--- title: "The Uncertainty Principle from Wave Packets" description: "A localised wave packet combines a range of wave numbers. Position and momentum standard deviations obey ΔxΔp ≥ ħ/2, so narrowing position raises the momentum lower bound. This state-spread limit is s" 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: 2026-11-12 --- > **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 A localised wave packet combines a range of wave numbers. Position and momentum standard deviations obey ΔxΔp ≥ ħ/2, so narrowing position raises the momentum lower bound. This state-spread limit is separate from instrument error. 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 superposing waves with different wave numbers can make a localised packet - Estimates a momentum-spread lower bound from a stated position standard deviation - Distinguishes the prepared state's spread relation from instrument error and measurement disturbance ## Lesson: Why pinning down waves spreads them You have used |ψ|² to predict position probabilities. A plane-wave component with signed wave number k has momentum p = ħk. Its magnitude is constant across space, so it is not a normalised, localised state on the whole line. A packet combines different k values with chosen amplitudes and phases. They reinforce near its centre and mostly cancel farther away. Δx and Δp are standard deviations of position and momentum along the same axis, across identically prepared copies of a state. Each measures spread about its mean. For example, Δx = √(⟨x²⟩ - ⟨x⟩²), where brackets mean averages. For finite spreads, ΔxΔp ≥ ħ/2, with ħ = h/(2π). A smaller Δx raises the lower bound on Δp; the product need not equal that bound. **Example.** Rearrange: Δp_min = ħ/(2Δx). Use ħ = 1.056e-34 J s. If Δx = 6.00e-10 m, the bound is 8.80e-26 kg m/s. For a nonrelativistic electron of mass 9.11e-31 kg, Δp = mΔv along this axis. Dividing gives Δv ≥ 9.66e4 m/s, to three significant figures. These bounds do not specify the actual spreads. *(drawing: Heights show momentum lower bounds in units of ħ/(2L). These are not packet shapes or actual momentum spreads.)* The relation constrains the state before measurement. Instruments may add error or disturb a particle, but those effects are separate. Better equipment cannot prepare a state below ħ/2. A Gaussian packet without a quadratic position-dependent phase attains equality. Other states can have larger products. Halving Δx doubles Δp_min = ħ/(2Δx). A hypothetical nucleus-sized Δx therefore gives a larger bound than an atom-sized Δx. Actual spreads can lie above their bounds, so they need not change inversely between two arbitrary states. To check a proposed state, multiply its spreads and compare with 0.50ħ. A product of 0.75ħ passes this check without attaining equality. **Recap.** Position and momentum standard deviations obey ΔxΔp ≥ ħ/2. Narrower position raises the momentum lower bound. Instrument error and disturbance are separate. *Lesson, questions and flashcards written for this dataset using standard wave-packet and uncertainty-relation methods.* ## 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)