### The Dirac Equation, Spin, and Open Access to the Royal Society Transactions and Proceedings

Hans Bethe on spin implied by angular momentum commutator
Here's a cool thing about spin and the Dirac equation I hadn't seen until I read Hans Bethe's book "Intermediate Quantum Mechanics".  Commuting the Hamiltonian of the Dirac equation with the orbital angular momentum of a particle indicates that total angular momentum isn't conserved[2].

If you're new to, or not in quantum mechanics, the commutator determines how two quantities behave in a multiplication when the order of the multiply is reversed.  In multiplication with real numbers, A times B is the same as B times A, (the commutative property).  Quantum mechanics uses matrices and in matrix multiplication, A times B is not always the same as B times A.  The commutator, [A,B] just subtracts B times A from A times B.  If the two quantities commute the result will be zero.

One last note for the non-QM inclined.  In quantum mechanics, if you take the commutator of an operator matrix with something called the Hamiltonian of the system, you wind up with the time derivative, (a measure of how a quantity changes with time) of the operator.  If something is conserved, angular momentum for example, then it shouldn't change with time, you should always have the same amount of it and the time derivative should be zero.  The best explanation I've seen of this is in Landau and Lifshitz' "Mechanics".

Bethe, like every other QM author I've ever read, just blurts out the commutator above.  I worked through it for practice, and although all the steps aren't here, I'm including my work for folks, (like me), who find themselves looking for an example

We start out with

The term involving beta is a matrix, but it's a diagonal matrix of real numbers.  Because of this, it will commute with everything and can be ignored.  The resulting multiplication and subtraction give

Each of the alphas is a vector of three matrices.  Dot producting the alpha vector with the momentum vector p, we get

Where the terms that are circled are the only ones that will matter, and yes, I'll tell you the two reasons why. Every component of momentum, p-sub-x, p-sub-y, and p-sub-z, commutes with every other component of momentum. Different components of momentum and position commute, so the y coordinate of position commutes with the x coordinate of momentum for example.  The same coordinates of position and momentum don't commute though, (Heisenberg uncertainty principal and whatnot).  The circled terms are the only four terms where a position coordinate is combined with the same coordinate of momentum.  All the other terms will commute and go to zero.  Just saving the four terms mentioned, we wind up with the following last few steps.

The terms from the lines labeled 1 and 3 can be combined as

You'll notice there's been some substantial re-ordering of terms here.  As long as the order of the terms that don't commute is maintained, everything else is fair game, so I slid y and p-sub-y around so I could see the relationship I was looking for more easily.  Then, once I had y and p-sub-y arranged into their own commutator, I substituted the result for [y, p-sub-y]=i times hbar.

Now, we do the same thing for the terms from lines 2 and 4.

Then evaluate the whole expression, (excuse my mistake in the middle), and we're there!

So, as Dr. Bethe pointed out, orbital angular momentum isn't conserved.  But then, he posits that the whole thing might be cleaned up by including spin with the orbital angular momentum in the form of the Pauli matrices.  I'm running out of time, so I'll let Bethe speak for himself[2]:

And there you have it!  The Hamiltonian from the Dirac equation forces you to add spin to the orbital angular momentum of the electron in order to conserve total angular momentum!

Viewing old Royal Society Transactions and Proceedings articles
For what it's worth, I've noticed a lot of the old articles in the Proceedings of the Royal Society and the Transactions of the Royal Society are available open access.  The only difference between the free version and the pay versions is the ability to select and copy text from the pdf.  If you have the volume, issue number, and page number of the article your looking for, then what often works for me is to type in the web address for the proceedings as:

articles in the transactions can be accessed as

Google scholar also seems to pull up a pointer to the free version of these articles if there is one.

References:
1.  Directory of Dirac's publications on Google Scholar

2.  Bethe's Book
H. A. Bethe and R. Jackiw, Intermediate Quantum Mechanics, 2nd ed. (Benjamin/Cummings, New York, 1964)

3.  Dirac on spin
open access:
http://rspa.royalsocietypublishing.org/content/117/778/610
non-open access:
http://dx.doi.org/10.1098%2Frspa.1928.0023
Dirac P.A.M. (1928). The Quantum Theory of the Electron, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 117 (778) 610-624. 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…