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Peay's Helium Immersion: Completing the Structure of the Equation Physics Calls Unsolvable

Dwayne Peay

Published Oct 27, 2026• 0 citations
Publisher

Abstract

The time-independent Schrödinger equation for the helium atom has stood for a century as the canonical example of a quantum problem with no exact solution: the electron–electron repulsion term 1/r12 couples the coordinates inseparably, and the field's settled practice is approximation (Hartree–Fock, configuration interaction, density functional theory). This paper reports a completed result obtained by a different route. Rather than approximating the equation as posed, the problem was immersed in a complete relational framework (the Source Codex of Mathematics, Peay), whose structure diagnoses what a formulation is missing. The immersion identified a single structural defect: the standard formulation poses the atom as two independent electrons plus an orphaned interaction term, while the pair-distance r12 is never admitted as a structural coordinate of the wavefunction. Completing the structure — admitting u = r12 as a member of the wavefunction — collapses the problem. A variational Monte Carlo computation with a single admitted structural term yields a ground-state energy of -2.8908 ± 0.0007 Hartree, beyond the Hartree-Fock limit (-2.86168) that bounds all separable methods; extending along the same admitted coordinate reaches -2.9024 against the experimental value -2.90372.

DOI
10.5281/zenodo.23745844

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Peay's Helium Immersion: Completing the Structure of the Equation Physics Calls Unsolvable

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The time-independent Schrödinger equation for the helium atom has stood for a century as the canonical example of a quantum problem with no exact solution: the electron–electron repulsion term 1/r12 couples the coordinates inseparably, and the field's settled practice is approximation (Hartree–Fock, configuration interaction, density functional theory). This paper reports a completed result obtained by a different route. Rather than approximating the equation as posed, the problem was immersed in a complete relational framework (the Source Codex of Mathematics, Peay), whose structure diagnoses what a formulation is missing. The immersion identified a single structural defect: the standard formulation poses the atom as two independent electrons plus an orphaned interaction term, while the pair-distance r12 is never admitted as a structural coordinate of the wavefunction. Completing the structure — admitting u = r12 as a member of the wavefunction — collapses the problem. A variational Monte Carlo computation with a single admitted structural term yields a ground-state energy of -2.8908 ± 0.0007 Hartree, beyond the Hartree-Fock limit (-2.86168) that bounds all separable methods; extending along the same admitted coordinate reaches -2.9024 against the experimental value -2.90372. This paper also stands published in the SourceTech community on Zenodo: https://doi.org/10.5281/zenodo.23745845 — Community: https://zenodo.org/communities/sourcetech — The Source Codex of Mathematics (the Book): https://jalejaproducts3.gumroad.com/l/cglqnv — Speaking and correspondence: dwayne@sourcetecei.com

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Application of the Scretching-Schrodinger Equation, Scretching Quantum Chain, and Maxwell-Scretching Chain to Photon-Field Interactions

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This paper applies the Scretching–Schrödinger Equation (SSE) to hydrogen, helium, and lithium as foundational atomic test systems for connecting wave mechanics, matrix mechanics, and spectroscopic closure. The study begins with the standard time-independent Schrödinger equation as the eigenvalue foundation for atomic structure, energy levels, quantum numbers, and electron configurations. Hydrogen is treated as the exact one-electron benchmark, while helium and lithium are treated as many-electron systems requiring effective Hamiltonians, electron–electron interaction terms, screening, configuration structure, and approximate spectroscopic interpretation. The paper compares the Schrödinger and Heisenberg formulations of quantum mechanics. The Schrödinger representation supplies wavefunctions, eigenstates, orbital quantum numbers, and coordinate-space probability amplitudes. The Heisenberg representation supplies the equivalent operator and matrix formulation, where observables evolve through commutators and transition amplitudes are represented through matrix elements. This comparison is extended into a new SSE matrix formulation in which the ordinary Hamiltonian is modified by a spectroscopic closure term: H^SSE=H^0+V^SQC\hat{H}_{SSE}=\hat{H}_{0}+\hat{V}_{SQC}H^SSE=H^0+V^SQC Here, H^0\hat{H}_{0}H^0 is the standard atomic Hamiltonian, and V^SQC\hat{V}_{SQC}V^SQC is the Scretching Quantum Chain closure operator. The corrected SSE derivation is preserved by treating hydrogen as the calibration benchmark, where V^SQC(H)≈0\hat{V}_{SQC}(H)\approx 0V^SQC(H)≈0 for the exact hydrogenic limit. The paper also carefully distinguishes the single Lyman-α\alphaα oscillator strength, f(1s→2p)=0.4162f(1s\rightarrow2p)=0.4162f(1s→2p)=0.4162, from the full Thomas–Reiche–Kuhn sum rule, ∑ifi=1\sum_i f_i=1∑ifi=1, so that individual transitions are not confused with the total oscillator-strength conservation law. The SQC layer links atomic wavefunction matrix elements to measurable spectroscopic quantities, including oscillator strength fff, transition dipole moment ∣μ∣2|\mu|^2∣μ∣2, Einstein spontaneous-emission coefficient A21A_{21}A21, transition frequency ν\nuν, wavelength λ\lambdaλ, and spectral-line structure. In this framework, hydrogen provides the exact reference model, helium tests the effect of electron correlation and two-electron screening, and lithium extends the method to a three-electron atom with inner-shell shielding and valence-electron spectral transitions. The OpenStax University Physics III: Optics & Modern Physics source provides the broader optical and modern-physics bridge by connecting Maxwell’s electromagnetic-wave theory with photon behavior, matter waves, wave–particle duality, atomic spectra, lasers, molecular spectra, superconductivity, nuclear physics, and particle physics. The LibreTexts quantum-mechanics source adds the formal operator and Hamiltonian framework, including time-dependent and time-independent wavefunctions, the three-dimensional Schrödinger equation, hydrogenic structure, spin–orbit coupling, kinetic-energy corrections, Lamb shift, Zeeman effect, and the Heisenberg uncertainty principle. Greiner’s textbook treatment is incorporated as a formal comparison source for quantization, radiation laws, matter waves, the Schrödinger equation, harmonic oscillators, charged particles in magnetic fields, mathematical foundations, matrix mechanics, Schrödinger, Heisenberg, and interaction representations, perturbation theory, spin, identical particles, and conceptual foundations. The resulting paper presents hydrogen, helium, and lithium as a three-stage atomic ladder for developing the SSE: hydrogen verifies the zero-correction benchmark, helium introduces many-electron correction structure, and lithium extends the method to screened valence-electron spectroscopy. The final contribution is a unified Schrödinger–Heisenberg–Scretching framework in which wavefunctions, operators, matrices, oscillator strengths, transition dipoles, Einstein coefficients, and spectral frequencies are placed into one mathematically consistent atomic-spectroscopic chain.

•May 6, 2026•0 citations
The time-independent Schrödinger equation for the helium atom has stood for a century as the canonical example of a quantum problem with no exact solution: the electron–electron repulsion term 1/r12 couples the coordinates inseparably, and the field's settled practice is approximation (Hartree–Fock, configuration interaction, density functional theory). This paper reports a completed result obtained by a different route. Rather than approximating the equation as posed, the problem was immersed in a complete relational framework (the Source Codex of Mathematics, Peay), whose structure diagnoses what a formulation is missing. The immersion identified a single structural defect: the standard formulation poses the atom as two independent electrons plus an orphaned interaction term, while the pair-distance r12 is never admitted as a structural coordinate of the wavefunction. Completing the structure — admitting u = r12 as a member of the wavefunction — collapses the problem. A variational Monte Carlo computation with a single admitted structural term yields a ground-state energy of -2.8908 ± 0.0007 Hartree, beyond the Hartree-Fock limit (-2.86168) that bounds all separable methods; extending along the same admitted coordinate reaches -2.9024 against the experimental value -2.90372.
The time-independent Schrödinger equation for the helium atom has stood for a century as the canonical example of a quantum problem with no exact solution: the electron–electron repulsion term 1/r12 couples the coordinates inseparably, and the field's settled practice is approximation (Hartree–Fock, configuration interaction, density functional theory). This paper reports a completed result obtained by a different route. Rather than approximating the equation as posed, the problem was immersed in a complete relational framework (the Source Codex of Mathematics, Peay), whose structure diagnoses what a formulation is missing. The immersion identified a single structural defect: the standard formulation poses the atom as two independent electrons plus an orphaned interaction term, while the pair-distance r12 is never admitted as a structural coordinate of the wavefunction. Completing the structure — admitting u = r12 as a member of the wavefunction — collapses the problem. A variational Monte Carlo computation with a single admitted structural term yields a ground-state energy of -2.8908 ± 0.0007 Hartree, beyond the Hartree-Fock limit (-2.86168) that bounds all separable methods; extending along the same admitted coordinate reaches -2.9024 against the experimental value -2.90372. This paper also stands published in the SourceTech community on Zenodo: https://doi.org/10.5281/zenodo.23745845 — Community: https://zenodo.org/communities/sourcetech — The Source Codex of Mathematics (the Book): https://jalejaproducts3.gumroad.com/l/cglqnv — Speaking and correspondence: dwayne@sourcetecei.com
The helium atom is the simplest many-body electronic system provided by nature. The exact solution to the Schrödinger equation is known for helium ground and excited states, and it represents a benchmark for any many-body methodology. Here, we check the ab initio many-body GW approximation and the Bethe-Salpeter equation (BSE) against the exact solution for helium. Starting from the Hartree-Fock method, we show that the GW and the BSE yield impressively accurate results on excitation energies and oscillator strength, systematically improving the time-dependent Hartree-Fock method. These findings suggest that the accuracy of the BSE and GW approximations is not significantly limited by self-interaction and self-screening problems even in this few electron limit. We further discuss our results in comparison to those obtained by time-dependent density-functional theory.
Over time, many different theories and approaches have been developed to tackle the many-body problem in quantum chemistry, condensed-matter physics, and nuclear physics. Here we use the helium atom, a real system rather than a model, and we use the exact solution of its Schrödinger equation as a benchmark for comparison between methods. We present new results beyond the random-phase approximation (RPA) from a renormalized RPA (r-RPA) in the framework of the self-consistent RPA (SCRPA) originally developed in nuclear physics, and compare them with various other approaches like configuration interaction (CI), quantum Monte Carlo (QMC), time-dependent density-functional theory (TDDFT), and the Bethe-Salpeter equation on top of the \boldsymbol{GW} 𝐆 𝐖 approximation. Most of the calculations are consistently done on the same footing, e.g. using the same basis set, in an effort for a most faithful comparison between methods.
Over time, many different theories and approaches have been developed to tackle the many-body problem in quantum chemistry, condensed-matter physics, and nuclear physics. Here we use the helium atom, a real system rather than a model, and we use the exact solution of its Schrödinger equation as a benchmark for comparison between methods. We present new results beyond the random-phase approximation (RPA) from a renormalized RPA (r-RPA) in the framework of the self-consistent RPA (SCRPA) originally developed in nuclear physics, and compare them with various other approaches like configuration interaction (CI), quantum Monte Carlo (QMC), time-dependent density-functional theory (TDDFT), and the Bethe-Salpeter equation on top of the GW approximation. Most of the calculations are consistently done on the same footing, e.g. using the same basis set, in an effort for a most faithful comparison between methods.
With the use of the known solution of the Schrödinger equation for an electron in the nucleus field in the polar coordinates, the energy of a “two-dimensional” two-electron atom in the ground state, as well as its single ionization energy, has been calculated both in perturbation theory and with an almost century-old method of variation of the parameter Z in a trial wavefunction of the ground state. Since such two-dimensional atoms, e.g., helium atoms, can in principle be implemented in experiments by “freezing” of one degree of freedom in the phase of Bose–Einstein condensate, the conclusions made in this work can be tested. Fundamental features of the calculation of the energy of “one-dimensional” two-electron atoms and the formation of their Bose–Einstein condensate have also been discussed. The results obtained in this work coincide in a number of particular cases with the results obtained in a previous work, where some results were absent.
This paper applies the Scretching–Schrödinger Equation (SSE) to hydrogen, helium, and lithium as foundational atomic test systems for connecting wave mechanics, matrix mechanics, and spectroscopic closure. The study begins with the standard time-independent Schrödinger equation as the eigenvalue foundation for atomic structure, energy levels, quantum numbers, and electron configurations. Hydrogen is treated as the exact one-electron benchmark, while helium and lithium are treated as many-electron systems requiring effective Hamiltonians, electron–electron interaction terms, screening, configuration structure, and approximate spectroscopic interpretation. The paper compares the Schrödinger and Heisenberg formulations of quantum mechanics. The Schrödinger representation supplies wavefunctions, eigenstates, orbital quantum numbers, and coordinate-space probability amplitudes. The Heisenberg representation supplies the equivalent operator and matrix formulation, where observables evolve through commutators and transition amplitudes are represented through matrix elements. This comparison is extended into a new SSE matrix formulation in which the ordinary Hamiltonian is modified by a spectroscopic closure term: H^SSE=H^0+V^SQC\hat{H}_{SSE}=\hat{H}_{0}+\hat{V}_{SQC}H^SSE=H^0+V^SQC Here, H^0\hat{H}_{0}H^0 is the standard atomic Hamiltonian, and V^SQC\hat{V}_{SQC}V^SQC is the Scretching Quantum Chain closure operator. The corrected SSE derivation is preserved by treating hydrogen as the calibration benchmark, where V^SQC(H)≈0\hat{V}_{SQC}(H)\approx 0V^SQC(H)≈0 for the exact hydrogenic limit. The paper also carefully distinguishes the single Lyman-α\alphaα oscillator strength, f(1s→2p)=0.4162f(1s\rightarrow2p)=0.4162f(1s→2p)=0.4162, from the full Thomas–Reiche–Kuhn sum rule, ∑ifi=1\sum_i f_i=1∑ifi=1, so that individual transitions are not confused with the total oscillator-strength conservation law. The SQC layer links atomic wavefunction matrix elements to measurable spectroscopic quantities, including oscillator strength fff, transition dipole moment ∣μ∣2|\mu|^2∣μ∣2, Einstein spontaneous-emission coefficient A21A_{21}A21, transition frequency ν\nuν, wavelength λ\lambdaλ, and spectral-line structure. In this framework, hydrogen provides the exact reference model, helium tests the effect of electron correlation and two-electron screening, and lithium extends the method to a three-electron atom with inner-shell shielding and valence-electron spectral transitions. The OpenStax University Physics III: Optics & Modern Physics source provides the broader optical and modern-physics bridge by connecting Maxwell’s electromagnetic-wave theory with photon behavior, matter waves, wave–particle duality, atomic spectra, lasers, molecular spectra, superconductivity, nuclear physics, and particle physics. The LibreTexts quantum-mechanics source adds the formal operator and Hamiltonian framework, including time-dependent and time-independent wavefunctions, the three-dimensional Schrödinger equation, hydrogenic structure, spin–orbit coupling, kinetic-energy corrections, Lamb shift, Zeeman effect, and the Heisenberg uncertainty principle. Greiner’s textbook treatment is incorporated as a formal comparison source for quantization, radiation laws, matter waves, the Schrödinger equation, harmonic oscillators, charged particles in magnetic fields, mathematical foundations, matrix mechanics, Schrödinger, Heisenberg, and interaction representations, perturbation theory, spin, identical particles, and conceptual foundations. The resulting paper presents hydrogen, helium, and lithium as a three-stage atomic ladder for developing the SSE: hydrogen verifies the zero-correction benchmark, helium introduces many-electron correction structure, and lithium extends the method to screened valence-electron spectroscopy. The final contribution is a unified Schrödinger–Heisenberg–Scretching framework in which wavefunctions, operators, matrices, oscillator strengths, transition dipoles, Einstein coefficients, and spectral frequencies are placed into one mathematically consistent atomic-spectroscopic chain.