This synthesis keeps the logical route of Chapter 7 visible: physical interactions motivate the addition problem; the local and coupled bases organize the states; Clebsch-Gordan coefficients connect the bases; the same machinery is then used in the Zeeman effect and in Hund's rules.
The chapter begins with Hamiltonians containing more than one angular momentum: exchange interactions, spin-orbit coupling, local anisotropy and the Zeeman interaction.
Once the Hamiltonian contains two angular momenta, it becomes useful to compare the local basis with the coupled basis.
The local basis keeps the individual projections \(m_1\) and \(m_2\) visible:
The coupled basis keeps the total angular momentum and its projection visible:
The coupled basis must preserve the Hilbert-space dimension. For two angular momenta, the allowed values are
The projection condition is
For more than two angular momenta, sequential and non-sequential pairwise constructions are possible. Detailed \(3+\) coupling constructions remain in Chapter 7.
Tensor products expand local operators into the total Hilbert space:
Clebsch-Gordan coefficients then form the change-of-basis matrix \(\hat U\), and operators transform as
| Chapter step | Main object | Typical exercise action |
|---|---|---|
| Interactions | \(\hat H\) with two angular momenta | identify the angular momenta and the physical interaction |
| Commutation relations | local and coupled commuting sets | choose the basis compatible with the observables |
| Coupled vectors | \(|l_1,l_2,l,m\rangle\), \(|l_{12},l_{13},l,m\rangle\), \(|l_{12},l_{34},l,m\rangle\) | list allowed intermediate and total values |
| Hilbert expansion | tensor products with \(1^{(n)}\) | put operators and vectors in the same total dimension |
| Clebsch-Gordan coefficients | \(\langle l_1,l_2,m_1,m_2|l_1,l_2,l,m\rangle\) | convert local vectors into coupled vectors |
| Applications | \(\vec{\hat\mu}_J\), \(g_J\), Hund rules | read magnetic-moment projections and atomic term symbols |
Start with the Hamiltonian, decide whether local or coupled labels expose the physics, expand operators when several Hilbert subspaces are present, use Clebsch-Gordan coefficients for the basis change, and then read the physical result in the coupled basis.
The app keeps the central equations and representative examples; extended coupling schemes, long calculations and additional examples remain in Chapter 7.
This page supports compact guided exercises on: the complete Chapter 7 route from angular-momentum interactions to coupled-basis applications.
Use these anchors only within the material introduced on this page.