Pauli Approximation to the Dirac Equation

Peter Woit

Jeng-Yih Su

With the use of a simple similarity transformation which brings the radial wave equations of the Dirac-Coulomb problem into a form nearly identical to those of the Schr\"odinger and Klein-Gordon equations, we derive simplified solutions to the Dirac-Coulomb equation for both the bound and continuum states following the familiar standard procedure adopted in the derivation of the conventional solutions. We show that to obtain the desired form of the second-order radial equations we can still work with a first-order partial differential equation rather than with the second-order Dirac equation widely employed in the derivation of the simplified solutions, and thus we can avoid the task of reducing the solutions of the second-order equations to those of the original Dirac equation. The transformed Dirac-Coulomb radial equations are so simple that one can apply the WKB method to them in the same way that one applies the WKB approximation to the Schr\"odinger radial equation, without making the further approximations commonly invoked, and the Sommerfeld-Dirac discrete spectrum follows immediately. For small Z\ensuremath{\alpha}, we also present approximate expressions for both the transformed and the original Dirac-Coulomb wave functions, which are valid for all energies and certainly valid in the quasirelativistic approximation which takes account of relativistic effects. The structure of these approximate expressions also suggests one of the methods to obtain relativistic approximate wave functions from the Pauli-spinor wave function for more general central potentials arising in atoms, molecules, and solids.

R. Gerritsma, Gerhard Kirchmair, F. Zähringer +3 more

Norbert J. Mauser

In this work we discuss relativistic corrections for the description of charge carriers in a quantum mechanical framework. The fundamental equation is the Dirac equation which takes into account also the electron′s spin. However, this equation intrinsically also incorporates positrons which play no role in applications in solid state physics. We give a rigorous derivation of the Pauli equation describing electrons in a first order approximation of the Dirac equation in the limit of infinite velocity of light. We deal with time‐dependent electromagnetic potentials where no rigorous results have been given before. Our approach is based on the use of appropriate projection operators for the electron and the positron component of the spinor which are better suited than the widely used simple splitting into ‘upper (large)’ and ‘lower (small) component’. We also systematically derive corrections at second order in 1/ c where we essentially recover the results of the Foldy‐Wouthuysen approach. However, due to the non‐static problem, differences occur in the term which couples the electric field with the spin.

Stefan Keppeler

Helge Kragh