The Difference Between a Probability and a Probability Density
Bibliographic record
Abstract
Learning introductory quantum physics is challenging, in part due to the different paradigms in classical mechanics and quantum physics. Classical mechanics is deterministic in that the equations of motion and the initial conditions fully determine a particle’s trajectory. Quantum physics is an inherently probabilistic theory in that only probabilities for measurement outcomes can be determined. Prior to studying quantum physics, students will typically have little experience with probabilistic analyses of physical systems, and thus probability may be a conceptual hurdle for introductory quantum physics students. This article describes two interactive simulations developed as part of the QuVis Quantum Mechanics Visualization Project that aim to bridge the gap between classical mechanics and quantum physics using probabilistic analyses of classical systems. The simulations illustrate how a probability density can be obtained for two classical systems well known to students. The key learning goals of the simulations are to introduce students to probability densities and to help students distinguish between a probability and a probability density. The simulations build on previous work by Bao and Redish, who developed an activity that used pseudo-random video frames of a glider in harmonic motion to derive a classical probability density for this system, and a University of Washington quantum mechanics tutorial focusing on probability and probability density for a classical system. Interactive simulations allow students to easily carry out experiments and change variables that would be difficult to do with real equipment, and help students connect multiple representations by showing explicitly how they are linked. The simulations described here only require basic knowledge of algebra and classical mechanics. They run on touchscreen devices as well as desktop computers, and can be run in a standard web browser from the QuVis website or downloaded for offline use.
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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 | 0.003 | 0.025 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.005 |
| Scholarly communication | 0.004 | 0.010 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.003 | 0.004 |
| Insufficient payload (model declined to judge) | 0.011 | 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".