--- title: "Commutators, Compatible Observables and the General Uncertainty Relation" description: "Two quantities can be known together only when their operators commute, and the commutator sets a floor on the product of their spreads. Position and momentum are one case of a relation that holds for" canonical: https://lightmysky.com/learn/science/commutators-compatible-observables-and-the-general-uncertainty-relation-mt_TKoUwu74K9 source: https://lightmysky.com/learn/science/commutators-compatible-observables-and-the-general-uncertainty-relation-mt_TKoUwu74K9.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`. # Commutators, Compatible Observables and the General Uncertainty Relation Two quantities can be known together only when their operators commute, and the commutator sets a floor on the product of their spreads. Position and momentum are one case of a relation that holds for any pair of observables. Subject: Science · Area: Quantum & Modern Physics · Ages 22 to 23 Page: https://lightmysky.com/learn/science/commutators-compatible-observables-and-the-general-uncertainty-relation-mt_TKoUwu74K9 ## Ready when they can - Evaluates a commutator and states whether the two observables share an eigenbasis - Derives the uncertainty bound for a general pair from their commutator - Applies the relation to a pair other than position and momentum ## Lesson: When two quantities refuse to be sharp together The commutator of two operators A and B is AB minus BA. When it is zero the pair commutes, and commuting observables share a full set of eigenstates, so both values can be sharp at once. When it is nonzero the pair cannot be known together. Evaluating a commutator is therefore the compatibility test: zero means compatible, nonzero means a tradeoff is mandatory. **Example.** Position and momentum are the famous incompatible pair. A state sharp in momentum is spread across position, since definite wavelength means an extended wave. Pinning down position needs many wavelengths superposed, which wrecks momentum sharpness. Pin down momentum and the state spreads over position, and vice versa. Both sharp at once is forbidden. The general uncertainty relation covers every pair: the product of the two spreads is floored by half the size of the average commutator. Position and momentum are one case of this rule, coming from their own commutator. To test a new pair, such as spin components, compute their commutator and read off the floor the same way. **Tip.** Treat the wave-packet argument as the prototype and the commutator rule as its generalization. The same derivation from the inner product structure covers every pair, which is what makes the relation feel like one law rather than many coincidences. Whenever a pair surprises you, write down AB minus BA before guessing. **Recap.** The commutator decides compatibility, and its size floors the product of the two spreads. ## Practice 8 questions on this page, each with its working shown. ## Needs first - [The Uncertainty Principle from Wave Packets](https://lightmysky.com/learn/science/the-uncertainty-principle-from-wave-packets-mt_DPYnoGksRd) - [Observables as Hermitian Operators and Their Spectra](https://lightmysky.com/learn/science/observables-as-hermitian-operators-and-their-spectra-mt_TXCN8uiWqs) ## Opens up - [Unitary Time Evolution and the Heisenberg Picture](https://lightmysky.com/learn/science/unitary-time-evolution-and-the-heisenberg-picture-mt_3JsTc9VYj8) - [Angular Momentum from Its Commutation Relations](https://lightmysky.com/learn/science/angular-momentum-from-its-commutation-relations-mt_u0y20XJh-v) - [Poisson Brackets and Constants of the Motion](https://lightmysky.com/learn/science/poisson-brackets-and-constants-of-the-motion-mt_UFzJyZRdOI)