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) (×10⁻²⁵ kg·m/s) |
|---|---|
| 0.5 | |
| 1 | |
| 2 |
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) to the gamma-confinement experiment's (nuclear scale, femtometre confinement, five orders of magnitude smaller Δx). Explain why confining a particle 100,000× more tightly produces a 100,000× larger minimum momentum uncertainty.
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.