Computing with the 1955 Spin Law: A Structure-Preserving Integrator and a Variationally Determined Density Family, Tested against (e,2e) Data
Year: 2026
Type: preprint
Abstract: For seventy years, Bohmian trajectories have served as an interpretation of quantum mechanics, rarely as a computational medium — and the spin-corrected law of motion published by Bohm, Schiller, and Tiomno in 1955, which removes the resting ground-state electron of the 1952 theory, has so far been used to draw trajectories from previously solved eigenstates, not as a computational medium in its own right. This work shows that it is one. Structure comes from the density family and circulation from the law; the laboratory test reported here is a test of the family. What is propagated is that law applied to per-electron beable spin vectors: for one electron in a spin eigenstate this is the published current exactly, so hydrogen calibrates it; for a closed-shell singlet the published current vanishes identically, so what is used there is a declared extension and is marked as one. A structure-preserving integrator is derived from the two exact invariants of the law; it is reversible to about 10⁻¹¹ Bohr over a thousand steps forward and back — the step alone; with the level projector, whose projection cannot be undone, 10⁻⁸ — and of second order along the trajectory (convergence factors 4.04, 4.03, 4.13), and a level projector — now a safeguard rather than a repair — holds the density-level theorem with a logged correction that falls at the order of the method. A spin relaxation term derived from the density family itself makes the triplet fixed point linearly unstable, which, with the beable spin sector identified with the spectroscopic one, the observed metastability of ortho-helium 2³S excludes; the term is withdrawn. On a variationally determined density family for hydrogen, helium, and lithium — whose inputs are the Coulomb interaction, spin, and the asymptotic decay rates fixed by the ionization energies, and in which no constant is fitted to data, though which channels the helium family carries was prompted by an earlier comparison (Section 4): every constant either source-pinned, internally stationary, or declared as structure — ensembles of spin trajectories are propagated in configuration space. Hydrogen calibrates the momentum pipeline to machine precision. For helium, compared with two (e,2e) measurements of the ionization ratio into excited ion states, the family has the dip below the Hartree–Fock level at small momentum, but not its depth: at the lowest point of the earlier data it lies a quarter above the datum, four standard deviations, and 40 % above the value the later work quotes at its lowest point; in the rise it meets both within about one standard deviation over ten points. A 1024-trajectory lithium ensemble keeps the 1s² 2s shell structure of the underlying density intact, held by two exact invariants of the motion rather than produced by it; the dynamics preserves the statics, the medians of the core radii moving by less than half a per cent over 600 atomic time units; and its core pair circulates in opposite senses — counted per walker, in 255 of 256, the remaining one by the same rule of spin direction and density gradient — one statement in this work that the stationary density itself cannot make, since a stationary density knows no direction of rotation. The momentum profile of the lithium 2s carries a node minimum at p_node = 0.9050 ± 0.0113 a.u.: a registered value, kept in three documented generations, whose budget covers parameter uncertainty at fixed structure. No momentum profile of atomic lithium has been measured — (e,2e) experiments on the free atom exist only at low impact energies, where the cross section does not map the bound momentum distribution — and the value does not separate the family from correlated theory. A documented correction protocol accompanies all claims. This is the manuscript as prepared for journal submission. Main text and supplement belong together; cross-pointers in the main text refer to the supplement. No third-party figures are reproduced; our readings of the (e,2e) measurements used for comparison are listed in Table S2 of the supplement and included in the reproducibility package. Everything in the package is in English: folder and file names, names in the code, comments, messages and data headers.
Author: Bäcker, Uwe
Language: English
Subfield: Atomic and Molecular Physics, and Optics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7215048985
Cites:
Cited by:
Related to: