Transformation of bound states of relativistic hydrogen-l ike atom into two-component form

TL;DR

This paper performs an exact single-step Eriksen transformation on relativistic hydrogen-like atom states, demonstrating that transformed functions are nearly identical to nonrelativistic states and validating the transformation's accuracy.

Contribution

The paper introduces an exact, single-step Eriksen transformation for relativistic hydrogen-like atom states and compares it with the linearized version, confirming their equivalence and accuracy.

Findings

Transformed functions retain initial state dominance (>86%) across all Z.

Continuum states with negative energies contribute significantly to the transformed functions.

Linearized Eriksen transformation agrees well with the exact transformation.

Abstract

A single-step Eriksen transformation of~$1S_{1/2}$,~$2P_{1/2}$ and~$2P_{3/2}$ states of the relativistic hydrogen-like 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~$Z \in [1\ldots 92]$ 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~$-m_0c^2-E_b$, 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.