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The Relativistic Dynamical Inversion technique, a novel tool for findinganalytical solutions to the Dirac equation, is written in explicitly covariantform. It is then shown how the technique can be used to make a change fromCartesian to spherical coordinates of a given Dirac spinor. Moreover the mostremarkable feature of the new method, which is the ease of performingnon-trivial change of reference frames, is demonstrated. Such a featureconstitutes a potentially powerful tool for finding novel solutions to theDirac equation. Furthermore, a whole family of normalizable analytic solutionsto the Dirac equation is constructed. More specifically, we find exactsolutions for the case of a Dirac electron in the presence of a magnetic fieldas well as a novel solution comprising of a combination of a sphericallysymmetric electric field and magnetic fields. These solutions shed light on thepossibility of separating the positive and negative energy parts of localizedDirac spinors in the presence as well as in the absence of magnetic fields. Thepresented solutions provide an illustration of the connection between thegeometrical properties of the spinor and spin-orbit coupling for normalizablespinorial wave functions.

Relativistic dynamical inversion in manifestly covariant form