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

Dwayne Peay

Published Oct 29, 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. 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

DOI
10.6084/m9.figshare.34858875

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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.

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Permit me to add some information to the article entitled, “Bohr’s molecular model, a century later.” The authors discuss the idea of “dimensional scaling,” approximating the spectra of atoms and molecules by employing perturbation theory to expand around the infinite-dimension/classical limit. That technique of approximating the Schrödinger equation, originally called the 1/N expansion, was developed and applied to numerous problems in the 1980s, including simple atoms, quarkonium, and the hydrogen molecule.11. L. D. Mlodinow, N. Papanicolaou, Ann. Phys. 128, 314 (1980); https://doi.org/10.1016/0003-4916(80)90323-1 L. D. Mlodinow, N. Papanicolaou, Ann. Phys. 131, 1 (1981). https://doi.org/10.1016/0003-4916(81)90181-0,22. C. M. Bender, L. D. Mlodinow, N. Papanicolaou, Phys. Rev. A 25, 1305 (1982). https://doi.org/10.1103/PhysRevA.25.1305,33. T. Imbo, U. Sukhatme, Phys. Rev. D 31, 2655 (1985); https://doi.org/10.1103/PhysRevD.31.2655M. Kumar, A. Srivastava, J. K. Bhattacharjee, K. Banerjee, Phys. Lett. A 117, 226 (1986). https://doi.org/10.1016/0375-9601(86)90081-2 It grew out of attempts in the 1970s to formulate quantum chromodynamics in the limit of a large number of colors. The use of the 1/N expansion in atomic physics was discussed by Edward Witten (Physics Today, July 1980, page 38), who pointed out that his discussion was based on work that applied the method to hydrogen and helium atoms.11. L. D. Mlodinow, N. Papanicolaou, Ann. Phys. 128, 314 (1980); https://doi.org/10.1016/0003-4916(80)90323-1 L. D. Mlodinow, N. Papanicolaou, Ann. Phys. 131, 1 (1981). https://doi.org/10.1016/0003-4916(81)90181-0The work in reference 11. L. D. Mlodinow, N. Papanicolaou, Ann. Phys. 128, 314 (1980); https://doi.org/10.1016/0003-4916(80)90323-1 L. D. Mlodinow, N. Papanicolaou, Ann. Phys. 131, 1 (1981). https://doi.org/10.1016/0003-4916(81)90181-0 made use of algebraic methods for the analysis, but the coordinate space method was also developed and was initially used to treat the strong-field Zeeman effect.22. C. M. Bender, L. D. Mlodinow, N. Papanicolaou, Phys. Rev. A 25, 1305 (1982). https://doi.org/10.1103/PhysRevA.25.1305 A nice description of the status of the field at the time was given by Laurence Yaffe (Physics Today, August 1983, page 50). Many additional applications of that method in the physics literature may be found in reference 44. A. Chatterjee, Phys. Rep. 186, 249 (1990). https://doi.org/10.1016/0370-1573(90)90048-7.REFERENCESSection:ChooseTop of pageREFERENCES <<1. L. D. Mlodinow, N. Papanicolaou, Ann. Phys. 128, 314 (1980); https://doi.org/10.1016/0003-4916(80)90323-1 , Google ScholarCrossref, ISIL. D. Mlodinow, N. Papanicolaou, Ann. Phys. 131, 1 (1981). https://doi.org/10.1016/0003-4916(81)90181-0, , Google ScholarCrossref, ISI2. C. M. Bender, L. D. Mlodinow, N. Papanicolaou, Phys. Rev. A 25, 1305 (1982). https://doi.org/10.1103/PhysRevA.25.1305, Google ScholarCrossref, ISI3. T. Imbo, U. Sukhatme, Phys. Rev. D 31, 2655 (1985); https://doi.org/10.1103/PhysRevD.31.2655, Google ScholarCrossref, ISIM. Kumar, A. Srivastava, J. K. Bhattacharjee, K. Banerjee, Phys. Lett. A 117, 226 (1986). https://doi.org/10.1016/0375-9601(86)90081-2, , Google ScholarCrossref, ISI4. A. Chatterjee, Phys. Rep. 186, 249 (1990). https://doi.org/10.1016/0370-1573(90)90048-7, Google ScholarCrossref, ISI© 2014 American Institute of Physics.

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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
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 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.
Permit me to add some information to the article entitled, “Bohr’s molecular model, a century later.” The authors discuss the idea of “dimensional scaling,” approximating the spectra of atoms and molecules by employing perturbation theory to expand around the infinite-dimension/classical limit. That technique of approximating the Schrödinger equation, originally called the 1/N expansion, was developed and applied to numerous problems in the 1980s, including simple atoms, quarkonium, and the hydrogen molecule.11. L. D. Mlodinow, N. Papanicolaou, Ann. Phys. 128, 314 (1980); https://doi.org/10.1016/0003-4916(80)90323-1 L. D. Mlodinow, N. Papanicolaou, Ann. Phys. 131, 1 (1981). https://doi.org/10.1016/0003-4916(81)90181-0,22. C. M. Bender, L. D. Mlodinow, N. Papanicolaou, Phys. Rev. A 25, 1305 (1982). https://doi.org/10.1103/PhysRevA.25.1305,33. T. Imbo, U. Sukhatme, Phys. Rev. D 31, 2655 (1985); https://doi.org/10.1103/PhysRevD.31.2655M. Kumar, A. Srivastava, J. K. Bhattacharjee, K. Banerjee, Phys. Lett. A 117, 226 (1986). https://doi.org/10.1016/0375-9601(86)90081-2 It grew out of attempts in the 1970s to formulate quantum chromodynamics in the limit of a large number of colors. The use of the 1/N expansion in atomic physics was discussed by Edward Witten (Physics Today, July 1980, page 38), who pointed out that his discussion was based on work that applied the method to hydrogen and helium atoms.11. L. D. Mlodinow, N. Papanicolaou, Ann. Phys. 128, 314 (1980); https://doi.org/10.1016/0003-4916(80)90323-1 L. D. Mlodinow, N. Papanicolaou, Ann. Phys. 131, 1 (1981). https://doi.org/10.1016/0003-4916(81)90181-0The work in reference 11. L. D. Mlodinow, N. Papanicolaou, Ann. Phys. 128, 314 (1980); https://doi.org/10.1016/0003-4916(80)90323-1 L. D. Mlodinow, N. Papanicolaou, Ann. Phys. 131, 1 (1981). https://doi.org/10.1016/0003-4916(81)90181-0 made use of algebraic methods for the analysis, but the coordinate space method was also developed and was initially used to treat the strong-field Zeeman effect.22. C. M. Bender, L. D. Mlodinow, N. Papanicolaou, Phys. Rev. A 25, 1305 (1982). https://doi.org/10.1103/PhysRevA.25.1305 A nice description of the status of the field at the time was given by Laurence Yaffe (Physics Today, August 1983, page 50). Many additional applications of that method in the physics literature may be found in reference 44. A. Chatterjee, Phys. Rep. 186, 249 (1990). https://doi.org/10.1016/0370-1573(90)90048-7.REFERENCESSection:ChooseTop of pageREFERENCES <<1. L. D. Mlodinow, N. Papanicolaou, Ann. Phys. 128, 314 (1980); https://doi.org/10.1016/0003-4916(80)90323-1 , Google ScholarCrossref, ISIL. D. Mlodinow, N. Papanicolaou, Ann. Phys. 131, 1 (1981). https://doi.org/10.1016/0003-4916(81)90181-0, , Google ScholarCrossref, ISI2. C. M. Bender, L. D. Mlodinow, N. Papanicolaou, Phys. Rev. A 25, 1305 (1982). https://doi.org/10.1103/PhysRevA.25.1305, Google ScholarCrossref, ISI3. T. Imbo, U. Sukhatme, Phys. Rev. D 31, 2655 (1985); https://doi.org/10.1103/PhysRevD.31.2655, Google ScholarCrossref, ISIM. Kumar, A. Srivastava, J. K. Bhattacharjee, K. Banerjee, Phys. Lett. A 117, 226 (1986). https://doi.org/10.1016/0375-9601(86)90081-2, , Google ScholarCrossref, ISI4. A. Chatterjee, Phys. Rep. 186, 249 (1990). https://doi.org/10.1016/0370-1573(90)90048-7, Google ScholarCrossref, ISI© 2014 American Institute of Physics.
For over a century, the quantum realm has been treated as a mathematically isolated domain where macroscopic laws break down, forcing legacy physics to rely on wave-function collapse, fundamental randomness, and instantaneous non-local connections (Einstein's "Spooky Action at a Distance"). This paper utilizes the Electron Flow Model (EFM) to replace quantum mysticism with continuous, deterministic fluid dynamics. We demonstrate that entangled particles are not communicating magically across a void, but are mechanically linked via incompressible "fluid tethers", allowing us to deterministically recover quantum correlation limits through geometric momentum transfer without violating local realism. Key Findings & Methodology: The Geometry of Spin and Polarity: Quantum spin is not an abstract mathematical property, but a literal, active mechanical response to hydrodynamic pressure. The physical orientation of the atom dictates its magnetic exhaust. Cryogenic Locking and Flux Pinning: Freezing thermal jitter locks the atomic piston into a rigid, stationary coordinate. This creates near-perfect thermodynamic impedance, allowing magnetic field lines to physically pin to the stable lattice. No Spooky, Just Action: Entangled particles are subatomic engines connected by a shared, continuous conduit of vacuum flux. The extreme density of this incompressible fluid tether allows for instantaneous hydrodynamic torque transfer. The CHSH Limit (2√2): The EFM successfully derives the exact quantum correlation limit strictly through N-body geometric momentum transfer collisions, proving that abstract mathematical spaces and non-locality are mathematically unnecessary. Included Resources: Appendix A contains the complete, reproducible Python mathematical engine used to simulate the N-Body fluid collisions and derive the CHSH correlation limit from first principles. The Unified Fluid Cosmos Framework: This paper serves as Manuscript 14 Version 1 in the comprehensive series, Marabūt's Theory of Gravity. It presents the Electron Flow Model (EFM), a generative, fluid-dynamic framework designed to fundamentally replace Albert Einstein's General Relativity and standard Quantum Mechanics. The EFM shatters classical circularity with a single, undeniable foundational premise: Gravity is Pressure, not curvature, and not an intrinsic mass property. Published Manuscripts in this Series: Manuscript 01: The Push-Pull Mechanics of Vacuum Flux (The Foundational Framework) — https://doi.org/10.5281/zenodo.18874694 Manuscript 02: Resolving the Weyl Potential Anomaly — https://doi.org/10.5281/zenodo.19784494 Manuscript 03: Mechanistic Resolution of Jet Deflection in Cygnus X-1 — https://doi.org/10.5281/zenodo.19790983 Manuscript 04: Quantum Mechanics of Stellar Chain Reactions and EVGS Formation — https://doi.org/10.5281/zenodo.19801398 Manuscript 05: The Heliospheric Grand Data Matrix — https://doi.org/10.5281/zenodo.20128776 Manuscript 06: Fluid-Dynamic Reinterpretation of Gravity, Black Holes, and Gravitational Waves — https://doi.org/10.5281/zenodo.20129518 Manuscript 07: Fluid-Dynamic Reinterpretation of the Kinematic Sunyaev-Zeldovich Effect — https://doi.org/10.5281/zenodo.20130259 Manuscript 08: Transcending the Singularity — Electron Void Gravitational Spheres — https://doi.org/10.5281/zenodo.20130801 Manuscript 09: The Final Parsec Problem Solved — https://doi.org/10.5281/zenodo.20134048 Manuscript 10: The Illusion of the Singularity — https://doi.org/10.5281/zenodo.20135533 Manuscript 11: The Illusion of Dark Energy — https://doi.org/10.5281/zenodo.20149141 Manuscript 12: The Illusion of Dark Matter — https://doi.org/10.5281/zenodo.20171372 Manuscript 13: The Illusion of Spacetime — https://doi.org/10.5281/zenodo.20172281 Manuscript 14: The Illusion of Spooky Action — https://doi.org/10.5281/zenodo.20173191 Manuscript 15: The Superluminal Mechanics of the Quantum Vacuum — https://doi.org/10.5281/zenodo.20173755 Manuscript 16: Empirical Validation of the Electron Flow Model — https://doi.org/10.5281/zenodo.20271062 Manuscript 17: Confronting the Cosmic Microwave Background — https://doi.org/10.5281/zenodo.20483713 Manuscript 18: Resolving the CMB Power Spectrum — https://doi.org/10.5281/zenodo.20493520 Manuscript 19: The Foundation of Stellar Dynamics — https://doi.org/10.5281/zenodo.20500168 Manuscript 20: Resolving the Black Hole Information Paradox — https://doi.org/10.5281/zenodo.20535631 Manuscript 21: Hydrodynamic Shockwaves and Wave-Particle Duality (Current Paper) — https://doi.org/10.5281/zenodo.20620456 Data & Code Availability: To accompany this manuscript, we have open-sourced the complete Python mathematical engine and a fully interactive 3D web-based simulation of the EFM Volumetric Profiler. Researchers, developers, and students can visually explore the hydrodynamic vacuum flux across the celestial bodies at: Interactive 3D EFM Volumetric Profiler: https://marabutmarabut.github.io/Theory-of-Gravity Official GitHub Repository: https://github.com/MarabutMarabut/Theory-of-Gravity