### 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…

### Lab Book 2014_07_10 More NaI Characterization

Summary: Much more plunking around with the NaI detector and sources today.  A Pb shield was built to eliminate cosmic ray muons as well as potassium 40 radiation from the concreted building.  The spectra are much cleaner, but still don't have the count rates or distinctive peaks that are expected.
New to the experiment?  Scroll to the bottom to see background and get caught up.
Lab Book Threshold for the QVT is currently set at -1.49 volts.  Remember to divide this by 100 to get the actual threshold voltage. A new spectrum recording the lines of all three sources, Cs 137, Co 60, and Sr 90, was started at approximately 10:55. Took data for about an hour.
Started the Cs 137 only spectrum at about 11:55 AM

Here’s the no-source background from yesterday
In comparison, here’s the 3 source spectrum from this morning.

The three source spectrum shows peak structure not exhibited by the background alone. I forgot to take scope pictures of the Cs137 run. I do however, have the printout, and…

### Unschooling Math Jams: Squaring Numbers in their own Base

Some of the most fun I have working on math with seven year-old No. 1 is discovering new things about math myself.  Last week, we discovered that square of any number in its own base is 100!  Pretty cool!  As usual we figured it out by talking rather than by writing things down, and as usual it was sheer happenstance that we figured it out at all.  Here’s how it went.

I've really been looking forward to working through multiplication ala binary numbers with seven year-old No. 1.  She kind of beat me to the punch though: in the last few weeks she's been learning her multiplication tables in base 10 on her own.  This became apparent when five year-old No. 2 decided he wanted to do some 'schoolwork' a few days back.

"I can sing that song... about the letters? all by myself now!"  2 meant the alphabet song.  His attitude towards academics is the ultimate in not retaining unnecessary facts, not even the name of the song :)

After 2 had worked his way through the so…