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arXiv:1003.5138

Calculation of Matrix Elements in the Foldy-Wouthuysen Representation

В. П. Незнамов, A.A. Sadovoi, Anton S Ulyanov

Published Mar 25, 2010• 0 citations
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Abstract

The paper compares the methods used to calculate matrix elements of the operator of radail electron coordinates in an arbitrary order in the Foldy-Wouthuysen representation and with the use of the Dirac equation for 1s-states of the hydrogen-like and helium-like ions of transuranic elements. The obtained analytical and numerical results for 1s-states of the hydrogen-like and helium-like ions proves that the wave function reduction requirement is met with transformation to the Foldy-Wouthuysen representation and confirms that matrix elements can be calculated using only one component (either upper, or lower) of the Dirac bispinor wave function.

DOI
10.48550/arxiv.1003.5138

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The paper compares the methods used to calculate matrix elements of the operator of radail electron coordinates in an arbitrary order in the Foldy-Wouthuysen representation and with the use of the Dirac equation for 1s-states of the hydrogen-like and helium-like ions of transuranic elements. The obtained analytical and numerical results for 1s-states of the hydrogen-like and helium-like ions proves that the wave function reduction requirement is met with transformation to the Foldy-Wouthuysen representation and confirms that matrix elements can be calculated using only one component (either upper, or lower) of the Dirac bispinor wave function.
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.
A single-step Eriksen transformation of~1S1/21S_{1/2}1S1/2​,~2P1/22P_{1/2}2P1/2​ and~2P3/22P_{3/2}2P3/2​ states of the relativistic hydrogen atom is performed exactly by expressing each transformed function (TF) as a linear combination of eigenstates of the Dirac Hamiltonian. The TFs are four-component spinors with vanishing two lower components, are calculated numerically and have the same symmetries as the initial states. For all nuclear charges~Z∈[1…92]Z \in [1\ldots 92]Z∈[1…92] a contribution of the initial state to TFs exceeds 86\% of the total probability density. Next large contribution to TFs comes from delocalized states with negative energies close to~−m0c2−Eb-m_0c^2-E_b−m0​c2−Eb​, where~EbE_bEb​ is the binding energy of initial state. Contribution of other states to TFs is less than~0.1%0.1\%0.1% of the total probability density. Other components of TFs are nearly zero which confirms both validity of the Eriksen transformation and accuracy of the numerical calculations. The TFs of~1S1/21S_{1/2}1S1/2​ and~2P1/22P_{1/2}2P1/2​ states are close to~1s1s1s and~2p2p2p states of the nonrelativistic hydrogen atom, respectively, but the TF of~2P3/22P_{3/2}2P3/2​ state differs qualitatively from the~2p2p2p state. Functions calculated with use of a linearized Eriksen transformation, being equivalent to the second order Foldy-Wouthuysen transformation, are compared with corresponding functions obtained by Eriksen transformation. A very good agreement between both results is obtained.
The Foldy-Wouthuysen formalism is presented in a representation-independent manner. It is shown that an algebraic transformation serves to remove the mass term from the Dirac equation for spin-\textonehalf{} particles, thereby providing decoupled two-component equations in a particular (Weyl) representation. Also it is shown that appropriate use of a constant algebraic transformation applied to a Dirac Hamiltonian in the usual (Pauli) representation leads to a nonrelativistic k. p equation used in solid-state theory for which the k-dependent spin-orbit term is absent.