Larmor diamagnetism and Van Vleck paramagnetism in relativistic quantum theory: The Gordon decomposition approach

We consider a charged Dirac particle bound in a scalar potential perturbed by a classical magnetic field derivable from a vector potential A(r). Using a procedure based on the Gordon decomposition of a field-induced current, we identify diamagnetic and paramagnetic contributions to the second-order perturbation-theory correction to the particle's energy. In contradiction to earlier findings, based on the sum-over-states approach, it is found that the resulting diamagnetic term is Ed⁽²⁾=(q²/2m)〈Ψ⁽⁰⁾|βA²Ψ⁽⁰⁾〉, where Ψ⁽⁰⁾(r) is an unperturbed eigenstate and β is the matrix associated with the rest-energy term in the Dirac Hamiltonian.