The application uses the addition of spin and orbital angular momenta to write the magnetic moment of an atom and the Zeeman Hamiltonian. The key step is the projection of \(\vec{\hat\mu}\) along \(\vec{\hat J}\).
The Zeeman Hamiltonian is first written as
Using dimensionless angular momenta, the section rewrites it as
The total magnetic moment is therefore
The total angular momentum is
Only the component of \(\vec{\hat\mu}\) along \(\vec{\hat J}\) contributes to the magnetic energy. The projection gives
By analogy with \(\vec{\hat\mu}_S\) and \(\vec{\hat\mu}_L\), write
In the coupled basis \(|s,l,j,m_j\rangle\), the Landé factor becomes
The Zeeman Hamiltonian can be written as
The magnetic moment is not parallel to the total angular momentum because the spin and orbital parts have different gyromagnetic factors. The energy keeps the projection of \(\vec{\hat\mu}\) along \(\vec{\hat J}\).
This page supports compact guided exercises on: magnetic moment, Zeeman Hamiltonian and the Lande factor.
- Use from this page: the definitions, physical setup, highlighted equations and quantum-number restrictions shown here.
- Keep in Chapter 7: the complete demonstrations, long examples, historical notes and extended exercise solutions.
- Boundary: do not introduce notation, Hamiltonians, coupling schemes or selection rules outside the sequence presented here.
Use these anchors only within the material introduced on this page.
- Focus: magnetic moment, Zeeman Hamiltonian and the Lande factor.
- Conceptual check: identify the basis or angular-momentum interaction before applying the equation.
- Key equation 1:
\[\vec{\hat\mu}_J=-g_J\mu_B\vec{\hat J}\]
- Key equation 2:
\[g\equiv g_J=1+\frac{j(j+1)-l(l+1)+s(s+1)}{2j(j+1)},\qquad \hat H_Z=g_J\mu_B\vec{\hat J}\cdot\vec B\]
- Typical task: apply one central equation and state what basis, quantum numbers or physical consequence it uses.