Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2610.02060

Stark effect of hydrogenic ions as a test of the indefinite-metric formalism of the eight-component relativistic wave equation for spin-$\frac{1}{2}$ particles

Paulus C. Tjiang, Sylvia H. Sutanto, Vincentius E. W. Tjia

physics.atom-phquant-phSubmitted Oct 1, 2026• Announced Oct 2, 2026
arXivHTMLPDF

Abstract

The eight-component relativistic wave equation for spin-$\frac{1}{2}$ particles of Robson and Staudte (FV$\frac{1}{2}$), a Feshbach--Villars linearization of the Feynman--Gell-Mann equation, reproduces the Dirac hydrogenic spectrum with a doubled solution space and an indefinite inner product. Its exact Stark shifts must equal the Dirac ones; whether its indefinite-metric perturbation theory delivers them is not guaranteed. We compute the first- and second-order Stark shifts of the $n=1$--3 levels of hydrogenic ions from H to U$^{91+}$ in the Schrödinger, Dirac and FV$\frac{1}{2}$ theories, evaluating all spectral sums, continuum included, by the Dalgarno--Lewis method. The FV$\frac{1}{2}$ and Dirac shifts coincide level by level, to working precision at first order (closed-form roots) and to 45 significant figures at second order. Both ingredients are needed: without the explicit spin--field coupling the first-order roots become complex, and without the negative-norm states the $(Zα)^2$ coefficient is wrong. Norm sign and parity are locked in the doubled space: a potential lifting the $2s_{1/2}$--$2p_{1/2}$ degeneracy needs its own spin--gradient term, and shifts placed by hand on the $κ$-labeled states give a spurious imaginary linear Stark shift. The relativistic linear Stark effect is smaller by finite factors, but a second-moment sum rule shows the non-relativistic linear Stark strength is conserved, the missing part reappearing in singular $(Zα)^{-2}$ second-order terms. For levels with a fine-structure partner, an exact intra-shell/regular split of the second-order shift gives the $(Zα)^0$ coefficients analytically and relates the $π^2$ terms of the $(Zα)^2$ coefficients to the singular ones. The Dirac weak/strong-field crossover for $n=2$ of hydrogen lies near 2.8~kV cm$^{-1}$, and the perturbative window set by the fine structure grows as $Z^5$.

Comments
21 pages, 2 figures, 8 tables

Related papers

Stark effect in hydrogenic ions: Schrödinger, Dirac and FV½ theories - Mathematica Notebook

Paulus C. Tjiang, Sylvia Hastuti Sutanto

The Mathematica code producing the numerical results used in the manuscript titled "Stark effect of hydrogenic ions as a test of the indefinite-metric formalism of the eight-component relativistic wave equation for spin-1/2 particles".

•Sep 19, 2026•0 citations

An eight-component relativistic wave equation for spin- particles II

D S Staudte

The eight-component relativistic wave equation for spin- particles derived in the preceding paper is studied from a mathematical viewpoint. It is shown that it is relativistically covariant, albeit not manifestly so. It has an enlarged solution space when compared to the Dirac equation. A derivation of the equation in an arbitrary gamma matrix representation is presented. Kronecker products are used to display the analogy between the eight-component equation and the two-component spin-0 equation. The linearization procedure used to obtain the eight-component equation from the second-order spin- equation is found to have a natural mathematical validation in the study of indefinite inner product spaces. The (relativistic) quantum mechanical formalism for the eight-component equation is constructed. This features the use of an indefinite inner product and some results are presented to show how the usual quantum mechanical formalism is generalized to account for this. The reason for the decoupling of the equation into two four-component equations in the Weyl representation of the gamma matrices is given. It is shown that only one of the decoupled parts has to be solved for any problem of a single particle coupled to an external electromagnetic field. Comments are made on the solution of the hydrogen atom presented in the preceding paper.

Journal of Physics A Mathematical and General•Jan 7, 1996•23 citations

On the Theory of the Stark Effect in Hydrogenic Atoms

Vladimir Rojansky

The Stark effect of the fine structure of hydrogenic atoms is discussed in the light of the quantum mechanics. Explicit formulas are given for the energy levels in either a weak or a strong electric field, including in each case both the relativity corrections and the Stark effect. (The conventional treatment neglects either one or the other.) The results are, except in form, similar to those recently given by Schlapp. In the new mechanics, in contradistinction to the old Bohr theory, there is a linear Stark effect even in weak fields, because of the identity of energy for the $\mathrm{ns}$ and $n{p}_{1}$ levels in the absence of external fields. This degeneracy is shown to account for the results of the recent experiments of Ornstein, Zernike, and Snoek, who found that the $2s$ level in $H$ is not metastable.The coefficients are found for the development of the "parabolic" eigenfunctions in terms of the "polar" ones, and a set of numerical values of these coefficients is tabulated.Methods are given for the calculation of the relative intensities in the Stark effect in terms of the already available theoretical fine structure intensities in the absence of fields.

Physical Review•Jan 1, 1929•43 citations

Finite Nuclear Size Effect on the Relativistic Hyperfine Splittings of 2s and 2p Excited States of Hydrogen-like Atoms

K.L. Franzke, U. Gerstmann

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.

Foundations•Oct 1, 2024•2 citations

The Eight Component Relativistic Wave Equation

Sylvia Hastuti Sutanto

The eight-component (8-C) relativistic wave equation for spin-1/2 particles was developed by B. A. Robson and D. S. Staudte in 1993 using the procedure analogous to the one used earlier on the Klein-Gordon equation. The relativistic covariance and the solution behaviour of the 8-C equation have been studied extensively by D. S. Staudte in 1993. The book provides the study of the 8-C equation, particularly its application to physical problems such as Compton scattering and transition probabilities in hydrogenic atoms. The application of the 8-C equation to the Compton scattering problem shows that it gives the same well-known Klein-Nishina cross section formula as that initially obtained using the Dirac equation. The 8-C equation has also been applied to the calculation of the transition probabilities for the components of both the Balmer and Lyman alpha-lines of hydrogenic atoms, and the results show the first indication that the 8-C equation and the Dirac equation are actually not equivalent. This book is intended for those who are interested in the comparative study between the 8-C equation and the well-known Dirac equation, as well as their theoretical consequences.

•May 19, 2010•0 citations

The Hydrogen Atom Wavefunction in the Scretching–Schrödinger Framework: Deterministic Spectroscopic Closure, Maxwell–Scretching Chain, and C*-Algebraic Extension

Daniel Scretching

This paper develops an advanced unified description of the hydrogen atom wavefunction across three major theoretical levels: nonrelativistic quantum mechanics, relativistic quantum mechanics, and the Scretching–Schrödinger Equation framework. The study begins with the standard Schrödinger hydrogen wavefunction, where the electron state is represented as a complex probability amplitude labeled by the quantum numbers nnn, lll, mlm_lml, and msm_sms. It then advances to the relativistic Dirac treatment, where the scalar wavefunction is replaced by a four-component spinor labeled by nnn, κ\kappaκ, and mjm_jmj, thereby incorporating spin, relativistic energy corrections, and the natural appearance of fine-structure splitting. The central contribution of the paper is the extension of the hydrogen atom into the Scretching–Schrödinger Equation (SSE) framework, where the ordinary Schrödinger structure is preserved but constrained by a deterministic spectroscopic closure condition from the Scretching Quantum Chain (SQC) and Maxwell–Scretching Chain (MSC). In this formulation, a valid hydrogen wavefunction must not only satisfy the wave equation but must also reproduce the full physical chain ε→f→∣μ∣2→A21→Kν,\varepsilon \rightarrow f \rightarrow |\mu|^2 \rightarrow A_{21} \rightarrow K_{\nu},ε→f→∣μ∣2→A21→Kν, linking molar absorptivity, oscillator strength, transition dipole moment, Einstein spontaneous-emission coefficient, and the SQC invariant. The invariant Kν=2πe2ε0mec3=7.42165600145×10−22 sK_{\nu}=\frac{2\pi e^2}{\varepsilon_0 m_e c^3} =7.42165600145\times 10^{-22}\ \text{s}Kν=ε0mec32πe2=7.42165600145×10−22 s is derived from first principles and treated as the fixed spectroscopic time-scale anchor of the hydrogen SQC closure. The paper verifies the SQC formulation against NIST hydrogen spectral data for the Lyman series (n=2–4)(n=2–4)(n=2–4) and Balmer series (n=3–7)(n=3–7)(n=3–7), obtaining relative errors no larger than 2.7×10−42.7\times 10^{-4}2.7×10−4, with the remaining discrepancy attributed to rounding limits. It also derives the corrected radial transition integral for the 1s→2p1s\rightarrow 2p1s→2p transition, Ir0=1.2902662 a0,I_{r0}=1.2902662\,a_0,Ir0=1.2902662a0, and uses this value to support the deterministic connection between hydrogen wavefunctions, transition moments, oscillator strengths, and Einstein coefficients. The manuscript further develops the hydrogen SSE Hamiltonian in C*-algebraic form, extends the wavefunction treatment to isotope-corrected protium, deuterium, and tritium, and demonstrates that KνK_{\nu}Kν remains invariant under isotope substitution. Fine-structure effects—including Lamb-shift corrections, hyperfine splitting, and Zeeman perturbations—are examined as higher-order refinements within the SSE context rather than violations of the underlying SQC invariant. Finally, the paper maps the full SQC-to-MSC projection, showing how quantum oscillator strength projects upward into macroscopic electromagnetic observables such as absorption, refractive behavior, dielectric response, and radiative transition strength. The result is a deterministic, parameter-free quantum–electromagnetic hydrogen framework in which NRQM, RQM, SSE, SQC, and MSC are treated as connected layers of the same physical closure chain.

•May 26, 2026•0 citations
Stark effect of hydrogenic ions as a test of the indefinite-metric formalism of the eight-component relativistic wave equation for spin-12\frac{1}{2}21​ particles
The eight-component relativistic wave equation for spin-12\frac{1}{2}21​ particles of Robson and Staudte (FV12\frac{1}{2}21​), a Feshbach--Villars linearization of the Feynman--Gell-Mann equation, reproduces the Dirac hydrogenic spectrum with a doubled solution space and an indefinite inner product. Its exact Stark shifts must equal the Dirac ones; whether its indefinite-metric perturbation theory delivers them is not guaranteed. We compute the first- and second-order Stark shifts of the n=1n=1n=1--3 levels of hydrogenic ions from H to U91+^{91+}91+ in the Schrödinger, Dirac and FV12\frac{1}{2}21​ theories, evaluating all spectral sums, continuum included, by the Dalgarno--Lewis method. The FV12\frac{1}{2}21​ and Dirac shifts coincide level by level, to working precision at first order (closed-form roots) and to 45 significant figures at second order. Both ingredients are needed: without the explicit spin--field coupling the first-order roots become complex, and without the negative-norm states the (Zα)2(Zα)^2(Zα)2 coefficient is wrong. Norm sign and parity are locked in the doubled space: a potential lifting the 2s1/22s_{1/2}2s1/2​--2p1/22p_{1/2}2p1/2​ degeneracy needs its own spin--gradient term, and shifts placed by hand on the κκκ-labeled states give a spurious imaginary linear Stark shift. The relativistic linear Stark effect is smaller by finite factors, but a second-moment sum rule shows the non-relativistic linear Stark strength is conserved, the missing part reappearing in singular (Zα)−2(Zα)^{-2}(Zα)−2 second-order terms. For levels with a fine-structure partner, an exact intra-shell/regular split of the second-order shift gives the (Zα)0(Zα)^0(Zα)0 coefficients analytically and relates the π2π^2π2 terms of the (Zα)2(Zα)^2(Zα)2 coefficients to the singular ones. The Dirac weak/strong-field crossover for n=2n=2n=2 of hydrogen lies near 2.8~kV cm−1^{-1}−1, and the perturbative window set by the fine structure grows as Z5Z^5Z5.
The eight-component relativistic wave equation for spin- particles derived in the preceding paper is studied from a mathematical viewpoint. It is shown that it is relativistically covariant, albeit not manifestly so. It has an enlarged solution space when compared to the Dirac equation. A derivation of the equation in an arbitrary gamma matrix representation is presented. Kronecker products are used to display the analogy between the eight-component equation and the two-component spin-0 equation. The linearization procedure used to obtain the eight-component equation from the second-order spin- equation is found to have a natural mathematical validation in the study of indefinite inner product spaces. The (relativistic) quantum mechanical formalism for the eight-component equation is constructed. This features the use of an indefinite inner product and some results are presented to show how the usual quantum mechanical formalism is generalized to account for this. The reason for the decoupling of the equation into two four-component equations in the Weyl representation of the gamma matrices is given. It is shown that only one of the decoupled parts has to be solved for any problem of a single particle coupled to an external electromagnetic field. Comments are made on the solution of the hydrogen atom presented in the preceding paper.
The Stark effect of the fine structure of hydrogenic atoms is discussed in the light of the quantum mechanics. Explicit formulas are given for the energy levels in either a weak or a strong electric field, including in each case both the relativity corrections and the Stark effect. (The conventional treatment neglects either one or the other.) The results are, except in form, similar to those recently given by Schlapp. In the new mechanics, in contradistinction to the old Bohr theory, there is a linear Stark effect even in weak fields, because of the identity of energy for the ns\mathrm{ns}ns and np1n{p}_{1}np1​ levels in the absence of external fields. This degeneracy is shown to account for the results of the recent experiments of Ornstein, Zernike, and Snoek, who found that the 2s2s2s level in HHH is not metastable.The coefficients are found for the development of the "parabolic" eigenfunctions in terms of the "polar" ones, and a set of numerical values of these coefficients is tabulated.Methods are given for the calculation of the relative intensities in the Stark effect in terms of the already available theoretical fine structure intensities in the absence of fields.
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.
The eight-component (8-C) relativistic wave equation for spin-1/2 particles was developed by B. A. Robson and D. S. Staudte in 1993 using the procedure analogous to the one used earlier on the Klein-Gordon equation. The relativistic covariance and the solution behaviour of the 8-C equation have been studied extensively by D. S. Staudte in 1993. The book provides the study of the 8-C equation, particularly its application to physical problems such as Compton scattering and transition probabilities in hydrogenic atoms. The application of the 8-C equation to the Compton scattering problem shows that it gives the same well-known Klein-Nishina cross section formula as that initially obtained using the Dirac equation. The 8-C equation has also been applied to the calculation of the transition probabilities for the components of both the Balmer and Lyman alpha-lines of hydrogenic atoms, and the results show the first indication that the 8-C equation and the Dirac equation are actually not equivalent. This book is intended for those who are interested in the comparative study between the 8-C equation and the well-known Dirac equation, as well as their theoretical consequences.
This paper develops an advanced unified description of the hydrogen atom wavefunction across three major theoretical levels: nonrelativistic quantum mechanics, relativistic quantum mechanics, and the Scretching–Schrödinger Equation framework. The study begins with the standard Schrödinger hydrogen wavefunction, where the electron state is represented as a complex probability amplitude labeled by the quantum numbers nnn, lll, mlm_lml, and msm_sms. It then advances to the relativistic Dirac treatment, where the scalar wavefunction is replaced by a four-component spinor labeled by nnn, κ\kappaκ, and mjm_jmj, thereby incorporating spin, relativistic energy corrections, and the natural appearance of fine-structure splitting. The central contribution of the paper is the extension of the hydrogen atom into the Scretching–Schrödinger Equation (SSE) framework, where the ordinary Schrödinger structure is preserved but constrained by a deterministic spectroscopic closure condition from the Scretching Quantum Chain (SQC) and Maxwell–Scretching Chain (MSC). In this formulation, a valid hydrogen wavefunction must not only satisfy the wave equation but must also reproduce the full physical chain ε→f→∣μ∣2→A21→Kν,\varepsilon \rightarrow f \rightarrow |\mu|^2 \rightarrow A_{21} \rightarrow K_{\nu},ε→f→∣μ∣2→A21→Kν, linking molar absorptivity, oscillator strength, transition dipole moment, Einstein spontaneous-emission coefficient, and the SQC invariant. The invariant Kν=2πe2ε0mec3=7.42165600145×10−22 sK_{\nu}=\frac{2\pi e^2}{\varepsilon_0 m_e c^3} =7.42165600145\times 10^{-22}\ \text{s}Kν=ε0mec32πe2=7.42165600145×10−22 s is derived from first principles and treated as the fixed spectroscopic time-scale anchor of the hydrogen SQC closure. The paper verifies the SQC formulation against NIST hydrogen spectral data for the Lyman series (n=2–4)(n=2–4)(n=2–4) and Balmer series (n=3–7)(n=3–7)(n=3–7), obtaining relative errors no larger than 2.7×10−42.7\times 10^{-4}2.7×10−4, with the remaining discrepancy attributed to rounding limits. It also derives the corrected radial transition integral for the 1s→2p1s\rightarrow 2p1s→2p transition, Ir0=1.2902662 a0,I_{r0}=1.2902662\,a_0,Ir0=1.2902662a0, and uses this value to support the deterministic connection between hydrogen wavefunctions, transition moments, oscillator strengths, and Einstein coefficients. The manuscript further develops the hydrogen SSE Hamiltonian in C*-algebraic form, extends the wavefunction treatment to isotope-corrected protium, deuterium, and tritium, and demonstrates that KνK_{\nu}Kν remains invariant under isotope substitution. Fine-structure effects—including Lamb-shift corrections, hyperfine splitting, and Zeeman perturbations—are examined as higher-order refinements within the SSE context rather than violations of the underlying SQC invariant. Finally, the paper maps the full SQC-to-MSC projection, showing how quantum oscillator strength projects upward into macroscopic electromagnetic observables such as absorption, refractive behavior, dielectric response, and radiative transition strength. The result is a deterministic, parameter-free quantum–electromagnetic hydrogen framework in which NRQM, RQM, SSE, SQC, and MSC are treated as connected layers of the same physical closure chain.