Alternative Treatment of Magnetic Interactions

The nonrelativistic Hamiltonian can—as we have shown—be derived by elimination of the small component from the relativistic Hamiltonian, including magnetic terms. 

However, the nonrelativistic term in arises from the relativistic second-order perturbation expressions given above like the correspondence between the nonrelativistic and relativistic kinetic energy discussed in section 11.2, there is a projection operator involving the small component (or in this case, a resolvent). If the small component is poorly represented, the cancellation between the sums over positive- and negative-energy states could show considerable error. 

There is of course a corresponding transformation to the wave function, 

The parameter X has been introduced so that an expansion can be made, but it can also be considered as a field strength parameter. 

We now use the expansion of the exponential operators in a commutator series (see section 8.1), and collect terms of various orders to define perturbed operators  

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