---
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: 2026-09-12
---

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# 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, and these tilts decide how lines split in a magnet. 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, atoms in a magnetic field split their lines, one tilt per extra line. That Zeeman splitting is direct proof the quantum numbers are real. Counting the split lines tells you l, and their spacing measures the moment.

**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

- [Angular Momentum and Its Conservation](https://lightmysky.com/learn/science/angular-momentum-and-its-conservation-mt_9v3S9-rhwc)
- [Separating the Schrödinger Equation for Hydrogen](https://lightmysky.com/learn/science/separating-the-schrodinger-equation-for-hydrogen-mt_Yk2qHB0_v2)

## 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)