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First Principles of PhysicsLimits & Uncertainty · Rule 30

Limits & Uncertainty

Rule 30Chapter V

You cannot know everything at once

Why this rule exists

Certain pairs of quantities — position and momentum, energy and time — are linked so that sharpening knowledge of one necessarily blurs the other. This is not a limit of instruments but a property of waves: a wave packet localized in space is built from a broad spread of wavelengths, hence momenta. Since quantum objects are described by such packets, the trade-off is built into what they are.

First Principles of Physics

In practice

Use Δx·Δp ≳ ℏ/2 to estimate the minimum spread you cannot escape. Given a confinement length, bound the momentum and thus a minimum kinetic energy — this sets ground-state energies and atomic sizes. Treat it as a floor on simultaneous precision, not merely on measurement, and stop looking for a state that beats it.

Example

Worked example — why atoms don't collapse✓

Confine an electron to a box of size Δx ≈ a.
Uncertainty gives a minimum momentum:

Δp ≳ ℏ / (2Δx) ≈ ℏ / (2a)

Kinetic energy KE ≈ (Δp)² / 2m ≈ ℏ² / (8m·a²)

Smaller a costs more energy — this balances
attraction and fixes the atom's size.

When it doesn't apply

The bound is negligible when actions are enormous compared to ℏ, which is why it never troubles everyday objects. It constrains only conjugate pairs; compatible observables can be known together exactly.

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