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Volume 9 (1999), Issue 4, Pages 415-426

Rigorous Derivation of the Pauli Equation With Time-dependent Electromagnetic Field

Univ. Wien, Inst. f. Math., Strudlhofg.4, A-1090 Wien, Austria and Univ. Nice, Lab. J.A. Dieudonné, URM 6621 du CNRS, Parc Valrose, Nice F-06108, France

Received 13 August 1997; Revised 1 December 1998

Copyright © 1999 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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.