The one-electron Hamiltonian including the relativistic spin-orbit coupling term (but excluding other relativistic corrections) adds the term in eq261 to eq45 to give:
or
Since
where .
From eq106 and eq107, we know that . We have also shown in an earlier article (Pauli matrices) that
and
. Consequently,
and
.
Question
Show that ,
,
and
, where
.
Answer
Substituting in
, expanding the expression and using eq104, we have
. Similarly,
. Clearly,
. Finally, expanding the RHS of
, noting that
and
act on different vector spaces and using eq100, eq101, eq166 and eq167,
.
Since ,
,
and
all commute with
, we can select a complete set of eigenfunctions in the form of
for all operators. The eigenvalue equation of
is solved by treating
as a perturbation.
Question
Evaluate using the first order perturbation theory.
Answer
For a one-electron system, . So,
. Using this equation and eq133, eq160 and eq205a,
where .