Stereographic projection to and from the Bloch sphere: Visualizing solutions of the Bloch equations and the Bloch–Riccati equation
Bibliographic record
Abstract
Stereographic projection mapping is typically introduced to explain the point at infinity in the complex plane. After this brief exposure in the context of complex analysis, students rarely get an opportunity to fully appreciate stereographic projection mapping as an elegant and powerful technique on its own with many fruitful applications in the physical sciences. Here, using a classical description of nuclear magnetic resonance in the rotating frame, I show how stereographic projection mapping to and from the Bloch sphere can be used for visualizing solutions to Bloch's equation and the Bloch–Riccati equation, respectively. After developing the fundamentals of stereographic projection mapping using examples drawn from nuclear spin precession in the rotating frame, the method is then applied to visualizations of composite pulse excitation of a spin-1/2 system and to radiation damping in a system of isolated spins-1/2. In the case of the radiation-damped system, these visualizations provide particularly vivid illustrations of loxodromic Möbius transformation dynamics.
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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 | 0.001 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 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".