--- title: "Orbital Angular Momentum and the Quantum Numbers" description: "Angular momentum is quantised in both size and direction, so only certain projections along a chosen axis are allowed. The shapes labelled s, p and d are the angular solutions drawn out." canonical: https://lightmysky.com/learn/science/orbital-angular-momentum-and-the-quantum-numbers-mt_OXlbat0vat source: https://lightmysky.com/learn/science/orbital-angular-momentum-and-the-quantum-numbers-mt_OXlbat0vat.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`. # Orbital Angular Momentum and the Quantum Numbers Angular momentum is quantised in both size and direction, so only certain projections along a chosen axis are allowed. The shapes labelled s, p and d are the angular solutions drawn out. Subject: Science · Area: Quantum & Modern Physics · Ages 21 to 22 Page: https://lightmysky.com/learn/science/orbital-angular-momentum-and-the-quantum-numbers-mt_OXlbat0vat ## Ready when they can - States the allowed values of orbital angular momentum and its projection for a given level - Connects the angular quantum numbers to the shapes of the orbitals - Relates the orbital magnetic moment to the splitting of lines in a magnetic field ## Lesson: Orbits with set tilts Solving hydrogen quantises the electron circulation. The orbital number l fixes the magnitude of orbital angular momentum at the square root of l times l plus 1, times h bar. Only whole values l equals 0, 1, 2 and so on occur, which is why both the size of circulation and its allowed tilts come in steps. The magnetic number m fixes the projection on one chosen axis at m times h bar, with m running from minus l to plus l in whole steps. That gives 2l plus 1 tilts for each l. For n equals three, l is zero, one, or two, so the largest projection is two h bar and l equals two holds five tilts. **Example.** The angular shapes labelled s, p, and d are these rules drawn out: s is round for l equals 0, p is a dumbbell for l equals 1, d is cloverlike for l equals 2. A circling electron is a tiny current loop, so its magnetic moment stays locked to its angular momentum: bigger circulation means a stronger magnet. Because the moment is quantised, a magnetic field shifts each sublevel by m times one fixed step, so a level splits into 2l plus 1 sublevels. Zeeman splitting proves the tilts are real. Ignoring spin, a jump changes m by at most one, so each line splits into three, whatever l is. **Recap.** Size and tilt both come in steps, and magnets reveal the steps as split lines. ## Practice 8 questions on this page, each with its working shown. ## Needs first - [Separating the Schrödinger Equation for Hydrogen](https://lightmysky.com/learn/science/separating-the-schrodinger-equation-for-hydrogen-mt_Yk2qHB0_v2) ## Optional connections - [Angular Momentum and Its Conservation](https://lightmysky.com/learn/science/angular-momentum-and-its-conservation-mt_9v3S9-rhwc) ## Opens up - [The Nuclear Shell Model and the Magic Numbers](https://lightmysky.com/learn/science/the-nuclear-shell-model-and-the-magic-numbers-mt_1tiXMm0sA2) - [Electron Spin and the Stern-Gerlach Experiment](https://lightmysky.com/learn/science/electron-spin-and-the-stern-gerlach-experiment-mt_HVFLoNe6wk) - [Angular Momentum from Its Commutation Relations](https://lightmysky.com/learn/science/angular-momentum-from-its-commutation-relations-mt_u0y20XJh-v)