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

From the above commutators
and
, the **components of total angular momentum do commute** with
.

The Dirac equation naturally

We will need another conserved quantity for the solution to the Hydrogen atom; something akin to the
in
we used in the NR solution.
We can show that
for

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 and .

So for the Hydrogen atom, , , , and form a complete set of mutually commuting operators for a system with four coordinates , , and electron spin.

Jim Branson 2013-04-22