### Matrix Operations and Ladder Operators

The short version.  If you're curious about how the x matrix for a harmonic oscillator moves an element within the oscillator state column matrix, this post may hurry things along for you.  I was curious and spent bits and pieces of a few days trying to work it all out.  Here's what I found hastily jotted down so I won't forget and so I can get back to studying for finals.

We're rapidly plowing towards finals week, so this may be rather disjointed, but I wanted to capture a few notes on actually using matrix operations before they escaped.  In Hecht's book when calculating the perturbtion of an harmonic oscillator due to an x ubed, or an x to the fourth potential he makes a rather light reference to using the 'known' matrix elements of the x operator of the harmonic oscillator.  I've seen the references quite frequenlty and decide for once rather than just using the elelments the book wrote down, seeminly from nowhere, that I'd go calculate them.  It was a bit of work!  It basically invovled three days of pondering and pulling hair out that I couldn't really spare.  I won't waste your time or mine with all the mistakes I made since I don't want either of us to remember how to do things incorrectly.

Here's what I came up with after all that.  I'll just run through my notes up to the x squared operator.  First, the x operator, (see first picture), is the sum of the raising and lowering ladder operators.  There's a factor out in front there, but we'll just ignore that since it won't effect the matrix operations.

The raising operator raises the eigenstate index of an oscillator state by one, (to a higher energy), and the lowering operator lowers it one to a lower energy.  If we write them out in matrices, (second picture), they look like this:

The above picture of the matrix is what took the majority of my time by the way.  While I had a number of books that quaintly said to 'simply use the matrix elements', none of them actually bothered to write the elements down.

You can see what's going on just by looking at the elements of the matrix.  I ultimately wanted to work with the x squared operator, (applying the matrix on the left twice), so I placed my single state, n, in the middle of the column matrix to give the raising and lowering operators room to move it up/down twice without falling out of the matrix.  Take a look at the first row.  The only non-zero element is in the second column.  That element will operate on the second element in the state vector column.  The result will be deposited in the first row of the state column matrix.  The matrix multiply itself 'lowered' the element in the second row into the first row after operating on it.  All the matrix elements above the diagonal, (marked in red), will do this, they're the lowering operators.  The matrix elements below the diagonal, (marked in green), do the opposite.  They correspond to raising operators and will move an element in the column state matrix up in energy, (down one row in the column matrix).

And that's all for now.

### 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…