The Uncertainty Tradeoff at Everyday Atomic Scale
1 · Predict
At the scale of a single atom (roughly a nanometre), how does the minimum momentum uncertainty compare across a range of confinement sizes?
- Δp_min is inversely proportional to Δx — doubling the confinement region halves the minimum momentum uncertainty.
- Δp_min is directly proportional to Δx.
- They aren't related in a simple way.
2 · Set Up
- Open the position-momentum-tradeoff preset and press Reset. This preset explores confinement near atomic (nanometre) scale.
- Enable the minimum-momentum-uncertainty readout.
- Set the confinement region Δx for each trial and record the minimum possible momentum uncertainty.
3 · Collect Data
| Confinement region Δx (nm) | Momentum uncertainty Δp (×10⁻²⁵ kg·m/s) |
|---|---|
| 0.50 | |
| 1.00 | |
| 2.00 |
Plot Δp (y-axis) against 1/Δx (x-axis) for your three trials. Is the line straight through the origin?
4 · Analyze
- For one trial, compute Δp_min = ℏ/(2Δx) using ℏ = 1.055×10⁻³⁴ J·s. Compare to the table.
- Confirm your data shows a clean inverse proportionality: doubling Δx exactly halves Δp_min. Explain why this specific inverse relationship (rather than, say, inverse-square) follows directly from the uncertainty principle's Δx·Δp ≥ ℏ/2 form.
5 · Extend
- Early (pre-quantum) models imagined electrons orbiting a nucleus on precise, well-defined paths — like tiny planets. Explain why the uncertainty principle at atomic scale makes such a definite-position, definite-momentum orbit fundamentally impossible, motivating the 'electron cloud' picture instead.
- Compare your Δp values here (atomic scale, nanometre confinement, Δx ~ 1 nm) to the gamma-confinement experiment's (nuclear scale, femtometre confinement, Δx ~ 1 fm) — six orders of magnitude smaller. Explain why confining a particle that much more tightly produces a correspondingly larger minimum momentum uncertainty, and why the exact ratio between any two specific trials also depends on which Δx values you picked in each experiment, not on the scale difference alone.
The Physics Behind This Experiment
Uncertainty Principle Scaling
Because Δp_min = ℏ/(2Δx), the minimum momentum uncertainty scales as exactly 1/Δx — tightening the position confinement by any factor loosens the momentum certainty by that same factor.
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