Modern Physics
Position–momentum tradeoff
Wide Δx vs low K — explore the tradeoff between position and momentum spread.
Position–momentum tradeoff — interactive Modern Physics simulation. Wide Δx vs low K — explore the tradeoff between position and momentum spread. Free browser-based virtual physics lab with live SI measurements and a guided tutorial.
Position–momentum
Wide Δx vs low K — explore the tradeoff between position and momentum spread.
With a wide position box (Δx ≈ 1 nm) and low electron energy (small K), momentum spread Δp is modest — the tradeoff between knowing where the particle is and how fast it moves. Compare with the tight heisenberg-electron preset: narrowing Δx forces Δp up so the uncertainty product stays above ℏ/2.
- ΔxΔp ≥ ℏ/2
- Spread tradeoff
Investigation brief
Plan the question before you open the lab
The brief mirrors the prerendered page: driving question, competing predictions, variable roles, governing laws, setup, analysis and extension prompts remain visible and in this order.
Driving question
At the scale of a single atom (roughly a nanometre), how does the minimum momentum uncertainty compare across a range of confinement sizes?
Predictions to weigh
- 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.
Variable roles
What you set:
- Confinement region Δx (nm)
What you measure:
- Momentum uncertainty Δp (×10⁻²⁵ kg·m/s)
How the investigation runs
- 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.
Governing equation
Uncertainty Principle Scaling — Δx·Δp ≥ ℏ/2
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.
What the printable worksheet asks students to work out
- 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.
Where this shows up beyond the lab
- 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.
- AP Physics 2 — Unit 15: Modern Physics
- IB Physics — E.2 Quantum physics
- General High School Physics — Modern physics intro
- NGSS High School Physics — Wave-particle duality of light
- Welcome to Position Momentum Tradeoff
- Select the uncertainty setup
- Press Play
- Sharper position, broader p
- Open the Properties panel
- You did it!
Open the interactive simulation to build the scene, press Play, and explore with live measurements and a guided tutorial.