Introduction: Breaking the Degeneracy
The Zeeman effect is the splitting of spectral lines in a magnetic field. It breaks the \(m\)-degeneracy and was crucial evidence for angular momentum quantization.
The Interaction
A magnetic moment in field \(\vec{B}\) has energy:
\[H_Z = -\vec{\mu}\cdot\vec{B} = \frac{e}{2m}(\vec{L} + 2\vec{S})\cdot\vec{B}\]The factor of 2 for spin is the electron's g-factor.
Weak Field (Anomalous Zeeman)
When Zeeman energy << spin-orbit energy, \(j\) remains good quantum number:
\[E = g_j m_j \mu_B B\]where \(g_j\) is the Landé g-factor.
Strong Field (Paschen-Back)
When Zeeman >> spin-orbit, \(m_l\) and \(m_s\) are good individually:
\[E = (m_l + 2m_s)\mu_B B\]The Quantum Connection
Zeeman splitting is used in atomic clocks, MRI imaging, and astrophysics (measuring stellar magnetic fields from spectral line splitting). The pattern of splitting directly reveals the quantum numbers of atomic states.