Transformation of bound states of relativistic hydrogenlike atoms into a two-component form

Abstract

A single-step Eriksen transformation of 1S 1/2 , 2P 1/2 and 2P 3/2 states of the relativistic hydrogenlike atom is performed exactly by expressing each transformed function (TF) as a linear combination of eigenstates of the Dirac Hamiltonian. The transformed functions, which 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] a contribution of the initial state to TFs exceeds 86% of the total probability density. Next large contribution to TFs comes from continuum states with negative energies close to −m0c 2 − E b , where E b is the binding energy of initial state. Contribution of other states to TFs is less than 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 1S 1/2 and 2P 1/2 states are close to 1s and 2p states of the nonrelativistic hydrogen-like atom, respectively, but the TF of 2P 3/2 state differs qualitatively from the 2p 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.