### Notes on Working with Schrodinger's Equation

I got a quick lesson on the importance of graphing my work this afternoon. After diligently calculating the expectation position of a particle as predicted by the Schrodinger equation:

I came up with the value 1/2λ.

After a little bit of thought, it occurred to me that the Schrodinger equation was symmetric about 0 and that if I was guessing the expectation, or average, value for the position of a particle predicted by the equation, I'd guess zero. So, I graphed the Schrodinger equation for the potential. Just for good measure, I went ahead and graphed the integrand of the expectation integral as well. Sure enough, the Schrodinger equation is symmetric about 0 and the areas under the expectation integrand curve to the left and right of (x=0) cancel out.

Upon checking my work I realized that I was missing a factor of x in my original integration. I multiplied the x back in and everything worked out just like the graph said it should! Not bad for taking a 20 year break between quantum mechanics study sessions. I went ahead and added the integrand for the square of x, used to calculate the standard deviation, to the graph above as well.

NOTE: Had I graphed the integrands first, I could have gotten out of doing the integral for the expectation value altogether. The areas to the left and right of 0 obviously cancel.

NOTE 2: The graphs above were created with Google Docs which is free. No pricey graphing packages required these days! No more excuses!

Have fun with it!

### 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…