Expand the energy term on the left of the equation for the non-relativistic case.

We will be attempting to get the correct Schrödinger equation to order , like the one we used to calculate the fine structure in Hydrogen. Since this energy term we are expanding is multiplied in the equation by , we only need the first two terms in the expansion (order 1 and order ).

The **normalization condition** we derive from the Dirac equation is

We've defined , the 2 component wavefunction we will use, in terms of so that it is properly normalized, at least to order . We can

This equation is correct, but not exactly what we want for the Schrödinger equation. In particular, we want to

We have only kept terms to order . Now we must

Plugging this back into the equation, we can cancel several terms.

Now we can

This **``Schrödinger equation'', derived from the Dirac equation**, agrees well with the one we used to understand
the fine structure of Hydrogen.
The first two terms are the kinetic and potential energy terms for the unperturbed Hydrogen Hamiltonian.
Note that our units now put a
in the denominator here.
(The
will be absorbed into the new formula for
.)
The third term is the relativistic **correction to the kinetic energy**.
The fourth term is the correct **spin-orbit interaction**, including the **Thomas Precession** effect that we did not
take the time to understand when we did the NR fine structure.
The fifth term is the so called **Darwin term** which we said would come from the Dirac equation;
and now it has.

This was an important test of the Dirac equation and it passed with flying colors.
The **Dirac equation naturally incorporates** relativistic corrections, the interaction with electron spin,
and gives an additional correction for s states that is found to be correct experimentally.
When the Dirac equation is used to make a quantum field theory of electrons and photons, Quantum ElectroDynamics,
we can calculate effects to very high order and compare the calculations with experiment, finding good agreement.

Jim Branson 2013-04-22