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Position Space Wave Packet |Ψ(x)|Spread Δx = 3.0Momentum Probability Density |Φ(p)|²Spread Δp = 1.7

Live Telemetry & Summary

Observe the Fourier relationship. When you narrow the Position Spread (Δx) to localize the particle, the wave packet compresses in space, causing the Momentum Spread (Δp) to expand. Knowing exactly where the particle is makes its momentum highly uncertain!

Position Uncertainty (Δx):3.00
Momentum Uncertainty (Δp):1.667
Uncertainty Product (Δx · Δp):5.00 (≥ ħ/2 = 5.0)

Variable Adjuster

Position Spread (Δx)3
17

Heisenberg Uncertainty

UNCER

The Heisenberg Uncertainty Principle states that it is impossible to simultaneously measure a particle's position (x) and momentum (p) with infinite precision. A highly localized wave packet (narrow position spread Δx) requires a superposition of many different wavelengths, resulting in a wide spread of momentum states (large Δp).

Δx⋅Δp≥ℏ2\Delta x \cdot \Delta p \ge \frac{\hbar}{2}

Whiteboard Solver Steps

Step 1

Heisenberg Uncertainty Relation

The fundamental limit to the precision with which certain pairs of physical properties of a particle can be known.

Δx⋅Δp≥ℏ2\Delta x \cdot \Delta p \ge \frac{\hbar}{2}
Step 2

Minimum Momentum Uncertainty (Δp)

As the position of the particle becomes more localized (smaller Δx), its momentum becomes highly uncertain (larger Δp).

Δp≥ℏ2Δx=10.02⋅3.00=1.667\Delta p \ge \frac{\hbar}{2\Delta x} = \frac{10.0}{2 \cdot 3.00} = 1.667