--- title: "Angular Momentum from Its Commutation Relations" description: "Hermitian angular-momentum operators obey commutation relations that determine finite ladders of states. Raising and lowering operators change the projection quantum number by one. The allowed j value" canonical: https://lightmysky.com/learn/science/angular-momentum-from-its-commutation-relations-mt_u0y20XJh-v source: https://lightmysky.com/learn/science/angular-momentum-from-its-commutation-relations-mt_u0y20XJh-v.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`. # Angular Momentum from Its Commutation Relations Hermitian angular-momentum operators obey commutation relations that determine finite ladders of states. Raising and lowering operators change the projection quantum number by one. The allowed j values are nonnegative integers or half-integers; orbital angular momentum uses integers. Subject: Science · Area: Quantum & Modern Physics · Ages 22 to 24 Page: https://lightmysky.com/learn/science/angular-momentum-from-its-commutation-relations-mt_u0y20XJh-v ## Ready when they can - Builds the ladder of states from the commutators without solving a differential equation - Explains why the eigenvalues step by one and may be half-integer - Matches the algebraic result to the spherical harmonics found the analytic way ## Lesson: Ladders built from commutators Start with Hermitian angular-momentum operators satisfying [Jx,Jy]=i hbar Jz and cyclic versions. Define J²=Jx²+Jy²+Jz² and J±=Jx±iJy. J² commutes with each component. Label joint eigenstates of J² and Jz by |j,m>, with eigenvalues hbar²j(j+1) and hbar m. The component commutators start the algebraic derivation without an orbital differential equation. The j=0 state has all components sharp. The relation [Jz,J±]=±hbar J± fixes the size of a nonzero ladder step. Acting on |j,m>, JzJ± gives hbar(m±1) times the new vector. Thus J+ raises m by one and J- lowers it by one. The change in the Jz eigenvalue is ±hbar, not ±hbar/2. At an endpoint, the relevant ladder action gives zero instead of a new state. † means adjoint. Hermiticity gives J+†=J-. Thus ||J±|j,m>||²=. Expand J∓J±=J²-Jz²∓hbar Jz. In normalized states, insert the eigenvalues to get hbar²[j(j+1)-m(m±1)]. Norms cannot be negative. J+ annihilates the top m=j; J- annihilates the bottom m=-j. These endpoints and unit steps determine a finite ladder's projections, without an orbital differential equation or a line spectrum. **Example.** To list a ladder, start at m=j, subtract one repeatedly, and stop at m=-j. Include both endpoints. There are 2j steps and 2j+1 states. For j=1, the list is 1,0,-1 and the count is three. Apply the same listing rule to half-integer j; successive entries still differ by one. The number of steps from j to -j is 2j, so 2j must be a nonnegative integer. Thus j may be integer or half-integer. Orbital angular momentum has integer j, usually written l. Intrinsic spin may be integer or half-integer. An electron has spin 1/2; an integer-only rule would exclude its observed two spin projections. For orbital motion, spherical harmonics Y_lm have L² and Lz eigenvalues hbar²l(l+1) and hbar m, with integer l and m=-l,...,l. They realize the integer orbital ladders. They do not describe intrinsic spin states or disprove half-integer representations. Intrinsic spin is a different representation of the same angular-momentum algebra. **Recap.** Commutators build ladders in unit steps, giving integer and half-integer angular momentum. *Original lesson, practice questions and flashcards authored for this dataset.* ## Practice 8 questions on this page, each with its working shown. ## Needs first - [Orbital Angular Momentum and the Quantum Numbers](https://lightmysky.com/learn/science/orbital-angular-momentum-and-the-quantum-numbers-mt_OXlbat0vat) - [Commutators, Compatible Observables and the General Uncertainty Relation](https://lightmysky.com/learn/science/commutators-compatible-observables-and-the-general-uncertainty-relation-mt_TKoUwu74K9) ## Optional connections - [Electron Spin and the Stern-Gerlach Experiment](https://lightmysky.com/learn/science/electron-spin-and-the-stern-gerlach-experiment-mt_HVFLoNe6wk) ## Opens up - [Adding Angular Momenta and the Coupled Basis](https://lightmysky.com/learn/science/adding-angular-momenta-and-the-coupled-basis-mt_eSBNPKm9C4) - [The Klein-Gordon and Dirac Equations](https://lightmysky.com/learn/science/the-klein-gordon-and-dirac-equations-mt_njCW9OEevW)