Spin-$\frac{1}{2}$ Dirac Equation

Abstract

What the Schrödinger equation is to nonrelativistic physics, the Dirac equation is to relativistic physics. We begin this chapter by describing three alternative ways of deriving this spin-$\frac{1}{2}$ wave equation—the more the merrier, in order to develop as much intuition as possible about this fundamental dynamical tool. Next we formulate the Dirac equation in a manifest covariant manner and emphasize the structure of γ-matrix algebra and the positive and negative free-particle solutions. The Dirac equation in the presence of external fields is then generated by minimal replacement, and the resulting electron bound-state energies are obtained for the one-particle Coulomb atom and for a constant external magnetic field. We pay particular attention to the difference between the Dirac atom and the fine-structure level shifts in the Schrödinger atom. Finally, we develop free-particle Dirac equations for spin-$\frac{1}{2}$ massless neutrinos and spin-$\frac{3}{2}$ massive particles.