The Uncertainty Principle from Wave Packets
Localising a particle needs many wavelengths added together, so a narrow position spread forces a wide momentum spread. The limit is a property of waves, not a comment on clumsy apparatus.
What a learner can do afterwards
- Explains how adding waves of many wavelengths makes a localised packet
- Estimates a minimum momentum spread from a stated position spread
- Argues why the limit is not caused by disturbance from the measuring instrument
1 · Read
A single wavelength sprawls everywhere, so building a localised packet means adding many wavelengths that cancel everywhere except one spot. But each wavelength carries its own momentum, since momentum follows from wavelength. A tight position spread therefore forces a wide momentum spread, and the limit reads delta x times delta p at least h-bar over two.
Knowing an electron's speed to 0.001 m/s leaves its position uncertain by about 5.8 cm, which is huge on atomic scales. The same speed precision for a 6.0 kg bowling ball leaves only about 8.8e-33 m, far below any detector. Squeezing an electron into an atom-sized 0.1 nm box predicts tens of eV (about 38 eV), the same scale as the real ionisation energy near 10 eV.
This limit lives in the quantum state, not in your tools. A better microscope cannot beat it, because the spread belongs to the packet itself. The bell-shaped Gaussian packet hits the minimum exactly, with delta x times delta p equal to h-bar over two.
Estimate minimum spreads by inverting the relation: delta p is at least h-bar over twice delta x. Smaller boxes mean larger momentum spreads, and shorter-lived states mean wider energy spreads. Never blame the apparatus for a spread the state itself demands.
Localising a particle costs wavelengths, each carrying momentum, so tight position means wide momentum by h-bar over two.
2 · Watch
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Where it sits
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.