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College Physics

Discussion on non-relativistic approximation of Dirac equation by means of Foldy-Wouthuysen transformation

Yun‐De Li

Published Jan 1, 2013• 0 citations
OpenAlex

Abstract

The non-relativistic approximation of Dirac equation is discussed by means of method of Foldy-Wouthuysen transformation.The recurrence fomiulas of Hamiltonian and wave function are given out.

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The paper describes conditions for transformation from the Dirac representation to the Foldy-Wouthuysen representation. The necessary condition is the block-diagonal transformation of Hamiltonian relative to the upper and lower components of the wave function. The sufficient condition is the wave function transformation law described by relation (6). It has been demonstrated that the unitary transformations offered by the authors of the papers [14], [15], [16], [17] do not satisfy the sufficiency condition (6) and, hence, they are not the Foldy-Wouthuysen transformations. In applications, the matrix elements of any operator in the FW representation can be calculated, according to (6), using the normalized two-component wave functions in the Dirac representation known for the given problem.
The Dirac Hamiltonian for a particle in a nonexplicitly time-dependent field is converted to an even Dirac matrix by means of a single canonical transformation. When the interaction term is an odd Dirac matrix, the transformed Hamiltonian is expressed in a very simple form. An exact transformation is also found for two-particle wave equations of Breit's type. The transformed Hamiltonian is then a uU\mathrm{uU}uU-separating matrix, in Chraplyvy's sense.In the nonrelativistic limit expansions in powers of 1m\frac{1}{m}m1​ or 1c\frac{1}{c}c1​ are made. The approximate wave equations are in agreement with previous transformation results.
The spectrum of the Dirac Hamiltonian H D has a positive and a negative part, the first corresponds to the energy levels of the electron, the second to minus the energy levels of the positron. The Foldy-Wouthuysen transformation is a tool to obtain a decomposition of H D in a direct sum H D =H e- (+)-H e+ , where H e- is the Hamiltonian of the electron and H e+ is that of the positron. Unfortunately, this decomposition has been obtained in a closed form only when the external field is purely magnetic, while, in the presence of an electric field, only a perturbative expansion is available. The authors give a path integral representation of the semigroup phi (x, t)=exp(-(t/)H e- ) phi 0 (x) in an external electromagnetic field. Their formula is the relativistic version, for Dirac particles, of the well known Feynman-Kac-Ito formula.