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Abstract

The Relativistic Dynamical Inversion technique, a novel tool for finding
analytical solutions to the Dirac equation, is written in explicitly covariant
form. It is then shown how the technique can be used to make a change from
Cartesian to spherical coordinates of a given Dirac spinor. Moreover the most
remarkable feature of the new method, which is the ease of performing
non-trivial change of reference frames, is demonstrated. Such a feature
constitutes a potentially powerful tool for finding novel solutions to the
Dirac equation. Furthermore, a whole family of normalizable analytic solutions
to the Dirac equation is constructed. More specifically, we find exact
solutions for the case of a Dirac electron in the presence of a magnetic field
as well as a novel solution comprising of a combination of a spherically
symmetric electric field and magnetic fields. These solutions shed light on the
possibility of separating the positive and negative energy parts of localized
Dirac spinors in the presence as well as in the absence of magnetic fields. The
presented solutions provide an illustration of the connection between the
geometrical properties of the spinor and spin-orbit coupling for normalizable
spinorial wave functions.