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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 byexpressing each transformed function (TF) as a linear combination ofeigenstates of the Dirac Hamiltonian. The transformed functions, which arefour-component spinors with vanishing two lower components, are calculatednumerically and have the same symmetries as the initial states. For all nuclearcharges~$Z \in [1\ldots 92]$ a contribution of the initial state to TFs exceeds86\% of the total probability density. Next large contribution to TFs comesfrom continuum states with negative energies close to~$-m_0c^2-E_b$,where~$E_b$ is the binding energy of initial state. Contribution of otherstates to TFs is less than~$0.1\%$ of the total probability density. Othercomponents of TFs are nearly zero which confirms both validity of the Eriksentransformation and accuracy of the numerical calculations. The TFsof~$1S_{1/2}$ and~$2P_{1/2}$ states are close to~$1s$ and~$2p$ states of thenonrelativistic hydrogen-like atom, respectively, but the TF of~$2P_{3/2}$state differs qualitatively from the~$2p$ state. Functions calculated with useof a linearized Eriksen transformation, being equivalent to the second orderFoldy-Wouthuysen transformation, are compared with corresponding functionsobtained by Eriksen transformation. A very good agreement between both resultsis obtained.

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