--- title: "The Uncertainty Principle from Wave Packets" description: "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." canonical: https://lightmysky.com/learn/science/the-uncertainty-principle-from-wave-packets-mt_DPYnoGksRd source: https://lightmysky.com/learn/science/the-uncertainty-principle-from-wave-packets-mt_DPYnoGksRd.md retrieved: 2026-09-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # 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. Subject: Science · Area: Quantum & Modern Physics · Ages 19 to 20 Page: https://lightmysky.com/learn/science/the-uncertainty-principle-from-wave-packets-mt_DPYnoGksRd ## Ready when they can - 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 ## Lesson: Why pinning down waves spreads them 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. **Example.** 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. **Tip.** 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. **Recap.** Localising a particle costs wavelengths, each carrying momentum, so tight position means wide momentum by h-bar over two. ## Practice 8 questions on this page, each with its working shown. ## Needs first - [Normalisation and Expectation Values](https://lightmysky.com/learn/science/normalisation-and-expectation-values-mt_6Y3DZ8P1qm) ## Opens up - [Fourier Analysis and the Wave Packet](https://lightmysky.com/learn/science/fourier-analysis-and-the-wave-packet-mt_1ZaojWUZEw) - [The Quantum Harmonic Oscillator and Zero-Point Energy](https://lightmysky.com/learn/science/the-quantum-harmonic-oscillator-and-zero-point-energy-mt_HyDxmqyReX) - [Commutators, Compatible Observables and the General Uncertainty Relation](https://lightmysky.com/learn/science/commutators-compatible-observables-and-the-general-uncertainty-relation-mt_TKoUwu74K9) - [The Time-Independent Schrödinger Equation](https://lightmysky.com/learn/science/the-time-independent-schrodinger-equation-mt_zeBVPery5T)