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.
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:
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
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
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: 10.1098/rspa.1928.0023