Microscopic, quantum derivation of cranking model for nuclear collective rotation: harmonic oscillator case
Bibliographic record
Abstract
In this article, the conventional semiclassical one-dimensional cranking model (CR), which is commonly used to investigate rotational structures of deformed nuclei, is derived from microscopic, quantum first principles for the harmonic oscillator case. The space-fixed particle coordinates are canonically transformed to an Euler angle and a set of 3N – 1 intrinsic coordinates to decompose the nuclear Hamiltonian into intrinsic and collective rotational components plus a Coriolis-centrifugal term that couples the intrinsic and rotational motions. To overcome the difficulties associated with finding an appropriate set of intrinsic coordinates, the rotational component in the transformed Hamiltonian is expressed in terms of the space-fixed coordinates and momenta by taking the commutator of the original Hamiltonian with the Euler angle, and by choosing an explicit expression for the Euler angle in terms of space-fixed particle coordinates. The intrinsic component in the transformed Hamiltonian is then the difference between the original Hamiltonian and the rotational component. The nuclear wave function is chosen as the product of an intrinsic function and an eigenfunction of the angular momentum operator (as in the unified rotational model). The Hamiltonian and Schrodinger equation for the intrinsic system become functions of the angular-momentum quantum number and intrinsic operators that are expressed in terms of space-fixed particles coordinates and momenta. The intrinsic Schrodinger equation is then reduced to that of a one-body operator using Hartree–Fock mean-field approximation. The intrinsic mean-field Hamiltonian is chosen to be an anisotropic harmonic oscillator Hamiltonian, and the Hartree–Fock mean-field equation is unitarily transformed to an equation resembling that of the CR but with oscillator frequencies and angular velocity that are microscopically and quantum mechanically determined. The unitary transformation is selected such that the model predicts the kinematic rigid-body moment of inertia, as does the CR when self-consistency condition is used.PACS Nos.: 21.60.Ev, 21.60.Fw, 21.60.Jz
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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 | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.002 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.004 | 0.001 |
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".