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 .