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 distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 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 teacher head, 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".