The Zeeman Effect (Perturbative)

A magnetic perturbation

An atom in a uniform magnetic field $\vec{B} = B\,\hat{z}$ feels a perturbation from the coupling of its magnetic moment to the field. The electron has both orbital and spin angular momentum, each contributing a magnetic moment, so

\hat{H}'_Z = \frac{\mu_B}{\hbar}\,(\hat{L}_z + g_s \hat{S}_z)\,B, \qquad \mu_B = \frac{e\hbar}{2m_e},

where $\mu_B$ is the Bohr magneton and $g_s \approx 2$ is the electron spin g-factor. The factor of 2 in front of spin is what makes the Zeeman effect richer than a naive "moment proportional to total angular momentum."

Weak field: perturb the fine-structure states

Whether $\hat{H}'_Z$ is treated as the small term depends on its size relative to the fine structure (the next lesson). In the weak-field (anomalous Zeeman) regime, fine structure dominates, so the good states are the fine-structure eigenstates $|n, l, j, m_j\rangle$ of total angular momentum $\vec{J} = \vec{L} + \vec{S}$ . We treat the Zeeman term as a perturbation on top.

Because $\hat{L}_z + 2\hat{S}_z = \hat{J}_z + \hat{S}_z$ is not simply proportional to $\hat{J}_z$ , we use the projection theorem to evaluate its expectation value within a fixed- $j$ multiplet. The result is the clean formula

E^{(1)}_Z = \mu_B\, g_J\, m_j\, B,

with the Landé g-factor

\boxed{\,g_J = 1 + \frac{j(j+1) + s(s+1) - l(l+1)}{2\,j(j+1)}\,.}

Reading the splitting

The energy depends linearly on $m_j$ , so a level of total angular momentum $j$ splits into $2j+1$ equally spaced sublevels separated by $\mu_B g_J B$ . The Landé factor $g_J$ sets the spacing per unit field and differs from term to term:

For a pure spin state ( $l = 0$ , $j = s = 1/2$ ): $g_J = 2$ .
For a pure orbital state ( $s = 0$ , $j = l$ ): $g_J = 1$ .
For hydrogen $^2P_{3/2}$ ( $l = 1$ , $s = 1/2$ , $j = 3/2$ ): substituting gives $g_J = 1 + \tfrac{3.75 + 0.75 - 2}{2(3.75)} = 1 + \tfrac{1}{3} = \tfrac{4}{3}$ .

The four sublevels $m_j = -\tfrac32, -\tfrac12, +\tfrac12, +\tfrac32$ are then spaced by $\tfrac{4}{3}\mu_B B$ .

Try it

Compute the Landé g-factor $g_J$ for the hydrogen $^2P_{3/2}$ term ( $l = 1$ , $s = 1/2$ , $j = 3/2$ ) using the boxed formula, and return it as a number.

Run your code to see the quantum state.