Archive for January, 2010

S03E13: The Bozeman Reaction

January 18, 2010

My favorite word in physics is Zitterbewegung.  Coming in at close seconds are Bremsstrahlung and Ansatz.  These technical terms, at least for physicists,  mean “rapid oscillation”,  “braking radiation” and “starting point” in German.  (I chose not to even mention Eigenvectorbecause I didn’t want to have to explain it.) However, viewers of tonight’s show saw  Sheldon use the most famous of all German physics words, Gedankenexperiment.    Thanks to Kai of the German BBT fansite “Big Bang Forum” for the audio files.

Like modern physicists, Sid Caesar employed German words.

Today’s modern physics is peppered with German words, a relic of the founding of modern physics in Germany, Austria and Switzerland about 100 years ago.   The German-speaking  heritage of  the giants of late 19th and early 20th century physics is not hard to discern from their names:   Ludwig Boltzmann,  Max Planck,  Albert Einstein, Erwin Shroedinger, Lise Meitner, Otto Stern, Werner Heisenberg, Wolfgang Pauli, James Franck, Max von Laue, …

These masters gave us the concept of a Gedankenexperiment, or a “thought experiment”.   Physics is at its deepest core an experimental science.  Questions that cannot be subject to experimental testing, at least in principle, are outside the realm of physics and are left, at best, to the philosophy department.  Now some experiments are too hard to do as a matter of practice, or  too unsavory.  But even if in principle an experiment could be conducted, the question of its outcome remains squarely in physics.  An experiment you can think about doing, but don’t need to actually perform, is a Gedankenexperiment.  Physicists use Gedankenexperiments for several purposes.   First, they allow teachers to isolate  physical effect perfectly in an explanation for students.  They allow presentation of possible paradoxes and their resolution for pedagogical purposes.   Second, they allow us to see if a set of physical rules are impossible.  If the result of a Gedankenexperiment contradicts the known laws of physics, then at least one of the principles upon which the Gedankenexperiment rests must be flawed.

Sheldon was using Gedankenexperiments for their third purpose, to see if a physics theory has any meaning at all.   If Sheldon has a physics theory, but there is not a single observation that would be different with versus without it, then his theory might as well not even exist.   Physicists are currently grappling with Gedankenexperiments to see if various interpretations of what it means to make a measurement in quantum mechanics (what Sheldon called “the quantum measurement problem”) have any meaning at all.

“I cannot define the real problem, therefore I suspect there’s no real problem, but I’m not sure there’s no real problem.” –Richard Feynman

Gedankenexperiments could someday illustrate if the quantum measurement problem really exists as a physics problem.  In tonight’s episode, Sheldon tells us he had “four out of the five Gedankenexperiments” that he thought would be necessary already written out on his laptop.  Was he close?  After what happened to his laptop, the world may never know.

Several famous Gedankenexperiments of this sort remain:

Shroedinger’s Cat was discussed at the end of Season 1.     In quantum mechanics, particles can exist in multiple observable states at once, in what is called a “superposition”.  To examine the implications further, Schroedinger put a Gedanken-cat (“thought cat”)  in a box and based on whether a certain radioactive atom decayed (or not), a bottle of poison would be opened (or not) and kill the cat (or not).   An experimentalist who has not yet looked in the box would have the treat the cat as a superposition of an alive and dead cat.  Schroedinger originally devised this Gedankenexperiment as a reductio ad absurdem argument.  He intended to show the concept to be ridiculous because, he argued, nothing as large as a cat could exist in a superposition of states.   However, the tables turned on Schroedinger since nothing has ever been done to show that such a state of a cat is impossible.  Instead, this Gedankenexperiment can now be used as an example of how quantum mechanics works.  Even the kitty litter would be both soiled and clean at once.

Einstein’s twin “paradox” was, like the Schroedinger’s cat Gedankenexperiment, developed to show a dramatic consequence of Einstein’s theory of relativity, but is only mislabeled as a paradox.  In his wonderful TV series Cosmos, Carl Sagan illustrates this Gedankenexperiment with  the story of two Italian boys (story starts at 21:50, ends 24:55), Paulo and little brother Vincenzo.  The boys and their friends are passing a nice day in a small Italian town.  Paulo decides to break off and spend some time riding through the Italian countryside at near the speed of light, all the more impressive considering he is riding a Vespa.   Einstein’s theory of relativity predicts that Paulo ages more slowly than his brother whom he left behind.  So when Paulo returns, all Paulo’s friends have grown old and died. Only Vincenzo, now a very old man, is left patiently waiting for Paolo in the piazza. Paulo, however, has experienced only a few minutes of time passing and remains a teenager.

A common misconception is that the “paradox” of this story lies in their ages because, the wrong argument goes,  a younger brother can never be older than the first-born brother.  Actually there’s not problem with that.  The real “paradox” raised was to say that since motion is relative, we cannot specify which brother is “really” moving and who was “really” at rest.  Each brother would see the other moving.  So in that case how could one brother age faster than the other since we can reverse the roles and say Paulo is on a stationary Vespa while Vincenzo was on a moving piazza.  The resolution of the paradox is that at some point Paulo had to turn his Vespa around.  Paulo accelerated but not Vincenzo.   The brothers can say who accelerated and the paradox is resolved, i.e. no paradox.   Sagan himself identifies the wrong paradox in this clip.  I am sure Sagan knew better–but we wouldn’t make that mistake on The Big Bang Theory.

After only a few minutes riding at near light-speed on his Vespa, young Paulo finds his little brother Vincenzo is 90 years old. (starts at 21:50 ends 24:55)

Maxwell’s Demon: is a famous Gedankenexperiment posed for the theory of heat.  In a hot gas, molecules are moving faster on average than in a cool gas.   The theory of heat (a.k.a. “thermodynamics”)  says that if you have two bottles of gas at equal temperature and connect them with a pipe, heat will not flow from one to the other.  However each gas has molecules with a variety of speeds around the average.  Imagine a gatekeeper, Maxwell’s Demon, who could preferentially allow fast molecules into one bottle and the slow ones into the other. One gas would heat up while the other cools without the work required by the theory of heat.  The resolution of this paradox is less obvious than Einstein’s twin paradox and physicists still can argue about it.

Wigner’s Friend is a  Gedankenexperiment proposed by the great physicist Eugene Wigner to explore the roles of consciousness in the quantum measurement problem.  It can be discussed as an added layer to the Schroedinger’s cat experiment.  Suppose Wigner leaves the room with the cat of unknown status  in the box while his friend looks in the box.  Typical theorist, he exits the room leaving the dirty work of cleaning up dead cats to an experimentalist friend.  He asks to be told about the experimental results later.   If it is Wigner’s friend’s consciousness that forces the cat to be 100% alive or 100% dead, then even for Wigner, who is out of the room and does not know the result, suddenly lives in a world where the outcome is 100% determined.  Alas, there is no more problem with Wigner living in a world with a superposition of dead/alive cats and corresponding sad/happy friends than there was with the original cat experiment.  I don’t know of anything fruitful that has been gleaned from this Gedankenexperiment.   In fact, I suspect Schroedinger chose a cat in the first place  to have a complex conscious being in the box. Wigner was no slouch, however, so perhaps I am missing something.  One thing is sure.  The next step will be for the theorist to report on the experimentalist’s hard-won findings to the newspapers.

At the same time as the German-speaking scientists of 100 years ago were developing modern physics, their Yiddish-speaking neighbors were writing comedic theater.   Just as physicists worldwide find funny old German words in our technical lexicon, viewers worldwide hear Yiddish words in the situation comedies of today.   Similarities between comedy and physics abound.


S03 Ep12: The Psychic Vortex

January 11, 2010

Hey Kids!  “Math Doesn’t Suck.”  That’s what we learned in tonight’s episode.

First, Sheldon takes us on a visit in his mind to Flatland.  Flatland is a small book written over 100 years ago about a people living in a world with only two space dimensions. Its inhabitants call their two directions  North/South and East/West.  They are unconcerned with our own three-dimensional world that includes a third direction; Up/Down.  Its a book that nearly every one of my physics friends, as well as those in math, engineering and computer science read as a teenager.   Except me…Flatland was the Moby Dick to my Zelig.

So the episode shamed me into finally reading Flatland.  I figured that by now it wouldn’t have anything new about dimensionality I hadn’t picked up somewhere else.   But I was wrong.  Something I had never considered is that being from a world that is one dimension larger allows you to peer inside everything in a world one dimension smaller.  For example, here is a picture of a Flatlander’s house:

A typical house in Flatland.

Since the Flatlanders (not to be confused with The Flatlanders) have no concept of  covering the “top” and “bottom” of their houses, we can see right inside their rooms and even their cabinets.    More abstractly put, we can see the inside of circles, which they cannot.  Similarly a four-dimensional being could see right inside one of our spheres. Four-dimensional people could see right inside every part of our bodies, organs and even cells as if they were all laid out on paper.

But one page in Flatland really blew me away.   Gravity cannot point “down” for them because they have no concept of down.  It points South in Flatland.   It is never explained, but this could be achieved by tilting Flatland relative to the Earth.  Flatland would be tilted in a dimension they cannot even comprehend.  Also gravity is much stronger in some regions of Flatland than others.   So it seems that Flatland is not so flat at all…it must  not only be tilted, but curved space  like so:

Flatland is really tilted and curved. Flatlanders just don't know it.

The increased “steepness” on one end means gravity has a stronger effect there.  Because Flatlanders have no idea of Up/Down, they have no concept of “steepness” either.  Therefore they have no explanation for the variation of gravity in their land.   In the language of modern mathematics we can say that Flatland is a two dimensional world, but it is “embedded” in a three dimensional one.  It would be decades after this book was published that  Albert Einstein described the possibility of gravity in our own world being related to curvature of space.   Like many science fiction writers,  Flatland’s author, Edwin Abbot, was a step ahead of the physicists.

Little books like this can give a pre-teenager or teenager a taste for mathematics that lasts a lifetime.   Which brings us the the second bit of mathematics in the episode:

I am sure the viewers noticed a bit of mathematics “stunt casting”.   At a university mixer, Raj meets Abby who is played by Danica McKellar.  Yes, she is the McKellar of the Chayes-McKellar-Winn Theorem   (“Percolation and Gibbs States Multiplicity for Ferromagnetic Ashkin-Teller Models on Z²”).   She wrote this paper while still an undergraduate math major (at my university, U.C.LA.)    Basically her theorem tells you how magnets must behave if you put them in a particular configuration.

Danica McKellar proceeded to write a wonderful set of books for pre-teen and teenage girls about mathematics, Math Doesn’t Suck followed by Kiss my Math about algebra and pre-calculus respectively.  A third book is on the way.

Mathematician Danica McKellar guest-starred as Abby in tonight's episode. Her books are terrific for middle school girls.

I bought them to take a look, and will be sending to my niece.  These are terrific books with real-world examples that connect directly to mathematics curricula.  They are fun and quirky.   But make no mistake…The content is serious math.  For example, she explains prime factorization,  in terms of a middle-school crush.  She gives little tricks to conquer the dreaded “word problems”.   She warns against common mistakes: 33 is not 9.  She includes testimonials from her friends,  women who used math to succeed, such as  a petroleum engineer and the finance director of a style magazine.   Other times she just gives friendly advice about growing up such as explaining why young women should not try to dumb themselves down to be popular.  The  first book leads up to and includes algebra. The second gives the reader all she needs for pre-calculus.  Somehow it still all looks fun.  I will see if my niece agrees.

The examples are intended  to resonate with young women.  I was having dinner with two female friends last month who remembered math from middle school.  They pointed out to me that many of the examples in middle-school math books use examples like football, that would  more likely interest boys than girls.  No one would accuse Danica McKellar of doing that in these books.

As we saw from Sheldon’s childhood love of Flatland, a small book can make a big impact on the  life of a pre-teen future mathematician, physicist or engineer.  So consider picking up Danica McKellar’s books for the tweenager you know.  Even if you don’t know any young women, you can click here and Danica McKellar  will send books to the library or school of your choice and send you an autograph.

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