### Heisenberg, Dirac, and Matrix Quantum Mechanics

I've been reading an excellent quantum mechanics book by K.T. Hecht this week.  The book is very complete, and heavy on worked examples.  The level of complexity is much higher than Griffiths, but there are many, many worked examples.  You don't run into the key bit of information you need being encrypted into a homework problem.  If you've read Griffiths, you know what I mean.

The coolest thing in the book so far is a historical reference to Heisenberg.  Hecht points out in less than a page how Heisenberg arrived at the matrix formulation of quantum mechanics.  Starting out with the model of the Bohr atom's energy levels, he first noted that the energy was dependent on the frequency of the emitted radiation.  This suggested a model similar to a Fourier series.  Heisenberg then noted that there were two indices, n and m, and posited that the Fourier model at the atomic scale would utilize a matrix of coefficients as opposed to a list.  The rest was just (lots and lots of) details.

If you're near a university that was a subscription to the Proceedings of the Royal Society and you'd like to see those details worked out in English rather than the original German, check out this great article from Dirac.  It very clearly spells out what Heisenberg was up to and as a bonus may be the first place the Dirac delta function was introduced.

### 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…