D.81 Dirac fine structure Hamiltonian

This note derives the fine structure Hamiltonian of the hydrogen atom. This Hamiltonian fixes up the main relativistic errors in the classical solution of chapter 4.3. The derivation is based on the relativistic Dirac equation from chapter 12.12 and uses nontrivial linear algebra.

According to the Dirac equation, the relativistic Hamiltonian and wave
function take the form

where

The Dirac equation is solvable in closed form, but that solution is
not something you want to contemplate if you can avoid it. And there
is really no need for it, since the Dirac equation is not exact
anyway. To the accuracy it has, it can easily be solved using
perturbation theory in essentially the same way as in derivation
{D.79}. In this case, the small parameter is
1/

So, following derivation {D.79}, take the Hamiltonian apart into
successive powers of

and similarly for the wave function vector:

and the energy:

Substitution into the Hamiltonian eigenvalue problem

The first, order

Plug that into the order

It follows from the first of those that the first order energy change must be zero because

where the summation index

Plug all that in the order

The first of these two equations is the nonrelativistic Hamiltonian eigenvalue problem of chapter 4.3. To see that, note that in the double sum the terms with

where

So the first part of the order

The energy

The sum multiplying

To find out the error in it, the relativistic expansion must be taken
to higher order. To order

Now if

To order

and that determines the approximate relativistic energy correction.

Now recall from derivation {D.79} that if you do a
nonrelativistic expansion of an eigenvalue problem

The first equation was satisfied by the solution for

but that is not right, since

It is maybe not that surprising that a two-dimensional wave function cannot
correctly represent a truly four-dimensional one. But clearly,
whatever is selected for the fine structure Hamiltonian

or using the properties of the Pauli matrices that

The approach will now be to show first that the final two terms are
the spin-orbit interaction in the fine structure Hamiltonian. After
that, the much more tricky first term will be discussed. Renotate
the indices in the last two terms as follows:

Since the relative order of the subscripts in the cycle was maintained in the renotation, the sums still contain the exact same three terms, just in a different order. Take out the common factors;

Now according to the generalized canonical commutator of chapter 4.5.4:

where

Note that the errant eigenvalue

and plugging that into the operator, you get

The term between the square brackets can be recognized as the

Get rid of

That leaves the term

in (D.59). Since

The final term is the claimed Einstein correction in the fine structure Hamiltonian, using

The first term,

is the sole remaining problem. It cannot be transformed into a decent physical operator. The objective is just to get the energy correction right. And to achieve that requires only that the Hamiltonian perturbation coefficients are evaluated correctly at the

for any arbitrary pair of unperturbed hydrogen energy eigenfunctions

Now if you simply swap the order of the factors in

Now, the potential

and the first term integrates away since

Get rid of