Put an atom in a magnetic field and each spectral line splits into several — Zeeman saw it in 1896. The field adds $\mu_{B}B(\hat{L}_{z}+2\hat{S}_{z})/\hbar$ to the Hamiltonian, and how it splits the levels depends on its strength against the spin–orbit coupling. In a weak field the spin–orbit coupling wins, the good states are those of definite $j$ and $m_{j}$, and each level splits into $2j+1$ equally spaced sublevels, $g_{J}\mu_{B}Bm_{j}$ from the centre. The Landé factor $g_{J}$ records how much of $\mathbf{J}$ is spin: it is 1 when $\mathbf{J}$ is all orbit and 2 when it is all spin. Because the spin’s $g$ is 2 and not 1, the pattern depends on $j$ and $l$: this was the anomalous Zeeman effect before spin was known. In a strong field the field wins, the good states are those of definite $m_{l}$ and $m_{s}$, and the shifts are $\mu_{B}B(m_{l}+2m_{s})$: the Paschen–Back effect.

Two dancers holding hands spin as a pair, and a gentle push acts on the pair as a whole; a hard shove separates them, and each answers for itself. A weak field acts on the coupled $\mathbf{J}$, a strong one on $\mathbf{L}$ and $\mathbf{S}$ apart.

Hydrogen’s $2p$ levels in a weak field, to first order: $2p_{3/2}$ above splits into four sublevels with $g_{J}=\tfrac43$, and below into two with . Energies are in units of the spin–orbit gap between the two levels, the field as in the same units; the straight lines hold while the shifts stay small beside the gap.

An atom in a field $\mathbf{B}$ along $z$ gains $\hat{V}=\frac{\mu_{B}B}{\hbar}(\hat{L}_{z}+2\hat{S}_{z})$. In a weak field, small beside the spin–orbit splitting, a level splits as

• The $2j+1$ sublevels are equally spaced, $g_{J}\mu_{B}B$ apart: the field removes the degeneracy in completely.

$\hat{L}_{z}+2\hat{S}_{z}=\hat{J}_{z}+\hat{S}_{z}$. Within one level of given , the Wigner–Eckart theorem makes every vector operator proportional to , so there acts as a number times . Taking the dot product with finds from , which the squares of , and give.

$\hat{\mathbf{J}}=\hat{\mathbf{L}}+\hat{\mathbf{S}}$.

Both are vector operators: by the Wigner–Eckart theorem their elements within one level differ only in the reduced element.

Insert $\sum_{m'}\ket{j\,m'}\bra{j\,m'}$ between and ; does not take a state out of the level.

Square $\hat{\mathbf{L}}=\hat{\mathbf{J}}-\hat{\mathbf{S}}$.

$\bra{j\,m}\hat{J}_{z}+\hat{S}_{z}\ket{j\,m}=(1+\Lambda)\hbar m$.

In a weak field a level $j$ splits into $2j+1$ sublevels, $g_{J}\mu_{B}Bm_{j}$ from its centre, with set by , and .

The field removes the degeneracy in $m_{j}$ completely: a level $j$ splits into $2j+1$ sublevels.

Without spin, $s=0$, the Landé factor is 1 for every level, and every line splits into the same three.

Within one level, the Wigner–Eckart theorem makes the spin act as a fixed multiple of $\mathbf{J}$.

Great work! You now know how to:

• derive the Landé factor from the Wigner–Eckart theorem
• find how a level splits in a weak field
• tell the weak-field pattern from the Paschen–Back one