### The Week in Preview, March 25th 2013

A quick review of what I'll be looking at over the course of the upcoming week.  This is as much to get my own thoughts in order as anything else.

Quantum Mechanics:
I'll be working on still more uncertainty and harmonic oscillator problems in QM this week.  What a surprise right :)  Specifically, this week, I'll be calculating matrix elements for both position and momentum squared using both the Hermite polynomial recursion operators and the ladder operators.  These are covered in chapters 5 and 10 in Merzbacher.  I was playing around with one of the recursion relations (picture 1)

for Hermitian polynomials earlier in the year and wound up with the following kind of interesting table.  You can see the n level of the wave function moved out of the way by the successive application of the recursion formula which amounts to the successive application of the x operator, or a sum of the raising and lowering operators (picture 2).

I know a lot of students, (including me), that think Jackson's EM book is merely a very thinly veiled math methods book.  While looking through old American Journal of Physics back issues, I found a review for another book by Jackson was apparently more appropriately titled than his 'EM' book.  This one is about the math required for quantum mechanics, "Mathematics for Quantum Mechanics: An Introductory Survey of Operators, Eigenvalues, and Linear Vector Spaces (Dover Books on Mathematics)".  At only \$6.50 on Amazon, at least the price is right, but as usual, I'll be using the library's copy for free.

Hirsch Based Superconductor Research:
There are a few things I have to look into regarding the upcoming experimental research I'm doing.  In the 1930's it was documented that lead, (Pb), had an intermediate state between its superconducting and normal states[1][2].  This is important information to have since what we'd ideally like to do is 'instantly' quench the Pb superconducting sample using a magnetic field.

The design of the superconducting magnet that will provide the quenching field will also get done this week, so look for a few graphs and calculations.  I got to take a look at the 1/18000 of an inch diameter superconducting wire that can carry 55 amps last week.  It was cool!

Casimir Research:
I have a presentation due on this on Friday at our local student research week exposition.  In addition to that, I'm working on calculating various aspects of the asymptotic Bessel function approximation for the fields contained in the annular wedge problem we're working on.

And now, off to work.

References:
1.  Intermediate state in lead superconductors
http://dx.doi.org/10.1098%2Frspa.1936.0129
Shoenberg D. (1936). The Magnetization Curves of a Supraconducting Sphere and Ring, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 155 (886) 712-726. DOI:

2.  More on the intermediate state using alternating current magnetic fields
http://dx.doi.org/10.1017%2FS0305004100077690
Shoenberg D. (1937). Superconductors in Alternating Magnetic Fields, Mathematical Proceedings of the Cambridge Philosophical Society, 33 (04) 559. DOI:

### 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…

### 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…