# Zeeman Effect¶

## Zeeman Effect¶

The Zeeman effect is the effect of splitting of a spectral line into several components in the presence of a static magnetic field.

In the magnetic field

$$\large \hat{H}'_z = - (\vec{\mu}_l + \vec{\mu}_s) \cdot \vec{B}$$

where

• $\vec{\mu}_l = -\frac{e}{2m}\vec{L}$
• $\vec{\mu}_s = -\frac{e}{m}\vec{S}$

### For the weak-field,¶

We want eigenstates of $\hat{J}^2$ and $\hat{J}_z$, and

$$\large E'_z \ = \ \frac{e}{2m} \vec{B} \cdots \langle \vec{L} + 2 \vec{S} \rangle \ = \ \frac{e}{2m} \vec{B} \cdot \langle \vec{J} \rangle \left[ 1 + \frac{j(j+1) - l(l+1) + 3/4}{2j(j+1)} \right]$$

This splits the ground state into two levels for $m_j = \pm 1/2$.

### For the strong-field,¶

We want eigenstates of $\vec{L}_z$ and $\vec{S}_z$, and

$$\large E'_z \ = \ \mu_B B (m_l + 2m_s)$$

where

• $\mu_B = \frac{e\hbar}{2m}$: Bohr magneton