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Molecular Physics

On the reduction of the Dirac equation to non-relativistic form: the Fermi contact interaction in the hydrogen atom

Elaine Ann Moore, R.E. Moss

Published Nov 1, 1975• 24 citations
Publisher

Abstract

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.

DOI
10.1080/00268977500102851

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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 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.

•Apr 8, 2016•0 citations
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.
Hyperfine splittings play an important role in quantum information and spintronics applications. They allow for the readout of the spin qubits, while at the same time providing the dominant mechanism for the detrimental spin decoherence. Their exact knowledge is thus of prior relevance. In this work, we analytically investigate the relativistic effects on the hyperfine splittings of hydrogen-like atoms, including finite-size effects of the nucleis’ structure. We start from exact solutions of Dirac’s equation using different nuclear models, where the nucleus is approximated by (i) a point charge (Coulomb potential), (ii) a homogeneously charged full sphere, and (iii) a homogeneously charged spherical shell. Equivalent modelling has been done for the distribution of the nuclear magnetic moment. For the 1s ground state and 2s excited state of the one-electron systems H1, H2, H3, and He+3, the calculated finite-size related hyperfine shifts are quite similar for the different structure models and in excellent agreement with those estimated by comparing QED and experiment. This holds also in a simplified approach where relativistic wave functions from a Coulomb potential combined with spherical-shell distributed nuclear magnetic moments promises an improved treatment without the need for an explicit solution of Dirac’s equation within the nuclear core. Larger differences between different nuclear structure models are found in the case of the anisotropic 2p3/2 orbitals of hydrogen, rendering these excited states as promising reference systems for exploring the proton structure.
We discuss the equivalent form of the Lévy-Leblond equation such that the nilpotent matrices are two-dimensional. We show that this equation can be obtained in the non-relativistic limit of the (2+1)-dimensional Dirac equation. Furthermore, we analyze the case with four-dimensional matrices, propose a Hamiltonian for the equation in (3+1) dimensions, and solve it for a Coulomb potential. The quantized energy levels for the hydrogen atom are obtained, and the result is consistent with the non-relativistic quantum mechanics.
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 author has previously derived an energy-momentum relationship applicable in a hydrogen atom. Since this relationship is taken as a departure point, there is a similarity with the Dirac’s relativistic wave equation, but an equation more profound than the Dirac equation is derived. When determining the coefficients and β of the Dirac equation, Dirac assumed that the equation satisfies the Klein-Gordon equation. The Klein-Gordon equation is an equation which quantizes Einstein's energy-momentum relationship. This paper derives an equation similar to the Klein-Gordon equation by quantizing the relationship between energy and momentum of the electron in a hydrogen atom. By looking to the Dirac equation, it is predicted that there is a relativistic wave equation which satisfies that equation, and its coefficients are determined. With the Dirac equation it is necessary to insert a term for potential energy into the equation when describing the state of the electron in a hydrogen atom. However, in this paper, a potential energy term is not introduced into the relativistic wave equation. Instead, potential energy is incorporated into the equation by changing the coefficient of the Dirac equation.
A single-step Eriksen transformation of~1S1/21S_{1/2}1S1/2​,~2P1/22P_{1/2}2P1/2​ and~2P3/22P_{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.