### QM, u Substitution, and Physics Home Rooms

The linked document below contains an integration 'trick' that's very interesting to me, but it arguably shouldn't be.  First, the interesting bit.  The problem shown is from a derivation of the Fourier transform of the ground state of the hydrogen atom from position space into momentum space  The trick here shows how to easily integrate theta dependency using simple 'u substitution'.  There' a second possibly much more interesting bit that I have to check out later, we wind up with sin(kr)/kr which is a prototype for a delta function, which is related to the whole Fourier transform process, but I digress.

While I'm very proud of my u substitution, I arguably shouldn't be.  I'll out myself.  I'm in grad school and I'm not an expert at integrals!  The shame!  After all, I learned the technique of u substitution in freshman calculus.

Here's a little bit of history followed by a suggestion I'd appreciate your thoughts on.  When I took freshman calc, it was a less than stunning experience, (except for the quarter I had Dr. Davis, thanks for all the pizza Dr. Davis!).  Even worse, we didn't really use hard core integrals in physics.  Consequently, two sleep deprived years later when I needed the hard core integral skills, they were gone.  Now I'm in grad school with (hopefully) the full understanding of the horrible, horrible thing I did by not memorizing every integral technique, which brings us to physics home rooms.

Would our undergrad experience be  more cohesive and useful if we had daily updates on what's actually important and why it's important?  Would the equivalent of what I always imagined, (we didn't have them in Ruidoso), a high school home room to be work? A simple half hour meeting every morning where the importance of the previous days classes was revealed.  My first impression was that the class shouldn't be graded, but perhaps grades are exactly what is needed.  In addition to emphasizing that integration by parts would be the favorite trick of each professor in three years, or that the silly top problem would ultimately lead to the precession of an electron in its orbit  what if there were quizzes to make sure the material was sticking?

Those are all the ideas I have for now, what do you think?  Does this affliction of 'lost knowledge' hit enough students to warrant an effort to fix it?  Would a home room kind of activity/environment help?  How could it be done better?

Integral Trick:

Picture of the Day:

 From 3/8/13

### Cool Math Tricks: Deriving the Divergence, (Del or Nabla) into New (Cylindrical) Coordinate Systems

The following is a pretty lengthy procedure, but converting the divergence, (nabla, del) operator between coordinate systems comes up pretty often. While there are tables for converting between common coordinate systems, there seem to be fewer explanations of the procedure for deriving the conversion, so here goes!

What do we actually want?

To convert the Cartesian nabla

to the nabla for another coordinate system, say… cylindrical coordinates.

What we’ll need:

1. The Cartesian Nabla:

2. A set of equations relating the Cartesian coordinates to cylindrical coordinates:

3. A set of equations relating the Cartesian basis vectors to the basis vectors of the new coordinate system:

How to do it:

Use the chain rule for differentiation to convert the derivatives with respect to the Cartesian variables to derivatives with respect to the cylindrical variables.

The chain rule can be used to convert a differential operator in terms of one variable into a series of differential operators in terms of othe…

### The Valentine's Day Magnetic Monopole

There's an assymetry to the form of the two Maxwell's equations shown in picture 1.  While the divergence of the electric field is proportional to the electric charge density at a given point, the divergence of the magnetic field is equal to zero.  This is typically explained in the following way.  While we know that electrons, the fundamental electric charge carriers exist, evidence seems to indicate that magnetic monopoles, the particles that would carry magnetic 'charge', either don't exist, or, the energies required to create them are so high that they are exceedingly rare.  That doesn't stop us from looking for them though!

Keeping with the theme of Fairbank[1] and his academic progeny over the semester break, today's post is about the discovery of a magnetic monopole candidate event by one of the Fairbank's graduate students, Blas Cabrera[2].  Cabrera was utilizing a loop type of magnetic monopole detector.  Its operation is in concept very simpl…

### Kids R Kapable

Just a little note to concerned ‘grownups’ everywhere.  If you look at a kid—and I mean really look—I don’t mean notice a person shorter than you, I mean make eye contact, notice their facial expression and observe their body language—If you look at a kid, don’t assume they need your help unless they’re obviously distressed, or ask for it.  You might think this is difficult call to make.  You might think, not having kids of your own, that you’re unable to make this determination.  You are.  You do in fact, already have the skills even if you’ve never been around kids  It’s a remarkably simple call to make, just use the exact same criteria you would for determining if an adult was in distress.  Because, guess what, kids and adults are in fact the same species of animal and communicate in the same way.  Honest.  If someone—adult or child—doesn’t need your help, feel free to say hello, give a wave, give a smile, but don’t—do not—try to force help on anyone that doesn’t want or need it.

Y…