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Transformation of localized states of relativistic hydrogen atom into two-component form

Tomasz M. Rusin

Published Apr 8, 2016• 0 citations
OpenAlex

Abstract

A single-step Eriksen transformation of~$1S_{1/2}$,~$2P_{1/2}$ and~$2P_{3/2}$ 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~$Z \in [1\ldots 92]$ a contribution of the initial state to TFs exceeds 86\% of the total probability density. Next large contribution to TFs comes from delocalized 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 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.

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A single-step Eriksen transformation of~1S1/21S_{1/2}1S1/2​,2P1/22P_{1/2}2P1/2​ and2P3/22P_{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.
Abstract The solution of a unitarily transformed Dirac equation for the hydrogenic electron in zero magnetic field is investigated here. The momentum‐space representation is adopted as a natural recourse. The spinor part of the transformed wavefunction in momentum space can be easily prescribed for a central potential. Hence, for the Coulomb potential, a pair of equations is obtained for the radial components in momentum space. It is shown that starting from these radial equations, one can recover the equations previously derived by Rubinowicz, Lévy, and Lombardi for the problem of the Dirac hydrogen atom in momentum space. This establishes equivalence among different approaches based on the momentum representation, including the current treatment. The recovery of the equations due to Rubinowicz permits the exact eigenvalues to be written down and exact expressions to be derived for the radial components of the transformed wavefunction in momentum space. A new approach is adopted to carry out a reduction to the nonrelativistic regime and the nonrelativistic limit. At first the transformed momentum‐space equation for the hydrogen atom is rewritten in terms of the hyperspherical coordinates. The zeroth‐order solutions of the new equation are recovered in the limit c → ∞ where c is the speed of light. These are manifestly separable into positive‐ and negative‐energy forms. For positive energy, these solutions have nonvanishing upper components that are two‐component spinors. The latter exactly correspond to the single‐component, nonrelativistic, momentum‐space solutions derived by Fock. It is shown that when the upper component is corrected through first order in v2/c2 but the separability is still maintained for the transformed wavefunction, one retrieves the Pauli equation in momentum space. It is also shown that for a hydrogen atom placed in a uniform magnetic field, the nonvanishing momentum‐space matrix elements representing the anomalous Zeeman effect have a simple form, namely, the product of a radial integral and an angular integral. These integrals are equal to the well‐known radial and angular integrals in coordinate representation. The matrix elements can be easily evaluated. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2003
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.
It is shown that the Dirac equation with the Coulomb potential can be solved using the algebra of the three spinor invariants of the Dirac equation without the involvement of the methods of supersymmetric quantum mechanics. The Dirac Hamiltonian is invariant with respect to the rotation transformation, which indicates the dynamical (hidden) symmetry SU(2) of the Dirac equation. The total symmetry of the Dirac equation is the symmetry SO(3)⊗SU(2). The generator of the SO(3) symmetry group is given by the total momentum operator, and the generator of SU(2) group is given by the rotation of the vector-states in the spinor space, determined by the Dirac, Johnson–Lippmann, and the new spinor invariants. It is shown that using algebraic approach to the Dirac problem allows one to calculate the eigenstates and eigenenergies of the relativistic hydrogen atom and reveals the fundamental role of the principal quantum number as an independent number, even though it is represented as the combination of other quantum numbers.
The second-order hyperfine energy in the 1s and 2s states of the hydrogen atom is calculated analytically using the Dirac equation for an electron in a Coulomb potential perturbed by a point nuclear magnetic moment. The (divergent) result is compared with the calculation using the non-relativistic form of the hamiltonian. It is emphasized that in this comparison all the contributions to the non-relativistic second-order energy must be included. It is shown that, at least for the Fermi contact interaction, the results are the same to order mc 2α6, so that for this case the reduction of the Dirac equation to non-relativistic form is not suspect, contrary to previous assertions.
The associated Laguerre polynomials are very popular in the Schrödinger equation of nonrelativistic quantum mechanics. However, the situation is quite different in the case of the Dirac equation of relativistic quantum mechanics. The exact solutions of the Dirac equation for three-dimensional (3D) hydrogen atom have been obtained using the wavefunction expansion method. The exact solutions of the radial wavefunctions are found in the form of a linear combination of the orthogonal Laguerre basis function spinors. According to the equality of coefficients in Laguerre series on both sides of the equations, the exact eigenenergies and exact expressions of the corresponding eigenstates of the Dirac equation for 3D hydrogen atom are obtained, which are the same as the power series solutions. Additionally, a comparison with the results from Dirac and Schrödinger equations for 3D hydrogen atom is discussed. Due to the relativistic effects, the Dirac bispinor occurs, and there are the tiny differences of the results between Dirac and Schrödinger equations. For the Dirac states with n r = 0, there is one set of the Dirac bispinor. However for the Dirac states with n r ≠ 0, there is a combination of two sets of the Dirac bispinor. For degenerate Dirac states, their exact expansion coefficients are commutative symmetric. These results show that the linear combination of the orthogonal Laguerre basis functions is the exact solutions of the Dirac equation for 3D hydrogen atom.
The solution of the Dirac equation with the Coulomb potential is used to analyze bound electron states in a hydrogen atom. The analysis is based on the fact that such states are characterized by a set of quantum numbers which describe definite values of the complete set of physical quantities that can be determined simultaneously. This set includes the energy, square of the total angular momentum, one of its component and a spinor invariant. The latter, spinor invariant gives two-valued quantum number which determines the sign of its eigenvalue. In addition to the known Dirac and Johnson–Lippman invariant, there exists a new one. Operators of these three spinor invariants do not commute between themselves which results in the degeneracy of the energy levels with respect to the two-valued quantum number. Three different systems of the eigenbispinor corresponding to the three spinor invariants are obtained and the generalized solution with free parameters is calculated. Variation of the free parameters transforms one particular solution into any other. It is shown that the electron probability densities and spin polarizations in an electron cloud depend essentially on the invariant set, demonstrating physical difference of the states corresponding to different spinor invariants.