Skip to main content

Posts

Showing posts from February, 2013

Series for ArcTan Using Integrals

I'm getting ready to present my superconductor research[1] at the Texas Academy of Science meeting this weekend, so today's post is just a quick note on an easy way to find a series expansion for the arctangent.

Arctan(x) looked a little daunting to get into a series at first.  The key to get started is to remember what the derivative of arctan is (picture 1)


At least the derivative is something that looks like it could be pretty easily worked into a series.  With a little more thought, it turns out that the anti-derivative, (the integral), of the result above is arctan.  So, the process will be to turn the derivative into a series first, and then take the integral of that series to wind up with the series for arctan.

The result of the derivative shown above is a prime candidate for a geometric series expansion(picture 2)


Now that we have the series, all that remains is to integrate and we get (picture 3):


So (picture 4),


References:
1.  http://arxiv.org/abs/1208.1870

Picture of…

Compton's Water Based Foucault Pendulum

A further review of the article[1] I mentioned yesterday, (Dr. Tatum on the Coriolis effect as a navigational aid for birds), turned up a very interesting reference to Arthur Compton of Compton scattering fame (picture 1)!


It turns out that Compton built an apparatus in 1915 at Wooster University, (now Wooster College) that accurately measured the rotation speed of the Earth allowing Compton to determine the length of a day, his latitude and longitude all without any astronomical references.  It was a sort of Foucault pendulum constructed with a tube of water.

To read all about Compton's apparatus and method in Scientific American, go to [5]:
http://books.google.com/books?id=3AoiAQAAMAAJ&vq=compton%202047&num=13&pg=PA196#v=onepage&q&f=false

or in the blog version of this post, just scroll to the bottom of the post for an embedded version of the article.

Picture 2, below, shows a simplified version of Compton's apparatus sitting directly on the axis of rotati…

Dr. J.B. Tatum Physics Profiles

I came across Dr. Tatum, (shown in the center in the picture to the left), because of the excellent sets of class notes he wrote for his class in electricity and magnetism at the University of Victoria.  At the time the picture was taken, he was a professor of astronomy at Radley College.  The history of Radley says the following about Dr. Tatum:
'And at this point the most distinguished astronomer ever to teach at Radley took up his post as Head of Physics. This was Jeremy B. Tatum, fresh from a senior post in Astrophysics at Victoria University in Canada.[3]"
Education
Dr. Tatum did his PhD research[9] at the University of London Observatory in 1960.  He did his postdoctoral work[10] at the Dominion Astrophysical Observatory before moving on to teach at Radley College[11].

Career
In addition to teaching electricity and magnetism, Dr. Tatum worked in astrophysics and was something of an expert on birds!  It had never occurred to me that a physicist could work in the field of o…

Binomial Expansion II

Use the Calculus Luke!

Yeah for Google+!!!  After posting yesterday's missive about figuring out the origin of the secant squared term, mathematician extraordinare and Google+er +John Baez pointed out to me that the derivative of tangent is secant squared and that I need not have fussed so much with the geometry.  John said: I wouldn't  call the introduction of sec^2(θ) a "substitution".  Introducing θ in the first place counts as a substitution, since you're trying to simplify an integral by replacing some other variable with this one.   But it's just a mathematical fact that the derivative of tanθ is sec^2(θ), so
d tan(θ) = sec^2(θ) dθ
I would use this equation automatically and unthinkingly, but it looks like you're trying to find a geometrical explanation of this equation.  If you're figuring it out for yourself because nobody told you the derivative of tan(θ) is sec^2(θ), that's very laudable!  The standard approach is to express tan in terms of sin and cos, and u…

Of Charged Discs, Trig Substitutions, Birds, and Fireballs

In studying form my EM midterm, I came across a practice problem involving finding the potential along the z axis due to a charged disc centered on the same axis.  After thinking about the problem a bit, I turned to what's becoming one of my favorite online references, Dr. J.B. Tatum's text on electricity and magnetism[2].  Sure enough, there was a solution that could be adapted to the task at hand.  It involved several 'clever' trig substitutions one of which were not immediately clear to me, so I've expanded upon it here.  After the clever trig trick, read on to find out more interesting stuff about Dr. Tatum, an emeritus professor at the University of Victoria[1].

The Basic Problem
The practice problem mentioned above is described by the following diagram from Dr. Tatum's text.  The first and handiest innovation in Dr. Tatum's treatment is to parameterize the problem using the angle marked as theta and the limit of that angle labeled as alpha.


To find the…

Flexagons and Other Things Found While Looking for Schwinger Lippmann

While searching for journal articles on the Schwinger-Lippmann scattering rule last night, I came across the following.

Maybe We Let Quaternions Go A Little Too Easily
http://aapt.scitation.org/doi/abs/10.1119/1.1934944
Relation of quaternions to four dimensional rotations[2]

If you're into fringe physics, McIntosh worked for RIAS who dabbled in shall we say  'gravity control' in the 1950s.
RIAS Inc. of Baltimore[3]
RIAS on antigrav[4]

McIntosh did a significant amount of group theory work and went on to publish a survey of flexagons in 2003[pdf][5].  This brings us to the really cool flexagon stuff!  First, watch the series of videos on hexaflexagons[10]!



Then, go read more about them at the American Mathematical Monthly[6] and Scientific American[7].  They were invented in 1939 and while Feynman didn't come up with the idea, he was involved.

from +Scientific American

Epstein again
The author that pointed out Israel Senitzky also wrote a paper on quantum mechanical scatter…

Chain Rule Notes

Not too surprisingly, I suppose, a homework problem in EM recently asked us to find the Laplacian in a different coordinate system.  The system of choice this time was parabolic cylindrical coordinates.  At first, I was unconcerned, I had my handy example on how to do this using the line element for the coordinate system[1].  As it turned out, that trick was disallowed by the assignment itself.  We were required to use the chain rule.  The parabolic coordinate system was defined as the set of equations to the left, (picture 1).


The chain rule of calculus for some odd reason has always been my arch-nemesis.  This is particularly problematic for studying physics because we use it to exhaustion.  I'd hoped to provide a more coherent presentation, but more midterms are coming up.  For now I'll just capture the solution for finding the Laplacian with a few notes as to what can be learned, and the motivations for various steps.

Step one
Write down the u and v differentials in terms …

Tesla, Meteors and Maps

I started out to write about parabolic coordinates and the chain rule this morning, but this was too much fun to resist. As I'm sure you've all seen over the last few days, a meteor came down over Russia last week.  Apparently the conspiracy theories have already started to crop up[1].  This got me thinking about the last Russian meteor with conspiracy theories, the Tunguska meteor strike[2].

The Tunguska Conspiracy
The story goes like this.  Tesla had realized that he was going to have to tear down Wardenclyffe, his ambitious project on Long Island to broadcast power to the world for free (picture 1).



He had time to make one last demonstration using the apparatus that the world would notice.  He told Admiral Byrd at a cocktail party, "When you get to the North Pole, look for my calling card."  Admiral Byrd saw nothing, and the whole event faded into obscurity, sort of.  Decades later, in the 1970's Russia finally released information on the Tunguska meteor strik…

Spheres, Special Relativity, and Rotations

I've been working though a set of ideas centered around explaining special relativity as rotations in four dimensional space.  +Jonah Miller 's excellent post on Minkowski space this week spurred me into finally capturing some of my nascent ideas here.  I should caution that what follows are the starting points for directions I intend to explore and are hypothetical thought processes, not (until they're mathematically verified or refuted), necessarily the way things actually work.  First though, we'll start out with a little bit of well established background on how special relativity does work.

Time Dilation and Four Velocity
Two different sources led to my ideas.  First, there's the concept of a four velocity[2].  The first place I saw this concept described was in a Brian Greene book[3].  The basic idea is that everything in the universe moves at a constant speed, the speed of light, in a four dimensional space, (hence the term four velocity), that consists of th…

Binomial Expansion

Quantum Mechanics Angular Momentum Flash Cards

I'm still studying away for quantum II.  I thought I'd at least get a sketch of a WKB derivation posted up here today, but that's not the case.  A while back, I wrote a little JavaScript flash card program.  I'm using it to study for my test.  Here are the angular momentum cards.  Apologies to the Google+ folks.  The cards have automated facebook login, and stodgy old login only for score and card tracking.  I still need to get them updated for Google+ login.  If the card stack is useful to you, (it shoudl continue to grow by the way), and you'd like to use them outside of this page, you can either go to:
http://copaseticflows.appspot.com/examhelp/flashqmiij.html

or, you can embed the flash cards in any site you want using the above address and an iframe

The little ads here and there will travel along with the cards when you embed them.  That's my get to APS meetings travel fund, (funding ain't what it used to be).


tan(x)=x The Overlooked Approximation and Other Notes

My quantum professor always knows exactly the right approximations to make, (I suppose it helps to be the person who wrote the homework).  Regarding the recent homework questions involving boundary conditions in shallow square potential wells which are just oozing with tangents of ratios of wave numbers, he pointed out a trig approximation that's often overlooked, (at least by students like myself).  At small angles (picture 1)


the tangent of x is just x.
Yup, small angles aren't just for sine anymore!  The tangent of a small angle is also equal to it's argument.  I had to go through the additional little thought process of (picture 2)


Here's a graph of tan and sin overlayed with x so you can see what's going on. (picture 3)


The Levi-Civita Symbol
Another mathematical tool that has come up repeatedly in the last few weeks is the Levi-Civita symbol.  If you've studied the cross product or the vector curl you've seen it.  Wikipedia defines the symbol like this…

Israel Senitzky and Coherent/Glauber States

Here are a few quick notes on what might become my next history of physics presentation.  Does anyone know more about the history of Dr. Senitzky and coherent states?

I'm still studyinng for my quantum II pre-midterm this week.  Looking for more material on quantum operators, I performed a search for quantum operators in the American Jouranl of Physics which led to an article about the harmonic oscillator wave packet.  Since harmonic oscillators are one of the favorite subjects in both of my quantum classes, I and II, I read on.  The author, Saul Epstein of the University of Nebraska, mentioned that while several authors covered the Gaussian wave packet as an example of a wave function that will not change shape in a harmonic oscillator potential well,  I. R. Senitzky[2] (picture 1) had shown using the Schrodinger formalism that there were actually an infinite number of such non-shape changing packet wave functions.

Switching over to Dr. Senitzky's article, in the Physical Re…

Guinness and the Physics of the Bubbly Flow

This month's lead paper in the American Journal of Physics[8] regards a subject that is dear to most grad students hearts, beer.  A team of scientists Limerick, Ireland report on research they did to determine why the bubbles near the wall of a pint glass of Guinness flow down instead of up.  The preprint[1] of the article can be found on arXiv.  The article inoduces fluid mechanics of the physics of the 'bubbly flow'.  In addition to the bubbly flow, the article introduces the 'anti-pint' (picture 1).  One wonders if the  phrase was coined while the research team[4] performed the experimental portion of their work, measuring the settling time of a Guinness Stout.

If you have a fear of fluid mechanics like I did until today, then this article is an excellent gentle introduction to the subject.  In terms of their relation to your Guinness, the article introduces the Reynolds number[2], the ratio of inertial to viscous forces in a fluid flow, and the Bond number[3],…

Renormalizing Basis Components in Quantum Wave Functions

I'm studying for our pre-midterm practice exam in quantum mechanics II this weekend, hence I've developed a bit of a fascination for all things, (even trivial things), quantum.  On our last homework, we had to decompose a wave function to its eigenfunction components, (sines and cosines in the infinite potential well that we were assigned), and then write down the time evolution for those components.  It was a bit of an exercise to remember how to normalize the components once I had them, so I'm recording the process here to hopefully make it easier to remember next time.

The wave function that had to be decomposed was




As you can see, a trig identity arose to make simple work of the problem once again.  No need to do a Fourier decomposition to sines and cosines when a simple little power-reduction formula[1] is readily available.

That leaves us with two components and two leading factors, or weights.  The mechanical normalization process is to integrate our wave function o…