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) 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.
- 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.