Perturbation theory of relativistic corrections
- Cite this article as:
- Kutzelnigg, W. Z Phys D - Atoms, Molecules and Clusters (1990) 15: 27. doi:10.1007/BF01436910
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Methods for perturbation theory of relativistic corrections for an electron in a Coulomb field are divided into three categories: (1) in terms of 4-component spinors; (2) in terms of the ‘large components’ of the Dirac spinor; (3) involving a Foldy-Wouthuysen type transformation, where one attempts to obtain a two-component spinor different from the ‘large component’. In methods of category 1 (the ‘direct perturbation theory’ of paper I of this series, the related approaches by Rutkowski as well as by Gesteszy, Grosse, and Thaller and a somewhat different one by Moore) the wave function, the energy and the Hamiltonian are analytic inc−2. No divergent terms arise. In methods of category 2 (that of the elemination of the small component as well as a similarity transformation in intermediate normalization) wave function and energy are still analytic inc−2, but the effective Hamiltonian no longer is. Regularized results can be obtained by controlled cancellation of divergent terms. In category 3 both the effective Hamiltonian and the wave function are highly singular and non-analytic inc−1. A controlled cancellation of divergent terms is at least very difficult. These pathologic feature survive in the non-relativistic limit and have hence little to do with relativistic effects. They are related to the fact that forr → 0 the sign of the quantum number κ rather than that of the energy determines which component of the Dirac spinor is large and which is small. In the limitr → 0 andc → ∞ the Foldy-Wouthuysen wave function of a 2p1/2 state is a 1p wave function. Hierarchies of transformations of the Dirac equation and its non-relativistic limit are presented and discussed. Finally the problem of the regularization of effective Hamiltonians on 2-component level ‘for electrons only’ is addressed.