Validity of the mass-velocity term in the Breit-Pauli hamiltonian
Bibliographic record
Abstract
The mass-velocity term, Hmv = –p4/8m3ec2, of the Breit-Pauli Hamiltonian is studied for its range and capability in providing relativistic corrections to the nonrelativistic kinetic energy, Hke = p2/2me. For this purpose, after a brief critical review of the characteristics of Hmv, the expectation values (Hkr), (Hmv), and (H0) are presented for a series of ground-state neutral atoms (Z = 1–92), where H0 is the relativistic kinetic energy, mec2[(1 + p2/m2ec2)1/2 – 1]. All expectation values are with respect to nonrelativistic wavefunctions. As expected, (Hke) has a larger value than (H0) for all atoms considered, whereas the mass velocity–corrected kinetic energy, (Hke) + (Hmv), is always (i.e., for all values of Z) smaller than (H0), even for the hydrogen atom. Our results also indicate that the efficiency of Hmv in bringing about relativistic corrections to (Hke), decreases linearly as a function of the atomic number, Z. This efficiency starts at 98.41% for Z = 1 (hydrogen) and decreases to 0.639% for Z = 60 (neodymium), beyond which the efficiency is negative and the inclusion of Hmv in the relativistic Hamiltonian produces more error than correction.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.004 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.002 | 0.001 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.003 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".