### Cool Math Tricks: Pulling Linear Factors out of Binomial Sums

While doing my statistical mechanics homework today I arrived at a sum that looked like

Sums like this come up frequently when you're working with random walks, or flipping coins, or counting states of quantum mechanical systems where only two energies are allowed, or in any other number of contrived situations.

It's almost a good looking sum because everything to the right of the factor of m looks like the sum for a binomial expansion:

which simply evaluates to:

It turns out that there's an easy way to get the factor of m out of the sum and get on with your life! First notice that:

so that the sum

can be re-written as:

So, we rather handily got rid of the factor of m. The extra factor of p can be taken outside of the sum since it has nothing to do with the summation index m. Furthermore, the order of summation and differentiation can be interchanged to arrive at:

Now the sum actually is a binomial expansion and after simplifying and performing the derivative that was introduce we arrive at:

I didn't come up with this trick, I'm merely passing it along. I found it in a Statistical Mechanics text that I'm very impressed with:

NOTE: I hope these little pointers are helping folks out, because they're definitely helping me. While preparing this post, I initially put in a summation index of i and found myself wondering what m had to do with anything. It turns out that I didn't have the concept firmly in my head yet, and upon re-investigating I corrected the summation index and got a much better understanding of what's going on. It just goes to show that what my childhood piano teacher said is true:

"The best way to learn something is to prepare yourself to be able to teach it to someone else."

Anonymous said…
Wow that is a nifty trick and I'll keep it in mind when I do stat mech. I've noticed that all the time a complex problem can be simplified by realizing that some expression is really the derivative of some other expression and it makes the problem trivial.
Hamilton said…
Thanks for the pointer! I always thought of derivatives for finding the rate of change. It's very cool that they can be used for simplifications as well!

### 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…

### Lost Phone

We were incredibly lucky to have both been in university settings when our kids were born.  When No. 1 arrived, we were both still grad students.  Not long after No. 2 arrived, (about 10 days to be exact), mom-person defended her dissertation and gained the appellation prependage Dr.

While there are lots of perks attendant to grad school, not the least of them phenomenal health insurance, that’s not the one that’s come to mind for me just now.  The one I’m most grateful for at the moment with respect to our kids was the opportunities for sheer independence.  Most days, we’d meet for lunch on the quad of whatever university we were hanging out at at the time, (physics research requires a bit of travel), to eat lunch.  During those lunches, the kids could crawl, toddle, or jog off into the distance.  There were no roads, and therefore no cars.  And, I realize now with a certain wistful bliss I had no knowledge of at the time, there were also very few people at hand that new what a baby…

### 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…