The Dirac Equation

Abstract

The solutions of the Dirac equation for the free electron and for hydrogenic atoms with stationary nuclei are primary building blocks for calculations on more complex many-electron systems. Section 3.1 introduces many useful notions, including plane wave solutions, the bilinear covariant expressions representing physical quantities such as the electron charge-current density, along with energy and spin projection operators. Charge conjugation relates electron and positron solutions. The separation of angular and radial amplitudes of central field Dirac spinors is explained in detail, permitting solution of the radial equations for bound and continuum states. Applications include relativistic Coulomb scattering and relativistic quantum defect theory. We show how to construct partial wave Green’s functions for the free and for hydrogenic Dirac electrons and sum the partial wave expansion for the free electron. Finally, we discuss the nonrelativistic limit and approximate relativistic Hamiltonians.