? is now available for download on the Kindle!]

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crayon

Very slightly more green than blue, “Tropical Rain Forest” can be thought of as a dark cyan.

My wife called me the other day and asked what my favorite color was.

“Hold on one second,” I said.  “I have it written down.”

She explained that she just needed a color in the most general terms, because she was buying me a case for my new iPhone.  So I said “blue.”  But I was disappointed that I didn’t get to be more specific.

You see, my actual favorite color is (currently) Tropical Rain Forest, formulated by Crayola in 1993.  Its RGB color code is (0, 117, 94).  If you want to read about the color, it’s the first variation on jungle green in the Wikipedia article of the same name.

But what’s an RGB color code?  Anyone familiar with computer graphics will recognize RGB as standing for Red/Green/Blue, which are taken to be the three primaries.  And therein lies a tale: for didn’t we all learn in kindergarten that red, blue, and yellow (not green) were the primary colors?  What’s going on?

Light comes in different wavelengths, or more commonly, combinations of multiple wavelengths.  Color is a purely biological phenomenon having to do with what we perceive with our eyes.  So when a kindergarten teacher says that “mixing red and blue make purple”, there’s really a whole lot of physics and biology that’s being glossed over.

In our retinas, we (generally) have three kinds of cones that react to incoming light.  These cones can detect many wavelengths of light, but each peaks in a different part of the spectrum.  Very simplistically, we can say that one peaks in the “red” part of the visible spectrum, one peaks in the “green” part of the spectrum, and one peaks in the “blue”.

Now, the “red” cones don’t just react to red light—it’s just that they react most strongly to red light.  But light in the “green” part of the spectrum might also stimulate a “red” cone to some degree.  The colors that we see depend on how our brains interpret three signals: how much each of the three kinds of cones is stimulated by incoming wavelengths of light.  For example, if a “red” cone and a “green” cone were stimulated about equally, your brain would interpret this as seeing yellow.  If all three cones were stimulated strongly, you’d “see” white.  (It’s weird to note that different combinations of wavelengths can actually cause the same sensation in your brain: there’s not necessarily a unique combination of wavelengths for any given color perceived.)

Here’s a chart to help you out (note that this is very simplistic and glosses over many issues which I will address later):

Kind(s) of cone stimulated            What you perceive

“Red”                                                                Red

“Green”                                                            Green

“Blue”                                                               Blue

Red & Green                                                    Yellow

Red & Blue                                                       Magenta

Green & Blue                                                   Cyan

Looking at this chart makes the notion of an “additive” primary easy to understand.  We declare red, green, and blue (RGB) to be the additive primary colors.  We can then build (most) other colors by adding these colors together.  This corresponds to multiple wavelengths of light stimulating one or more cones in the retina to varying degrees.  If you want an applet to play around with this kind of additive color mixing, try this.  Input (0, 117, 94) if you want to see Tropical Rain Forest.

colors

Additive color mixing

One caveat: the RGB scheme arbitrarily chooses three exact wavelengths of light to be “the” additive primaries, but this represents a judgment call on our part.  The degree to which different wavelengths of light stimulate the three kinds of cones is messy; the graphs of intensity (of cone response) vs. wavelength are not perfect bell curves, and have bumps and ridges.  Furthermore, it has long been known that if you try to limit yourself to only three “primary” additive colors then you cannot reproduce every possible color that humans can perceive.  We would say that the gamut of possible colors you can make with an RGB scheme does not encompass all possible perceived colors.  (For example, true violet as seen in the rainbow cannot be reproduced with RGB—it can only be approximated.  You can’t see true violet on a computer monitor!)

Now, tell a 6-year-old that Red + Green = Yellow, and they will look at you like you’ve grown a second head.  That’s because most experience we have with “color mixing” doesn’t involve mixing different kinds of light; it involves mixing different kinds of pigments.  And that’s a totally different ball of (crayon?) wax.

Suppose I have a flashlight that shines red light.  I have another flashlight that shines green light.  If I shine both flashlights into your eyes, you will see yellow, as we just discussed.  With two flashlights (two colors), more light has reached your eyes than would have with just one flashlight.

Pigments (such as crayons or paint) work in the opposite way.  “Red” paint is paint that takes white light (a combination of R,G, and B) and subtracts some of the light away, so that only the R reaches your eyes.  Green paint takes RGB light and lets only the G reach your eyes.  In other words, red paint “blocks” G and B, whereas green paint “blocks” R and B.

Can you guess what happens if we mix red and green paint?

The 6-year-old knows you get black.  That’s because two successive blockers have filtered out all the light, and nothing reaches your eyes at all.  And when no cones are stimulated, we perceive that as black.

When a teacher says that the “primary” colors are red, blue, and yellow, they are referring to so-called subtractive primaries.  By mixing those three kinds of pigments, you can make many of the colors we can see.  But not all the colors.  Try mixing red, blue, and yellow to make pink.  It cannot be done.  Like the additive primaries, the gamut of the subtractive primaries is limited.  And, like the choice of RGB as additive primaries, the choice of red, blue, and yellow as the subtractive primaries is arbitrary.  Arbitrary, and inferior.  It turns out that using yellow, magenta, and cyan as the subtractive primaries expands the gamut and increases the number of colors you can make by subtraction.

Why yellow, magenta, and cyan?  Well, those choices make sense if you’ve already picked RGB as your additive primaries.  Consider the chart above.  It’s clear that a paint that looks magenta must be blocking green, since you’re seeing an (additive) combination of red and blue.  Similarly, yellow paint blocks blue, and cyan paint blocks red.  So what happens if we mix, say, yellow and cyan?  Well, the mixture will block blue, and then block red, so what is left is green.  You can try this here.

Anyway, I spent some time at this site trying to determine exactly my favorite color.  I finally chose Tropical Rain Forest, RGB=(0, 117, 94).  I think it’s peaceful and organic.  I also enjoy Tyrian purple, RGB = (102, 2, 60).  Let me know which colors you favor.

Coming soon: some thoughts about the extra-spectral colors!

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maser

CNN has no love for the room-temperature maser.

It’s almost a new year, which of course is the time that everyone writes retrospectives of the year before.  I don’t really want to write a retrospective; I’d rather start a new tradition: criticizing someone else’s retrospective.  So to begin: I draw your attention to CNN’s Top 10 science stories of 2012.

Let me start by emphasizing that this list was not written by a scientist.  It was written by the CNN Health beat reporter.  So maybe I shouldn’t be so critical: maybe I should give that reporter a pass.  But come on, shouldn’t we expect at least half of the “top 10 science stories” to actually be science?  Is that so much to ask for?  Ideally, such a list should be written by several scientists, or at the very least one scientist.  Having the CNN Health reporter compile a list of the top 10 science stories is a kind of near miss—like having Bob Vila comment on the top 10 advances in mechanical engineering, or having Tiger Woods list the best Cricket players in Australia (be sure to mention Michael Clarke, Tiger).

I am a scientist, so I feel qualified to comment on CNN’s list.  Therefore, in the spirit of new year snarkiness, let’s evaluate each “top 10 science story” for import, for scientific value, and for “wow” factor.  And let’s see how the health reporter did.  Remember, that reporter got paid for their work (and I am not getting paid).  Go figure.

1. Curiosity lands, performs science on Mars

OK, this is cool, and maybe some science will be done eventually—I am not aware of any actual results published yet in a peer-reviewed journal.  But the Curiosity landing on Mars is in itself not science; it’s a remarkable feat of technology and engineering.  So it shouldn’t be on the list.

2. Higgs boson — it’s real

I don’t have a problem with this being on the list.  This is big, and important, and exciting to most physicists.  The one thing it is not is surprising: most physicists had faith in the Standard Model, and most expected the Higgs to be found in the 125 GeV/c2 range.  Now the real work begins: determining all the properties of the Higgs, and all the interactions that it might participate in.

3. James Cameron’s deep dive

Seriously?  How is this science?  Avatar-boy goes on a vanity jaunt to the heart of the ocean, and we pay attention why?

4. Felix Baumgartner’s record-breaking jump

This is an even more embarrassing entry than the previous one.  An idiot pushes the envelope, and we call it science?  Does the CNN Health reporter even know what science is?

5. Planet with four suns

Planethunters.org discovered a quadruple star system with a planet in a (somehow) stable orbit.  This is an interesting discovery and an impressive feat by an amateur collective.  Maybe someone will get a journal article out of this someday, but that’s it.  A bigger story is how many extrasolar planets have been discovered so far—854 by Dec. 24, 2012.

6. Nearby star has a planet

So Alpha Centauri B has a planet.  That’s nice.  But didn’t we already cover extrasolar planets in the previous entry in the list?  A good list should vary its entries: if you were listing your top 10 favorite comfort foods, and if #5 were pepperoni pizza, would #6 be sausage pizza?  I didn’t think so.

7. Vesta becomes a ‘protoplanet’

Sigh.  What’s with all the space stuff?  Hey CNN Health reporter: only a small percentage of physicists are astronomers, you know, and there are many other branches of science than just physics.  Did you consider asking a chemist what’s hot in chemistry?  Did you think of calling a geologist, or a neuroscientist, or a paleontologist, or a solid-state physicist?  I didn’t think so.

mayim

Hey CNN? Why didn’t you call Dr. Mayim Bialik?

8. Bye-bye, space shuttles

Again?  More space?  And this isn’t even remotely science.  This is about the retirement of a vehicle.  Good riddance, I say: imagine all the real science that could have been done if the space shuttle money had instead been used to send out hundreds of unmanned probes, to Europa, Titan, Callisto, Ganymede…

9. SpaceX gets to the space station, and back

And still more space?

Dear lord, you’d think from this list that space exploration is the only kind of science that anyone does.  And again: not a science story.  It’s a technology story.  Look up the difference, CNN.

10. Baby’s DNA constructed [sic] before birth

“For the first time, researchers at the University of Washington were able to construct a near-total genome sequence of a fetus, using a blood sample from the mother and saliva from the father.  The study suggested this method could be used to detect thousands of genetic diseases in children while they are still in the fetal stage.”

This is interesting, and may be important, but I have an issue with the list item as the CNN Health reporter wrote it.  The baby’s DNA wasn’t constructed before birth.  The DNA was present before birth, sure; fetuses do have DNA.  The baby’s genome was sequenced before birth, which is a completely different thing.  You’d think a health reporter would know better.

Final Score

So what’s the final tally?  By my count, only of 6 of the items on the list are science, and that’s being very generous.  Of those 6, the fields of science represented are astronomy, particle physics, astronomy, astronomy, astronomy, and biology/medicine.  Nice, balanced list there.  Way to go.

So what stories did deserve to be on the list?  I don’t know.  I am a (mostly) solid-state physicist, and I could mention that researchers successfully used neutrinos to communicate, built a room-temperature maser, and found Majorana fermions.  I’ll stop there, and let chemists, neuroscientists (that means you, Mayim), and geologists add items of their own.

Happy New Year, everyone!

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220px-Clomipramine.svg

The next time someone sneezes, don’t bless them. Take this instead.

[Note: I plagiarized this post.  From myself.  Over two years ago I “started” a blog and gave up a day later.  But that first post was OK, so here it is, with slight modifications.]

It’s the 21st century: science has successfully explained almost every aspect of the physical world (except that missing sock), and new successes are appearing every day. We have computers, cell phones, hand-held GPS devices, the Wii, velcro, and 8-track tape players. And instant pudding.

So why do people still cross their fingers for luck? Why is anyone still tossing spilled salt over their shoulder? Why do athletes still wear their “lucky” shorts? (See this for a list of the saddest athletes you’ve ever heard of.)

It boggles the mind.

Let this post be a rallying call to everyone that still has a shred of intellectual integrity. Let’s all agree to cast out the pernicious demon of superstition from our lives. Let’s all agree that there’s no such thing as your lucky number, that breaking a mirror won’t have any harmful effects (unless you break it with your bare hand), that Friday the 13th is nothing more than a bad movie franchise, and that crossing your fingers has about as much effect on the universe as taking a dump and wishing it were pancakes.

The next time you say “tomorrow’s going to be a good day”, refrain from knocking on wood.  Just don’t do it.  I mean, come on.  It’s silly.  Don’t do it.

Please.

Don’t do it.

And let’s not tolerate such bizarre, 13th century behavior in others: if someone is wearing their lucky Cubs jersey before the big game, call them on it. Say, “Hey Bob, you think wearing that will help? That’s ridiculous and frankly embarrassing. If you want to wear the jersey to support your team, then fine. But please, don’t tell me that wearing that shirt will have any effect on the outcome of the game.” And speaking of the Cubs, let’s all say it together: there is no such thing as a curse. The Cubs just haven’t been all that good in the past 100 years or so.

To bring the world into the 21st century, to promote a scientific and rational mindset, to remain skeptical in the face of irrational and pseudo-scientific claims—to do all these things requires your help. It all starts with you.

Seriously, you.

You can fire off a cannon shot in the superstition culture wars by just not being superstitious yourself. Continue the fight by making fun of people who are superstitious. (Shame: it’s a powerful weapon.) Start peer-pressuring people into being a little more rational. It’ll be good for them. They need to grow up. They can handle it; you know they can.

If not, there’s always clomipramine.

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mcfly

“I am your probability density”

In an earlier post I discussed my philosophy of teaching special relativity.  My main idea was that physics professors should keep the “weird stuff” at bay, and start with non-controversial statements; once students are on board, you can push under the grass and show them the seething Lynchian bugs beneath.

Well, what about quantum mechanics?  Does the same philosophy apply?

My answer is yes, of course.  Don’t start with Schrödinger’s cat.  Don’t mention the Heisenberg uncertainty principle, or wave collapse, or the EPR experiment, or Bell’s theorem, or the double slit experiment, or quantum teleportation, or many worlds, or Einstein’s dice.  Start with the problems of physics, circa 1900, and how those problems were gradually solved.  In working out how physicists were gradually led to quantum mechanics, students will build up the same mental framework for understanding quantum mechanics.  At least, that’s how it works in theory.

Now, my perspective is from the point of view of a professor who teaches only undergraduates.  I only get to teach quantum mechanics once a year: in a course called Modern Physics, which is sort of a survey course of 20th century physics.  (If I were to teach quantum mechanics to graduate students, my approach would be different; I’d probably start with linear algebra and the eigenvalue problem, but that’s a post for another day.)  As it is, my approach is historical, and it seems to work just fine.  I talk about the evidence for quantized matter (i.e. atoms), such as Dalton’s law of multiple proportions, Faraday’s demonstration in 1833 that charge is quantized, Thomson’s experiment, Millikan’s experiment, and so on.  Then I explain the ultraviolet catastrophe, show how Planck was able to “fix” the problem by quantizing energy, and how Einstein “solved” the problematic photoelectric effect with a Planckian argument.  Next is the Compton effect, then the Bohr model and an explanation of the Balmer rule for hydrogen spectra…

We’re not doing quantum mechanics yet.  We’re just setting the stage; teaching the student all the physics that a physicist would know up until, say, 1925.  The big breakthrough from about 1825-1925 is that things are quantized.  Things come in lumps.  Matter is quantized.  Energy is quantized.

The big breakthrough of 1925-1935 is, strangely, the opposite: things are waves.  Matter is waves.  Energy is waves.  Everything is a wave.

So then, quantum mechanics.  You should explain what a wave is (something that is periodic in both space and time, simultaneously).  Here, you will need to teach a little math: partial derivatives, dispersion relations, etc.  And then comes the most important step of all: you will show what happens when two (classical!) wave functions are “averaged”:

ψ1 = cos(k1x – ω1t)

ψ2 = cos(k2x – ω2t)

Ψ(x,t) = (1/2) cos(k1x – ω1t)  + (1/2) cos(k2x – ω2t)

Ψ(x,t) = cos(Δk·x – Δω·t) · cos(k·x – ω·t)

where Δk ≡ (k1 – k2)/2, k ≡ (k1 + k2)/2, etc.

[Here I have skipped some simple algebra.]

This entirely classical result is crucial to understanding quantum mechanics. In words, I would say this: “Real-life waves are usually combinations of waves of different frequencies or wavelengths.  But such ‘combination waves’ can be written simply as the product of two wave functions: one which represents ‘large-scale’ or global oscillations (i.e. cos(Δk·x – Δω·t)) and one which represents ‘small-scale’ or local oscillations (i.e. cos(k·x – ω·t)).

This way of looking at wave functions (remember, we haven’t introduced Schrödinger’s equation yet, nor should we!) makes it much easier to introduce the concept of group velocity vs. phase velocity: group velocity is just the speed of the large-scale wave groups, whereas phase velocity is the speed of an individual wave peak.  They are not necessarily the same.

It is also easy at this point to show that if you combine more and more wave functions, you get something that looks more and more like a wave “packet”.  In the limit as the number of wave functions goes to infinity, the packet becomes localized in space.  And then it’s simple to introduce the classical uncertainty principle: Δk·Δx > ½.  It’s not simple to prove, but it’s simple to make plausible.  And that’s all we want at this point.

We’re still not doing quantum mechanics, but we’re almost there.  Instead, we’ve shown how waves behave, and how uncertainty is inherent in anything with a wave-like nature.  Of course now is the time to strike, while the iron is hot.

What if matter is really made from waves?  What would be the consequences of that?  [Enter de Broglie, stage right]  One immediately gets the Heisenberg relations (really, this is like one line of algebra at the most, starting from the de Broglie relations) and suddenly you’re doing quantum mechanics!  The advantage of this approach is that “uncertainty” seems completely natural, just a consequence of being wave-like.

And whence Schrödinger’s equation?  I make no attempt to “prove” it in any rigorous way in an undergraduate course.  Instead, I just make it imminently plausible, by performing the following trick.  First, introduce complex variables, and how to write wave functions in terms of them.  Next, make it clear that a partial derivative with respect to x or t can be “re-written” in terms of multiplication:

d ψ /dx  →  ik ψ

d ψ /dt  →  –iω ψ

Then “proving” Schrödinger’s equation in a non-rigorous way takes 4 lines of simple algebra:

E = p2/2m

E ψ = (p2/2m)ψ

Now use the de Broglie relations E = ħω and p = ħk…

ħw ψ = (ħ2k 2/2m) ψ

iħ(∂ψ/∂t) = (–ħ2/2m) ∂2ψ/∂x2

There’s time enough for weirdness later.  Right now, armed with the Schrödinger equation, the student will have their hands full doing infinite well problems, learning about superposition, arguing about probability densities.  As George McFly said, “I am your density.”  And as Schrodinger said, probably apocryphally, “Don’t mention my cat till you see the whites of their eyes.”

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sideways

If you don’t go sideways,
you will return to Earth

Why is it that astronauts “float around” in space?

If you were to ask Bill O’Reilly or the Insane Clown Posse, the answer would be that it is a mystery.  If you were to ask the average person on the street, the answer would be that there’s no gravity in space.  Both answers are ridiculous, of course.  It’s not a mystery; we have a very firm working knowledge of the physics of orbits.  And there’s plenty of gravity in space: at 230 miles up, where the International Space Station is, the acceleration due to gravity is about 8.8 m/s2, which is only 10% less than its value at sea level.

So why does the general public still not understand this whole “floating astronaut” thing?

I submit that some of us physics professors are teaching it poorly.  Here’s an explanation from a physics book on my desk:

“All objects in the vicinity of, say, the space station are in free fall with the same acceleration, and so, absent nongravitational forces, they remain at rest relative to each other and their freely falling reference frame.” [Rex and Wolfson, Essential College Physics (2010) p. 215]

I don’t find this very helpful.  And many physics instructors teach “weightlessness” in the same non-helpful way: by hand waving and saying that astronauts are in free fall, and that they are only apparently weightless.  Unfortunately, to the novice this brings up a host of new questions: what’s the difference between apparent weightlessness and actual weightlessness?  More importantly, if you’re in free fall, why don’t you crash into the Earth?

Another book on my desk does a better job:

“Why don’t planets crash into the Sun [if they truly are in free fall]?  They don’t because of their tangential velocities.  What would happen if their tangential velocities were reduced to zero?  The answer is simple enough: their falls would be straight toward the Sun, and they would indeed crash into it.”  [Hewitt, Conceptual Physics, 10th edition (2006), p. 193]

Newton himself also got it right:

“We may therefore suppose the velocity to be increased, that it would describe an arc of 1, 2, 5, 10, 100, 1000 miles before it arrived at the Earth, till at last, exceeding the limits of the Earth, it should pass into space without touching it.”  [Isaac Newton, The System of the World, Section 3, translated by Motte, edited by Cajori (1946)]  [Note Isaac’s use of the word “till”!]

The key idea which we physics professors should emphasize is that astronauts are in free fall, but they don’t hit the Earth because they are moving very, very fast horizontally.  That’s it.  That’s the secret.  They are going so fast that they fall “around the curve of the Earth” so to speak.  I don’t think this horizontal motion is emphasized enough.  I’ll say it again: you need to go sideways to get into orbit.  The next time you’re piloting a spaceship, remember the old adage: that which goes (straight) up will surely come back down (unless you reach escape speed).  So don’t aim for infinity and beyond—aim for the horizon.

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