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Archive for December, 2012

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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mcdonalds-McRib-

I actually like this sandwich.

It’s time for McDonald’s to retire its slogan “i’m lovin’ it.”

I’m not sure what I thought of the slogan back in 2003, when it was first unveiled.  I’m sure I ignored it at the time; it seems innocuous enough, if a little too hip.  But as the years go by—and it has been over 9 years now, mind you—the slogan grates on my nerves more and more.  Maybe it’s the uncapitalized “i’m”, which is so reminiscent these days of lazy Facebook posts and lazy text messages (we can’t be bothered to capitalize!)  Maybe it’s the apostrophe after the “n”, as if McDonald’s customers are too busy eating McBeef sandwiches to pronounce a velar nasal.  Maybe it’s the totally slangy, scornful-of-correct grammar attitude that the slogan implies. Maybe it’s all of the above.   I’m sick of it.

It’s not hard to deconstruct the slogan, to get to the intentions of the Mad Men at Heye & Partner, the agency who came up with the slogan.  In fact, through the magic of “fiction”, I recreate their brainstorming session for you here:

BEAN COUNTER #1: We need a new slogan for McDonald’s.

BEAN COUNTER #2: Any ideas?

BEAN COUNTER #1: How about “You will love our food”?

BEAN COUNTER #2: Too formal.  Use a contraction.  They are hipper.

BEAN COUNTER #1: OK, “You’ll love our food”?

BEAN COUNTER #2: No, it sounds like someone is trying to convince you.  Someone is telling you that you’ll like it.  But as they always tell aspiring authors, show, don’t tell.

BEAN COUNTER #1: Meaning?

BEAN COUNTER #2: Meaning that it’s more convincing to see someone enjoy something, rather than have them tell you that you’ll enjoy it.

BEAN COUNTER #1: Well, we could say “I love our food,” something like that.

BEAN COUNTER #2: Too stiff.

BEAN COUNTER #1: “I love it.”

BEAN COUNTER #2: Better.  Still not hip enough.

BEAN COUNTER #1: “I’m loving it.”

BEAN COUNTER #2: Good, good.

BEAN COUNTER #1: You know, of course, that “I’m loving it” is really poor grammar.  It’s only used in the most informal English contexts.

BEAN COUNTER #2: All for the better.  It says McDonald’s is cool, iconoclastic.  It will hook the young people.  It says, “We’re not your parents’ McDonald’s.  We can’t be bothered with rules, with good English.  Yo, have a cheeseburger.”

BEAN COUNTER #1: Genius.

BEAN COUNTER #2: But we can go further.  Make “loving” a contraction.

BEAN COUNTER #1: “Lovin’”?

BEAN COUNTER #2: Yes!

BEAN COUNTER #1: “I’m lovin’ it.”

BEAN COUNTER #2: Perfect.

BEAN COUNTER #1: And you know, we can “dumb it down” further by not even capitalizing the “I”.

BEAN COUNTER #2: Oh, nice.  “i’m lovin it.”  Great.

BEAN COUNTER #1: I’ll write up a prospectus.  How much should we charge McDonald’s?

BEAN COUNTER #2: I’m thinking, $20,000,000?

BEAN COUNTER #1: i’m lovin’ it.

Photo credit: http://www.mcdonalds.com/us/en/food/product_nutrition.sandwiches.292.mcrib-.html

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Tony Montana

Tony Montana

I know, I know.  I said no more political blog posts until 2016…but in the wake of the Sandy Hook mass murder, I’d feel obligated to comment on the gun control “debate” in the United States.

I could cite data: I could arrange the proofs and figures in columns before me, and make an argument.  For example, this Washington Post article has done just that, much better than I ever could, and even resists the temptation to draw (many) conclusions.  The article simply lets the data speak for itself—a rhetorical device seldom employed these days.  But I won’t do this, in part because so many have already done this, and I don’t have much to add.

I could mention how the 2nd Amendment doesn’t say that citizens have a right to bear arms.  It says that citizens have the right to bear arms as part of a well-regulated militia.  But this leads to a debate about language, and what the founders intended.  And if you read the 2nd Amendment in its entirety, you can only conclude that the founders weren’t good writers.  The 2nd Amendment may be the most vague, poorly written sentence in all of jurisprudence.  But I don’t want to discuss this, because again this is well-trodden ground.

I simply want to put the discussion in as stark terms as possible.  As I see it, the gun control debate is just a debate as to wear to draw the line.

That there should be a line somewhere is indisputable.  Suppose someone invented a hand-held device that fired nuclear missiles.  Should we ban such a device?  In a way, it doesn’t really matter, because if such devices were widely available, civilization would collapse within hours.  What about flame throwers?  Rocket launchers?  I don’t think any reasonable person would be opposed to the (civilian) ban of such things.

At the other end of the spectrum, there are (say) knives.  A knife can be used as a weapon; a knife can even be used to commit multiple homicide (case in point: the Simpson case).  Does anyone want to ban knives?  Probably; but let’s say that most reasonable people are in favor of the right to bear knives.  After all, Chopped is such a fun show to watch.

So we have weapons or tools that can be used as weapons, some of which should be banned, some of which should not be.  Put all such weapons on a spectrum with knives at one side and flame throwers on the other.  There has to be a line somewhere.  (This follows, surprisingly, from the Squeeze Theorem of calculus).

Let’s just decide where to put the line.

The rest of this post is personal.  I will suggest a reasonable way to draw the line.  You may disagree.  But wherever you want to draw the line, you must admit that there has to be a line, and I hope you have logical, balanced, well-reasoned criteria for whatever you decide.  If you’re not logical, balanced, or well-reasoned, then honestly I don’t really want to talk to you.

So.  What line-drawing criteria should we use?  I would suggest that people have the right to hunt game animals (there are reasonable arguments against this, I will admit), and the right to reasonably defend themselves.  But I would say that any device that makes it easy to rapidly kill many, many people should be off limits to ordinary citizens.

Machine guns?  Automatic weapons?  Assault rifles?  Ban them all.  No one hunts with them, and no one needs them for protection.  If you need a machine gun to defend your house, then you’re either Tony Montana, or the zombie apocalypse has started.

On the other hand, keep your (non-automatic) pistol.  Keep your shotgun.  Enjoy that hunting trip.  Have fun (trying) to shoot that burglar.  (I could argue that an 80-year-old with a gun is more likely to have the gun pulled from his hand and be pistol-whipped with it, rather than use the gun successfully—but I won’t.)  I have drawn a line, and the line has a logical basis (the number of people that can easily be killed by the device).  You may disagree with my line.

Where would you put it?

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Comix 3

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War is nothing like chess.

War is nothing like chess.

I am a big chess fan.

I can name every chess world champion since Morphy; I could probably name around 17 of the world’s current top 20; I can checkmate a lone king with two bishops and a king; I have a good working knowledge of just about every opening there is.

(And by working knowledge, I don’t just mean I’ve “heard” of the Sicilian defense.  I don’t just mean that I know that 1. e4 c5 2. Nf3 Nc6 3. d4 cxd 4. Nxd4 Nf6 5. Nc3 e5 is the Sveshnikov.  I mean that I am fully aware of the differences between the 9. Nd5 and 9. Bxf6 Sveshnikov, and prefer the former.)

The problem is, I’m just not that good.

Oh, I can beat most casual players…the ones that begin a game by moving a rook pawn (to a4, say) and then move their rooks out vertically (to a3, say).  In USCF rating terms, my rating is around 1800, which (to my own surprise) is about the 85th percentile for tournament players.  So objectively, I am not bad at all.  But I am good enough to be aware of just how much better other players are.  I have a friend Shawn who is a master (here he is drawing a grandmaster).  I am in awe of his tactical strength, and his fine sense of dynamics.  I have beaten him dozens of times in speed chess, but for every game I win, he wins 10.

It has taken me a while to get to the point of this blog post, which is this: I like chess because of its icy logic and its mathematical purity.  For this reason, chess is a horrible metaphor for war, or for life.

Chess is used in books and movies for two basic purposes.  The first is to establish the intelligence of a character.  For example, Lisbeth Salander (in The Girl with the Dragon Tattoo) is an expert at chess.  This was a bad choice on the author’s part: Lisbeth is also an expert hacker and financial genius, has an eidetic memory, and is an incredible detective—why stretch credulity even further?  A lot of great chess players are certainly smart, but the correlation doesn’t go the other way: many smart people are terrible at chess.  Einstein was probably weaker than me.  Oppenheimer was even worse.  Comedian Howard Stern, a player of about my strength, would crush either one.

The other use for chess in books and movies is as metaphor.  In The Seventh Seal, Antonius Block plays a game of chess against Death.  In Harry Potter and the Philosopher’s Stone (the original title; not the dumbed-down American version) someone plays chess with someone else (like I remember?)  In both cases the chess itself is ludicrous.  For example, at one point Death captures Block’s queen; Block says that he “didn’t see that”.  (Really?  Did Block just learn the rules the day before?)  But I don’t want to evaluate the chess in such works per se; rather, I want to see how well chess works as a metaphor.

First, chess as war.  I can’t think of any examples off the top of my head, but there seems to be an assumption that skill at chess somehow equates to skill at war.  But this is ludicrous: in chess, every move is transparent; you can always see what your opponent is doing, and everyone starts on a level playing field.  In terms of game theory, chess is a perfect information game.  I’m no Colonel Dax, but I don’t think war works that way.  There is always a fog of war, and an element of chance, so war is about contingencies, and adaptability, and bluff, and extrapolation.

Second, chess as life.  I have to admit, I don’t really get this metaphor at all.  Is life therefore a game?  A perfect information game?  If chess represents life, does that mean that I struggle throughout my life against an opponent (Satan?  Howard Stern?) who is trying to thwart me at every turn?  And if I play well, but my opponent does too, then am I destined for a draw?  What is a draw, in life?  Is it retiring at 65 to play shuffleboard in Orlando?

As much as I like chess, I think backgammon is a much better metaphor for war or for life.  In backgammon, there is an element of chance, and so the “the best laid schemes o’ mice an’ men” will often go awry.  That is why a good backgammon player will weigh contingencies.  What move leaves me in the best position, based on what dice rolls are possible, and what might happen?  In backgammon, you’re not just playing against an opponent, you’re playing against the fates themselves (in the form of the dice) and this makes the game feel more “real” to me.

People who don’t play backgammon often think that luck is a major part of the game.  This is true, on the level of a single game, but backgammon is played in matches of multiple games, and luck is much less important at that level.  This is because of the doubling cube.  With the doubling cube, a master will almost always defeat a weaker player, in the same way that a Napoleon will almost always win a war against a General Mack, even if an individual battle is lost here or there.

And so, life.  The dice aren’t always going to go your way.  You should plan with that in mind.  Look at your current position, figure out the possible contingencies—the possible ways God might play dice with your universe—and set up your pieces accordingly.  Even if you get gammoned, tomorrow’s another day.

[Note: I subconsciously chose an inept Austrian general to be the foil against Napoleon’s military genius.  But I want to be balanced in my portrayal of Austrians.  So I will remind everyone that Lise Meitner was Austrian, and she was a super-smart physicist.  And strangely, her father was Philipp Meitner, a chess master and part of the immortal draw.]

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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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Anton_Bruckner

Anton Bruckner was Austrian…

So I’m new to this whole blogging thing.  When I started, back around Halloween in 2012, I had no expectations about how many hits I’d get per day, or from what parts of the world.  I don’t even think I was aware of how much of this information a blogger actually has access to.

At WorldPress.com, a blogger can look at a “stats” page and see from what country the IP addresses of hits have come from.  And now, just six weeks in, I have some interesting data to play with.

As I write this, at 2:20 pm on Thursday, Dec. 6, 2012, I have had 1366 hits to my blog.  This does not include hits from my own computer(s) as long as I’m logged in as the owner of the blog; otherwise, the number would be around 600 hits more (I edit my blogs obsessively, even days after I’ve posted them).  As I would have expected, most of the hits are from the United States (87%) but I nice 13% are from other countries.  It’s the specific countries that have hit upon my blog that have piqued my interest.

map

Now, at #3, I have 29 hits from Australia.  This is not surprising; I have a good friend in Australia who follows the blog.  Almost all of these hits are presumably attributable to him (thanks Rick!)

But at #2, with 86 hits, is Austria.  This is strange.  I don’t know anyone from Austria, nor honestly anyone who’s ever been to Austria.  I lived in Spain for four years, and traveled around Europe, but never made it to Austria, unfortunately.

Even stranger is the hit data from today specifically.  I have 36 hits from Austria today.  I can’t even think of a plausible explanation as to why a post about teaching quantum mechanics would suddenly be popular in Austria.

Except, I can think of explanations.  Maybe one of my blog followers is from Austria?  As of 2:34 pm today, there are 14 people who follow this blog; of those, 8 are known to me personally (and don’t live in Austria).  Of the other 6, at least one is obviously in the United States from his profile.  That leaves 5 possible Austrians.  This is the most plausible, if prosaic, explanation.

There may be another explanation.  It’s very possible to follow a blog without “following” it.  I’ve had Nate Silver’s 538 blog bookmarked for 4 years now, without ever having “signed up” to follow it.  I just go to the site and occasionally read what he’s written.  Similarly, maybe someone in Austria stumbled upon ManyWorldsTheory.com, liked it, bookmarked it, and comes back here every so often.  OK, that’s fine; but why 36 hits just today?  For that to be the work of one person, they’d have to visit the blog, then exit out, then visit it again, a total of 36 separate times.  Seems unlikely.

One (speculative) explanation is that there is an Austrian physics professor who reads this blog, liked today’s post, and then had everyone in her/his class read the post today.  But maybe you can think of a better explanation.  Maybe you live in Salzburg and are laughing at my feeble attempts at detective work.  Enlighten me, or not, as you will.  It’s fun either way.

Here’s some more blog statistics trivia, just for fun:

Average number of (unique) hits per day: 36

Record number of hits in a day: 396 (on Nov. 6, 2012, the day before the presidential election; I shared this day’s blog post on Facebook which drove up traffic)

Search engine term that sent the most number of people to my blog: missouri proposition b 2012 (39 times)

Post with most hits: Economics don’t matter (267)

Science post with most hits: Einstein’s “Imagination is more important than knowledge” (36)

Least favorite post (not including today’s and yesterday’s): Let’s ignore Grover Norquist (16)

Country #4 in terms of hits: Spain, with 7.

Country #5 in terms of hits: Canada, with 5.

Part of the fun of having a blog is reaching out to diverse people all over the world, and maybe affecting them in ways that would have been impossible even 20 years ago.  And let’s face it, playing with the demographic data is fascinating.  Keep this in mind if you’ve ever thought about blogging yourself.  Maybe you’ll get some mysterious Austrians following you, too.

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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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’til is not a word

romeo

For I ne’er saw true beauty till this night…

’til is not a word.

Please, use either until, or till.  Some people think that ’til is an abbreviation of until, but this is folk etymology.  “Till” is the older word by far, going back to at least Shakespeare’s day.  For example:

John 21:22 (KJV)     Jesus saith unto him, If I will that he tarry till I come, what to thee?  Follow thou me.

Romeo and Juliet: I, v     For I ne’er saw true beauty till this night.

Also note that “Till death us depart”, from the marriage liturgy in the Book of Common Prayer, dates back to 1549!  It became “Till death us do part” in 1662.

Seeing advertisements for the old Fox show ’til Death always grated on me like fingernails on a blackboard.  Luckily, like most Fox sitcoms, that show departed quite a while ago.

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