Copasetic Flow

While working on QM homework yesterday, I came across a very nice article [1] in the American Journal of Physics about a new perspective on solving for the energies of the bound states in a potential well.  The article points out that waves reflecting back and forth in a potential well have to remain self consistent (picture 1). Another way to say this is that they have to catch their own tails, or wind up back where they started after a round-trip transit of the well.  Using this as a basis, the following equations are quickly derived (picture 2) Equation 4 describes the particles wave function after traversing the well once immediately after the reflection.  Rho is the coefficient of reflectivity, the first exponent is the phase picked up by the waveform after travelling distance L, and the second exp is the wave function itself.  Equation five is the wave function immediately after the second bounce.  Note the extra reflection coefficient and phase g...

A quick physics question.  I'm reading an AJP article on finding the bound energies in an asymmetric rectangular potential well.  The method it proposes is really interesting and I hope to post a summary of the article soon, but for the moment, I'm stuck on the same step that always seems to stall me with Bohr-Sommerfeld quantization exercises. Starting with equation 9(picture 1) taking k approximately equal to the square root of U, we're supposed to arrive at (picture 2) I can get as close as (picture 3) but that's it.  I'd suspect the article of having a typo, but I know that's the programming equivalent of blaming the compiler, so I'm trying to hold off on that. Can anyone see where I've gone wrong? Excerpt from article (picture 4) References: 1.  http://ajp.aapt.org/resource/1/ajpias/v69/i11/p1177_s1 doi http://dx.doi.org/10.1119/1.1387043

Just a few random thoughts on why I wish I'd done a better job of memorizing my trig identities in high school. Here's an approximate high school conversation of mine regarding trigonometric identities with my teacher Mr. Tully, (who is awesome and by the way, who is also a rancher, and also by the way is not the guy pictured to the left(picture 1)... read on!) Me:  "Why do I have to memorize these 40 or so trig identities [2]?" (yes even then, I referenced my utterances" Mr. Tully:  "Because they'll be very important for what you want to do later." (he knew I wanted to be a physicist) Me, (typically not thinking 'later' might be after next week):  "Yeah... I'm not seeing it..." Fast forward a bit to grad school.  Twice in the last week, trig identities were make or break features of homework problems.  I didn't pick up on the necessary identities in electromagnetism, and I'll probably get a B instead of an A ...

In electromagnetism, we came across the following formula describing a potential on a plane (picture 1) In our problem, we were given a plane that looked like the following.  The potential inside the small square is V and the potential everywhere else is on the plane is zero (picture 2) We were to estimate why the potential on the plane didn't go to zero everywhere when the z in the pre-integral numerator  seemed to indicate that it did. Very roughly speaking, the points within the denominator blow up to infinity and conspire to cancel the numerators 0.  There's a more elegant and rigorous way, (courtesy of my professor), to show that everything is all right though.  Here's a sketch of the technique. What we want to do is show that when the denominator goes to infinity while the numerator is zero, things cancel nicely and we still get a finite answer.  When the denominator goes to infinity, the following conditions apply (picture 3) We want...

This week I've been reading about the apparent paradox [1]between the Lorentz force law, (the law that explains how an electric charge is affected by a magnetic field), and special relativity reported last year.  In the article describing the paradox  [2], I came across the phrase 'hidden momentum'.  I hadn't heard this term in relation to electromagnetism before, (neither had some of my professors), so I embarked on a little research project to learn more about it.  As an aside, if anyone would like to comment with a high level overview of the 'hidden momentum' concept, it would be greatly appreciated, (Thanks!) Stored Energy The first thing we need to know is that electric and magnetic fields store energy.  Put very basically, electric charges would rather rearrange so that an object is electrically neutral, and magnetic fields would rather not change.  Changing either a distribution of charges, or a magnetic field requires energy and the energy used...

I'm trying something new today.  I've been doing some background reading on the supposed violation of special relativity reported by M Mansuripur in PRL and pointed out by  +Cliff Harvey  earlier this week.  The embedded document below is a linked diagram of the background articles I've been reading.  An arrow indicates that the article referenced the pointed at article.  My goal in reading about this was to learn about the 'hidden momentum' issue mentioned in Mansuripur's article.  The articles listed here are the earliest and, in my opinion, the most basic articles I could find on the subject.

This is so cool!!! A few days ago I extolled the virtues of the law of cosines, taught in high schools the world over, and claimed that it turned up in all kinds of problems that you run into later in physics.  I gave one example of an electrostatics problem I was working on, but I had a nagging feeling in the back of my head that there were even cooler examples I'd forgotten about.  It turns out that there is a way cooler example of the importance of the law of cosines!  The law of cosines can be used to calculate the Legendre polynomials!!! OK, so what are the Legendre polynomials?  They turn up repeatedly in graduate physics classes.  First of all, they're used to solve electrostatics problems [4].  The most noticeable place I saw them was in quantum mechanics, where they were derived as a special case of spherical harmonics.  Spherical harmonics are used to describe the wavefunction of an electron in a hydrogen atom, and ult...