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

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 TFs, 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 $Z\in [1...92]$ a contribution of the initial state to TFs exceeds 86% of the total probability density. Next a large contribution to TFs comes from continuum states with negative energies close to $-m_0{c}^2-E_b$, where $E_b$ is the binding energy of the initial state. The contribution of other states to TFs is less than $0.1\%$ of the total probability density. Other components of TFs are nearly 0, which confirms both the validity of the Eriksen transformation and the accuracy of the numerical calculations. The TFs of the $1S_{1/2}$ and $2P_{1/2}$ states are close to the $1s$ and $2p$ states of the nonrelativistic hydrogenlike atom, respectively, but the TF of the $2P_{3/2}$ state differs qualitatively from the $2p$ state. Functions calculated with the use of a linearized Eriksen transformation, being equivalent to the second-order Foldy-Wouthuysen transformation, are compared with corresponding functions obtained by Eriksen transformation. Very good agreement between the two results is obtained.

Figure 1

Schematic matrix $\widehat{\mathcal{S}}=\widehat{\beta }\widehat{\lambda }+\widehat{\lambda }\widehat{\beta }$ in the basis of states ${\psi }_{nS}$ and discretized functions ${\mathrm{\Psi }}_{p_i}^{\pm }$ in Eq. (10). Diagonal matrix elements, which dominate over nondiagonal ones, are shown explicitly. Nondiagonal elements between bound states are indicated by $x$, while nondiagonal elements between bound and continuum states are indicated by $y$. Note the block-diagonal form of matrix $\widehat{\mathcal{S}}$ and vanishing matrix elements between states of positive and states of negative energies.

Figure 2

Matrix elements ${\widehat{\beta }}_{nS,p\pm }$ versus effective wave vector $K=(p/\hslash )\left(\alpha Z\right)$ for several values of $n$. Results for $n=1$ are obtained analytically from Eq. (B8).

Figure 3

Probability densities $|a_n{|}^2$ and $|A_{p_i}^{\pm }{|}^2$ calculated from Eqs. (15, 16, 17) for $Z=92$ versus the electron energy. The probability densities $|a_n{|}^2$ for the three lowest $n$'s are shown explicitly. Shaded areas indicate integrated probability densities $P^{\pm }$ for continuum states of positive and negative energies; see Fig. 4 for $Z=92$.

Figure 4

Contribution of various eigenstates in transformed function ${\chi }_{1S}$ versus nuclear charge $Z$. Solid line: probability density $|a_1{|}^2$ for the $1S_{1/2}$ state. Dot-dashed line: integrated density $P^{-}$ for continuum states of negative energies. Dashed line: integrated density $P^{+}$ for continuum states of positive energies. Dotted line: integrated densities $P_{nS}$ for bound states of $\widehat{\mathcal{H}}$ except the $1S_{1/2}$ state.

Figure 5

Electron wave functions prior to and after Eriksen and FW-like transformations for the $1S_{1/2}$ state of a relativistic hydrogenlike atom with nuclear charge $Z=92$. Negative values: functions $r{f}_{1S},r\widehat{\mathcal{U}}{f}_{1S}$, and $r{\widehat{\mathcal{U}}}^{\mathrm{FW}}{f}_{1S}$. According to the Eriksen theory the function $r\widehat{\mathcal{U}}{f}_{1S}$ should be identically 0. The FW-like transformed function $r{\widehat{\mathcal{U}}}^{\mathrm{FW}}{f}_{1S}$ is also close to 0, but its magnitude is larger than the magnitude of $r\widehat{\mathcal{U}}{f}_{1S}$. Positive values: functions $r{g}_{1S},r\widehat{\mathcal{U}}{g}_{1S},r{\widehat{\mathcal{U}}}^{\mathrm{FW}}{g}_{1S}$, and $r{g}_{1S}^{\mathrm{NR}}$. Note the small difference between exact $r\widehat{\mathcal{U}}{g}_{1S}$ and approximate $r{\widehat{\mathcal{U}}}^{\mathrm{FW}}{g}_{1S}$ functions.

Figure 6

Upper components of the $2P_{1/2}$ and $2P_{3/2}$ states of a relativistic hydrogenlike atom for $Z=92$ prior to and after Eriksen and FW-like transformations. Note the small differences in functions resulting from Eriksen vs FW-like transformations. The functions $rg,r\widehat{\mathcal{U}}g$, and $r{\widehat{\mathcal{U}}}^{FW}g$ for $j=1/2$ are close to the $2p$ function of the nonrelativistic hydrogenlike atom, but for $j=3/2$ they differ qualitatively from the $2p$ function.