### Fock, SO(4), and the Azimuthal Quantum Number

Our quantum professor worked out the Kepler-Coulomb equation earlier this week and ended with a cryptic remark about SO(4). A little bit of digging led to a presentation about the details from the University of Minnesota math department[pdf]. It turns out that Fock did a Fourier transform of Schrodinger's equation into momentum space and saw that the resulting integral kernel looked like the projection of a sphere onto a 4 dimensional hyperplane. He then saw that Schrodinger's equation is simply the Laplace equation in this space.

I hope in the future to write something more complete on the whole matter, but with finals coming up, I'll just add a few notes and references.

Fock's original article on all this is written in German.  However, the author of the above mentioned presentation, Jonas Karlsson was kind enough to point me to Linearity, Symmetry, and Prediction in the Hydrogen Atom (Undergraduate Texts in Mathematics).  Where the entire article is translated into English.  A book that I mentioned earlier in the week, Hecht's Quantum Mechanics (Graduate Texts in Contemporary Physics), also briefly discusses Fock's findings. One of these volumes, (apologies... I'm rather scattered and can't find the reference in any of them at the moment),  points out that the azimuthal quantum number while once considered 'mysterious', can be more simply seen as the projection of a fourth dimensional variable into our three dimensional space.  Scroll down for the Google Books excerpt from the Linearity book mentioned above.

tunafish said…
Great post. Anything to demystify the sometimes cryptic group symmetries in the world is appreciated!
Hamilton Carter said…
Thanks Tunafish! Hopefully more demystification will be coming in the new semester.

### 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…