Home / Topics / Quantum Mechanics
Addition of angular momenta
Advanced
Combining spins and orbits.
Combining spins and orbital momenta gives total J ranging from |l − s| to l + s. Controls the multiplet structure of spectra.
Two spin-½ particles: J = 0 (singlet) or 1 (triplet).
Use Clebsch–Gordan coefficients.
In more detail
Adding angular momenta in quantum mechanics combines orbital and spin contributions into a total angular momentum, with allowed values given by the triangle rule.
Real-world uses
- Spectroscopists use total angular momentum to label fine-structure levels.
- Nuclear shell models combine angular momenta of nucleons.
- Physicists combine the spins and orbital angular momenta of electrons to predict the fine structure of atomic lines.
Open the interactive lesson, with simulation and quiz →
Frequently asked questions
What is the triangle rule?
The total j runs from |l − s| to l + s in steps of one.
Why does it matter?
It determines the fine structure of atomic energy levels.
Why does the total angular momentum matter?
It determines the energy shifts due to spin–orbit coupling, which split spectral lines slightly.
Related topics in Quantum Mechanics
Fermions and bosons · Identical particles · Angular momentum in QM · Spin–orbit coupling · Zeeman effect · Stark effect