---
title: "Adding Angular Momenta and the Coupled Basis"
description: "Two angular momenta combine into a total whose allowed values run from the difference to the sum, and the coupled basis is a change of basis from the product one. Atomic fine structure and nuclear spi"
canonical: https://lightmysky.com/learn/science/adding-angular-momenta-and-the-coupled-basis-mt_eSBNPKm9C4
source: https://lightmysky.com/learn/science/adding-angular-momenta-and-the-coupled-basis-mt_eSBNPKm9C4.md
retrieved: 2026-09-12
---

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# Adding Angular Momenta and the Coupled Basis

Two angular momenta combine into a total whose allowed values run from the difference to the sum, and the coupled basis is a change of basis from the product one. Atomic fine structure and nuclear spin states are read off this way.

Subject: Science · Area: Quantum & Modern Physics · Ages 23 to 24
Page: https://lightmysky.com/learn/science/adding-angular-momenta-and-the-coupled-basis-mt_eSBNPKm9C4

## Ready when they can

- Lists the total angular momentum values available from two given ones
- Counts states in both bases and checks the totals agree
- Writes a coupled state as a combination of product states for a two-spin case

## Lesson: Adding two spins into a total

Two angular momenta combine into a total whose values run from the difference to the sum, step by step. The coupled basis is simply a change of basis from the product picture, where you label each part alone, to the total picture.

Both pictures hold the same states, so counting must agree. Multiply the counts in the product picture, then add up what each total value carries in the coupled picture, and check the two totals match.

**Example.** Two spin-half particles each carry two states, so the product picture holds four. The totals run from zero to one: the total one carries three states and the total zero carries one. Three plus one is four, so both bases agree.

**Tip.** Read atomic fine structure and nuclear spin states this way: list the allowed totals, count what each carries, and write each coupled state as a combination of product states.

**Recap.** Totals run from difference to sum, both bases count the same states, and two spin-halves give three plus one.

## Practice

8 questions on this page, each with its working shown.

## Needs first

- [The Exclusion Principle and Multi-Electron Atoms](https://lightmysky.com/learn/science/the-exclusion-principle-and-multi-electron-atoms-mt_4tLa-3Av0A)
- [Angular Momentum from Its Commutation Relations](https://lightmysky.com/learn/science/angular-momentum-from-its-commutation-relations-mt_u0y20XJh-v)

## Opens up

- [Identical Particles and the Symmetry of a Many-Body State](https://lightmysky.com/learn/science/identical-particles-and-the-symmetry-of-a-many-body-state-mt_hns-eEGeHU)