Wide Effectiveness of a Sine Basis for Quantum-Mechanical Problems in<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:math>Dimensions
Why this work is in the frame
A frame that forgets how it found something cannot be audited. These are the routes that admitted this work.
Bibliographic record
Abstract
It is shown that the spanning set for<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:msup><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">[</mml:mo><mml:mn>0,1</mml:mn><mml:mo stretchy="false">]</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:math>provided by the eigenfunctions<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M3"><mml:mo stretchy="false">{</mml:mo><mml:msqrt><mml:mn>2</mml:mn></mml:msqrt><mml:mi>sin</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mi>π</mml:mi><mml:mi>x</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:msubsup><mml:mrow><mml:mo stretchy="false">}</mml:mo></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>∞</mml:mi></mml:mrow></mml:msubsup></mml:math>of the particle in a box in quantum mechanics provides a very effective variational basis for more general problems. The basis is scaled to<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M4"><mml:mo stretchy="false">[</mml:mo><mml:mi>a</mml:mi><mml:mo>,</mml:mo><mml:mi>b</mml:mi><mml:mo stretchy="false">]</mml:mo></mml:math>, where<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M5"><mml:mrow><mml:mi>a</mml:mi></mml:mrow></mml:math>and<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M6"><mml:mrow><mml:mi>b</mml:mi></mml:mrow></mml:math>are then used as variational parameters. What is perhaps a natural basis for quantum systems confined to a spherical box in<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M7"><mml:mrow><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:math>turns out to be appropriate also for problems that are softly confined by U -shaped potentials, including those with strong singularities at<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M8"><mml:mi>r</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math>. Specific examples are discussed in detail, along with some bound<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M9"><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:math>-boson systems.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.001 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.001 | 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 it