We know that operators representing constants of the motion commute with the Hamiltonian.
The form of the Dirac equation we have been using does not have a clear Hamiltonian.
This is true essentially because of the covariant form we have been using.
For a Hamiltonian formulation, we need to separate the space and time derivatives.
Lets find the Hamiltonian in the Dirac equation.
Its easy to see the
commutes with the Hamiltonian for a free particle so that momentum will be conserved.
The components of orbital angular momentum do not commute with
The components of spin also do not commute with
However, the helicity, or spin along the direction of motion does commute.
From the above commutators
, the components of total angular momentum do commute with
The Dirac equation naturally conserves total angular momentum but not conserve the orbital or spin parts of it.
We will need another conserved quantity for the solution to the Hydrogen atom; something akin to the
we used in the NR solution.
We can show that
It is related to the spin component along the total angular momentum direction.
Lets compute the commutator recalling that
commutes with the total angular momentum.
It is also useful to show that
so that we have a mutually commuting set of operators to define our eigenstates.
This will be zero if
So for the Hydrogen atom,
form a complete set of mutually commuting operators
for a system with four coordinates
and electron spin.