Abstract
In this chapter we present an introduction to the Dirac operator describing a spin-1/2 particle in an external field. After a discussion of the free Dirac equation and the problems of relativistic quantum kinematics associated with the occurence of negative-energy solutions, we investigate the problem of relativistic invariance and the implementation of Lorentz transformations. External fields are introduced and classified according to their transformation properties. It turns out that the type of the external field (scalar, electric, magnetic, etc.) determines the spectral properties of Dirac operators in a crucial way. Dirac's theory contains the nonrelativistic Schrodinger equation as a limiting case. The lowest-order relativistic corrections to the Schrodinger eigenvalues are briefly discussed. We proceed to investigate spherically symmetric Dirac operators and the associated angular momentum eigenfunctions (spinor harmonics). Finally, we conclude this chapter with a presentation of the relativistic Coulomb problem.
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