Institut f. Mathematik
Universität Graz
A-8010 Graz, Austria
e-mail: bernd.thaller@kfunigraz.ac.at
Bernd Thaller
In this article we analyze the behavior of the Dirac equation in the nonrelativistic limit and derive the explicit form of the first order relativistic correction. There are two main reasons for this investigation. The first is purely conceptual: It is important to see how the relativistic theory contains the successful nonrelativistic theory as a limiting case. The second reason is a practical one: In some cases it is useful to replace the Dirac theory by the much simpler Schrödinger theory together with some relativistic corrections. The first order corrections can sometimes be calculated explicitly, but it is not always necessary and is often impossible to calculate higher order corrections or the exact solution of the Dirac equation. Besides, for a higher accuracy we expect quantum electrodynamical effects to play a certain role (e.g., the Lamb shift). These effects cannot be described by the Dirac equation alone.
We achieve the nonrelativistic limit by letting c, the relativistic bound for the propagation speed of signals, tend to infinity. Unfortunately, the Dirac operator H(c) itself, even after subtracting the rest mass, makes no sense for c = infinity. The correct way to analyze the parameter dependence of unbounded operators is to look at its resolvent. We shall prove norm-convergence as c tends to infinity of (H(c)-mc^2-z)^{-1} for one (and hence all) z with nonzero imaginary part. The nonrelativistic limit of the Dirac resolvent is the resolvent of a Schrödinger or Pauli operator times a projection to the upper components of the Dirac wavefunction. The Dirac resolvent is even analytic in 1/c.
From the explicit expansion of the resolvent in powers of 1/c we obtain complete information about the behavior of the relativistic energy spectrum in a neighborhood of c = infinity. The necessary background from perturbation theory is quickly reviewed. The Rayleigh-Schrödinger perturbation series for the eigenvalue of the resolvent determines its expansion into powers of 1/c. The corresponding expansion of the eigenvalue E(c) of the Dirac operator is easily obtained. It turns out that the eigenvalues of the Dirac operator are analytic in the parameter 1/c^2. We determine explicit formulas for the relativistic corrections to the nonrelativistic binding energies. For the corrections of order 1/c^2 only the knowledge of the corresponding nonrelativistic solutions (Schrödinger or Pauli equation) is required. The result can be written as a sum of expectation values which have a nice physical interpretation (spin-orbit coupling, etc.). We formulate the theory in terms of abstract Dirac operators in a "supersymmetric'' framework. In this way the calculations can be performed with the least amount of writing and the results are applicable to the widest range of concrete situations.
(This article is a version of the authors book "The Dirac Equation", Springer-Verlag 1992)
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