---
title: The Zeeman and Stark Effects
module: Approximation Methods for Bound States
moduleNumber: 10
lessonNumber: 3
order: 1003
summary: >
  An atom in an external field is a perturbation problem whose good basis
  depends on which interaction wins. A magnetic field competes with the
  internal spin–orbit coupling: the weak-field limit gives the anomalous Zeeman
  splitting set by the Landé g-factor, the strong-field limit gives the
  Paschen–Back pattern in the uncoupled basis, and the intermediate regime is a
  matrix diagonalization. An electric field gives a quadratic shift for the
  nondegenerate ground state and a linear splitting for the degenerate n = 2
  level.
topics: [Approximation Methods for Bound States]
sources:
  - book: Griffiths & Schroeter
    ref: "Ch. 7; §7.4 The Zeeman Effect (weak, strong, intermediate), §7.6 (Stark effect)"
  - book: Sakurai & Napolitano
    ref: "Ch. 5; §5.3 The Zeeman Effect and the competition with fine structure"
  - book: Cohen-Tannoudji, Diu & Laloë
    ref: "Complement D_XII — The Stark effect and static polarizability"
draft: false
---

Placing hydrogen in a static external field adds a perturbation that competes
with the atom's internal
[fine structure](/quantum-mechanics/approximation-methods/fine-structure-and-the-real-hydrogen-atom).
Which perturbation is treated as dominant sets the good zeroth-order basis, and
the answer changes with field strength. A magnetic field couples to the
magnetic moments of orbit and spin; an electric field couples to the charge
distribution's dipole. Both are textbook applications of the
[degenerate perturbation theory](/quantum-mechanics/approximation-methods/time-independent-perturbation-theory)
of the first lesson, and both turn on choosing the basis in which the total
perturbation is diagonal.

## The Zeeman Hamiltonian

An external magnetic field $\vec B = B\hat k$ couples to the orbital and spin
magnetic moments of the electron. With $\vec\mu_L = -(e/2m)\vec L$ and
$\vec\mu_S = -(e/m)\vec S$ (the spin moment carries the anomalous $g_s \approx 2$),
the interaction energy is

$$
\hat H_Z' = -(\vec\mu_L + \vec\mu_S)\cdot\vec B
= \frac{e}{2m}(\vec L + 2\vec S)\cdot\vec B
= \frac{\mu_B B}{\hbar}(\hat L_z + 2\hat S_z),
$$

where $\mu_B = e\hbar/2m = 5.788\times10^{-5}\ \text{eV/T}$ is the **Bohr
magneton**.[^codata] The factor $2$ on the spin, not $1$, is why the Zeeman
pattern is "anomalous": the naive classical prediction treats $\vec L$ and
$\vec S$ alike.

The physics is a competition of two energies: the Zeeman term, of size
$\mu_B B$, and the fine structure, of size $\alpha^4 mc^2 \sim 10^{-4}\ \text{eV}$.
For hydrogen the crossover field is about $1\ \text{T}$, so laboratory fields
span all three regimes.

$$
% caption: The Zeeman energy grows linearly with field while fine structure is
% fixed; their ratio sets three regimes, with the good basis switching from
% coupled to uncoupled as the field crosses the fine-structure scale.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[black, ->] (0,0) -- (6.2,0) node[right, black!70] {$B$};
\draw[black, ->] (0,0) -- (0,3.4) node[above, black!70] {energy};
% fine structure constant line
\draw[acc, very thick] (0,1.5) -- (6.0,1.5);
\node[anchor=south east, acc] at (6.0,1.5) {$E_{fs}$};
% zeeman linear
\draw[black, very thick] (0,0) -- (5.6,3.2);
\node[anchor=south, black!70] at (5.0,2.9) {Zeeman term};
% regimes
\node[anchor=north, black!70] at (1.2,-0.1) {weak};
\node[anchor=north, black!70] at (2.9,-0.1) {intermediate};
\node[anchor=north, black!70] at (5.0,-0.1) {strong};
\draw[black, dashed] (2.9,0) -- (2.9,1.5);
\end{tikzpicture}
$$

## The weak-field limit: the anomalous Zeeman effect

When $\mu_B B \ll E_{fs}$, fine structure dominates and the good basis is the
coupled $\ket{n,\ell,j,m_j}$. The Zeeman term is a perturbation on the
fine-structure levels, and its shift is the diagonal expectation
$E_Z^1 = \frac{\mu_B B}{\hbar}\braket{\hat L_z + 2\hat S_z}
= \frac{\mu_B B}{\hbar}\braket{\hat J_z + \hat S_z}$. Since $\braket{\hat J_z} = m_j\hbar$
is immediate, only $\braket{\hat S_z}$ remains. In the coupled basis $\vec S$ has
no definite $z$ component, but its expectation is fixed by the
**projection theorem**: within a fixed-$j$ multiplet, the expectation of any
vector operator is its projection along $\vec J$,

$$
\braket{\vec S} = \frac{\braket{\vec S\cdot\vec J}}{\braket{J^2}}\braket{\vec J},
\qquad
\vec S\cdot\vec J = \tfrac12(J^2 + S^2 - L^2).
$$

Carrying this through, $\braket{\hat L_z + 2\hat S_z} = g_J\, m_j\hbar$ with the
**Landé g-factor**

$$
g_J = 1 + \frac{j(j+1) - \ell(\ell+1) + s(s+1)}{2j(j+1)}.
$$

> **Theorem (Weak-field Zeeman splitting).** Each fine-structure level
> $\ket{n,\ell,j}$ splits into $2j+1$ equally spaced sublevels,
> $$E_Z^1 = \mu_B\, g_J\, B\, m_j, \qquad m_j = -j, \dots, +j,$$
> with spacing $\mu_B g_J B$ set by the Landé factor of that level.

The Landé factor encodes the different orbital and spin content of each level:
a pure orbital state ($s = 0$) has $g_J = 1$, a pure spin state ($\ell = 0$) has
$g_J = 2$, and mixed states fall between. Two levels of the same $j$ but
different $\ell$ split at different rates, which is how the anomalous pattern is
read off a spectrum.

$$
% caption: A vector-model reading of the Landé factor: L and S precess rapidly
% about the conserved J, so only their projections on J survive averaging, and
% the magnetic energy responds to that projection rather than to L and S
% separately.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
% J vertical
\draw[->, black, very thick] (0,0) -- (0,3.2) node[anchor=south] {$J$};
% L and S at angles about J
\draw[->, acc, very thick] (0,0) -- (0.9,2.4) node[anchor=west] {$L$};
\draw[->, black, very thick, dashed] (0,0) -- (-0.7,2.7) node[anchor=east] {$S$};
% precession hint
\draw[black] (0,2.55) ellipse (0.9 and 0.22);
\node[anchor=west, black!70] at (1.2,1.4) {$L, S$ precess about $J$};
\end{tikzpicture}
$$

> **Worked example.** The Landé factors of the $n = 2$ levels follow directly. For
> $2S_{1/2}$ ($\ell = 0, j = \tfrac12$), $g_J = 1 + (3/4 - 0 + 3/4)/(3/2) = 2$; for
> $2P_{1/2}$ ($\ell = 1, j = \tfrac12$), $g_J = 1 + (3/4 - 2 + 3/4)/(3/2) = 2/3$; for
> $2P_{3/2}$ ($\ell = 1, j = \tfrac32$), $g_J = 1 + (15/4 - 2 + 3/4)/(15/2) = 4/3$.
> In a weak field of $1\ \text{T}$ the sublevel spacing is $\mu_B g_J B$: for
> $2P_{3/2}$ this is $(5.79\times10^{-5}\ \text{eV/T})(4/3)(1\ \text{T}) = 7.7\times10^{-5}\ \text{eV}$,
> the same order as the fine-structure splitting itself. A $1\ \text{T}$ field is
> therefore already the intermediate regime for hydrogen, and the clean weak-field
> pattern requires fields well below a tesla.

## The strong-field limit: Paschen–Back

When $\mu_B B \gg E_{fs}$ the ordering reverses: the Zeeman term dominates and
the good basis is the uncoupled $\ket{n,\ell,m_\ell,m_s}$, in which
$\hat L_z + 2\hat S_z$ is already diagonal. The **Paschen–Back** shift is

$$
E_Z^1 = \mu_B B\,(m_\ell + 2 m_s).
$$

Fine structure is now the perturbation, added on top as a diagonal correction
in the uncoupled basis. The splitting pattern is a grid in $(m_\ell, m_s)$ rather
than the $m_j$ ladder of the weak-field case: the field has decoupled $\vec L$
and $\vec S$, and each precesses independently about $\vec B$. The number of
distinct lines is smaller than the weak-field count because different
$(m_\ell, m_s)$ pairs share the same $m_\ell + 2m_s$.

$$
% caption: The two Zeeman limits split a level differently: the weak field
% resolves the coupled multiplet into 2j+1 m-sub-levels spaced by the Landé
% factor, the strong field decouples L and S into a grid spaced by m-ell plus
% twice m-s.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
% weak field ladder (left)
\node[anchor=south, acc] at (1.0,3.0) {weak $B$};
\foreach \y in {0.4,1.0,1.6,2.2} \draw[acc, very thick] (0.3,\y) -- (1.7,\y);
\node[anchor=west, black!70] at (1.8,2.2) {$m_j = +\frac{3}{2}$};
\node[anchor=west, black!70] at (1.8,0.4) {lowest $m_j$};
% strong field grid (right)
\node[anchor=south, black!70] at (6.0,3.0) {strong $B$};
\foreach \y in {0.6,1.4,2.2} \draw[black, very thick, dashed] (5.3,\y) -- (6.7,\y);
\node[anchor=west, black!70] at (6.8,2.2) {decoupled grid};
\end{tikzpicture}
$$

## The intermediate regime

When $\mu_B B \sim E_{fs}$ neither basis diagonalizes the total perturbation
$\hat H' = \hat H_{fs}' + \hat H_Z'$, and the two must be diagonalized together
within each degenerate $n$ shell. Choose either basis, build the matrix of
$\hat H'$, and diagonalize. For $n = 2$ the calculation reduces to a set of
small blocks labeled by $m_j = m_\ell + m_s$, the one quantum number both terms
conserve. The two-dimensional block for a given $\lvert m_j\rvert$ produces
eigenvalues that interpolate smoothly between the weak-field lines
($\propto g_J m_j B$ at small $B$) and the strong-field lines
($\propto (m_\ell + 2m_s)B$ at large $B$).

$$
% caption: A correlation diagram tracks each sublevel from the low-field Landé
% pattern into the high-field Paschen–Back pattern; levels of equal conserved
% m-j connect and never cross, matching weak- and strong-field labels at the
% two ends.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[black, ->] (0,0) -- (6.2,0) node[right, black!70] {$B$};
\draw[black, ->] (0,-1.6) -- (0,2.2) node[above, black!70] {energy};
% weak field bunched (left), strong field spread (right)
\draw[acc, very thick] (0.2,0.4) .. controls (2.5,0.9) and (4,1.5) .. (6.0,1.8);
\draw[acc, very thick] (0.2,0.15) .. controls (2.5,0.3) and (4,0.6) .. (6.0,0.7);
\draw[acc, very thick] (0.2,-0.15) .. controls (2.5,-0.3) and (4,-0.6) .. (6.0,-0.7);
\draw[acc, very thick] (0.2,-0.4) .. controls (2.5,-0.9) and (4,-1.5) .. (6.0,-1.5);
\node[anchor=east, black!70] at (0.1,0.3) {low $B$};
\node[anchor=west, black!70] at (6.05,1.8) {high $B$};
\end{tikzpicture}
$$

## The quadratic Stark effect for the ground state

An external electric field $\vec E = E\hat k$ adds
$\hat H_S' = eE\hat z$ (the electron's potential energy in the field). The ground
state $\ket{100}$ is nondegenerate, so nondegenerate perturbation theory applies.
The first-order shift vanishes by parity: $\hat z$ is odd, $\lvert\psi_{100}\rvert^2$
is even, so $\braket{100 | \hat z | 100} = 0$. An atom in a nondegenerate
spherically symmetric state has no permanent dipole moment.

The leading effect is second order. Every term in the second-order sum is
negative for the ground state, giving a downward shift quadratic in the field,

$$
E_{100}^2 = e^2 E^2 \sum_{m\neq 100}\frac{\lvert\braket{m | \hat z | 100}\rvert^2}{E_1^0 - E_m^0}
\equiv -\tfrac12\,\alpha_p\, E^2,
$$

which defines the **static polarizability** $\alpha_p$. The field induces a
dipole $\vec p = \alpha_p\vec E$ proportional to itself, and the interaction
energy of an induced dipole is quadratic. The sum can be evaluated exactly for
hydrogen (the Dalgarno–Lewis method converts it to a differential equation),
giving

$$
\alpha_p = 18\pi\epsilon_0 a_0^3 = \frac{9}{2}\,(4\pi\epsilon_0)\,a_0^3 .
$$

> **Definition (Static polarizability).** The coefficient $\alpha_p$ relating the
> induced dipole to the applied field, $\vec p = \alpha_p\vec E$, equivalently the
> quadratic energy response $\Delta E = -\tfrac12\alpha_p E^2$. It is positive:
> the atom polarizes along the field and lowers its energy.

$$
% caption: The ground state has no first-order (linear) Stark shift because it
% has no permanent dipole; its energy falls quadratically with field through the
% induced dipole, the parabola set by the polarizability.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[black, ->] (-3.0,0) -- (3.0,0) node[right, black!70] {$E$};
\draw[black, ->] (0,0.4) -- (0,-2.6) node[below, black!70] {energy};
\draw[acc, very thick] plot[domain=-2.7:2.7, samples=80] (\x, {-0.3*\x*\x});
\node[anchor=south west, acc] at (1.4,-1.5) {quadratic shift};
\fill[acc] (0,0) circle (2pt);
\end{tikzpicture}
$$

## The linear Stark effect for n = 2

The $n = 2$ level is four-fold degenerate (ignoring spin): $\ket{200}$,
$\ket{210}$, $\ket{21,\!+\!1}$, $\ket{21,\!-\!1}$. Degenerate perturbation theory
requires the $4\times 4$ matrix of $\hat H_S' = eE\hat z$ in this subspace. Two
selection rules gut it. The operator $\hat z$ is odd under parity, so it connects
only states of opposite parity ($\Delta\ell = \pm 1$); and $\hat z$ commutes with
$\hat L_z$, so it connects only states of equal $m$ ($\Delta m = 0$). The single
surviving matrix element couples the $m = 0$ states of different $\ell$,

$$
\braket{200 | \hat z | 210} = -3 a_0,
\qquad
\braket{200 | \hat H_S' | 210} = -3 e a_0 E .
$$

The two $m = \pm 1$ states have no partner and stay unshifted. The $m = 0$ block
is a $2\times 2$ matrix with zero diagonal and off-diagonal $-3ea_0 E$, whose
eigenvalues are $\pm 3ea_0 E$ with good states $(\ket{200} \mp \ket{210})/\sqrt2$.

> **Theorem (Linear Stark effect in hydrogen).** The $n = 2$ level splits into
> three: two levels shifted by $\pm 3 e a_0 E$, linear in the field, and one
> unshifted level (doubly degenerate, the $m = \pm 1$ states). The shifted
> states are $s$–$p$ hybrids carrying a permanent electric dipole moment of
> magnitude $3 e a_0$.

The linear (rather than quadratic) response is a direct consequence of the
degeneracy: because $\ket{200}$ and $\ket{210}$ share an energy, the field can mix
them without cost and build a state of definite dipole orientation. Only the
accidental $\ell$ degeneracy of the Coulomb problem permits it; in any potential
that splits $s$ from $p$, the linear Stark effect becomes quadratic.

$$
% caption: The four-fold degenerate n = 2 level splits under the field into a
% raised and a lowered s–p hybrid carrying opposite permanent dipoles, with the
% two m = plus-or-minus-one states left unshifted between them.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
% degenerate level
\draw[black, line width=2.2pt] (0,1.2) -- (1.8,1.2);
\node[anchor=east, black!70] at (0,1.2) {$n = 2$};
% split into three
\draw[acc, very thick] (4.0,2.2) -- (5.8,2.2);
\draw[acc, very thick] (4.0,1.2) -- (5.8,1.2);
\draw[acc, very thick] (4.0,0.2) -- (5.8,0.2);
\node[anchor=west, acc] at (5.9,2.2) {$+3ea_0 E$};
\node[anchor=west, black!70] at (5.9,1.2) {unshifted (two states)};
\node[anchor=west, acc] at (5.9,0.2) {lowered};
\draw[black, dashed] (1.8,1.2) -- (4.0,2.2);
\draw[black, dashed] (1.8,1.2) -- (4.0,1.2);
\draw[black, dashed] (1.8,1.2) -- (4.0,0.2);
\end{tikzpicture}
$$

## Which effect and which basis

The two field effects are the same calculation with different symmetry. The
distinction that governs both is whether the unperturbed level is degenerate in
a way the perturbation can exploit.

| Situation | Degeneracy exploited | Leading shift | Good states |
| --- | --- | --- | --- |
| Ground state, $E$ field | none | quadratic, $-\tfrac12\alpha_p E^2$ | $\ket{100}$ unchanged |
| $n = 2$, $E$ field | $s$–$p$ ($\ell$) | linear, $\pm 3ea_0 E$ | $s$–$p$ hybrids |
| Weak $B$ field | lifted by fine structure | linear, $\mu_B g_J B m_j$ | coupled $\ket{n\ell j m_j}$ |
| Strong $B$ field | lifted by field | linear, $\mu_B(m_\ell + 2m_s)B$ | uncoupled $\ket{n\ell m_\ell m_s}$ |

Reading the field strength against the internal scale selects the basis, and the
[degenerate perturbation theory](/quantum-mechanics/approximation-methods/time-independent-perturbation-theory)
of the first lesson supplies the machinery. The remaining lessons of the module
turn from perturbations with a small parameter to methods with none: the
[variational method](/quantum-mechanics/approximation-methods/the-variational-method)
bounds a ground state without any expansion, and the
[WKB approximation](/quantum-mechanics/approximation-methods/the-wkb-approximation)
handles slowly varying potentials semiclassically.

[^griffiths-zs]: **Griffiths & Schroeter**, _Introduction to Quantum Mechanics_ 3rd ed., §7.4 — the Zeeman effect in the weak, strong, and intermediate regimes, the Landé g-factor from the projection theorem; §7.6 and Problems — the quadratic and linear Stark effects. Cambridge, 2018.
[^sakurai-zs]: **Sakurai & Napolitano**, _Modern Quantum Mechanics_ 3rd ed., §5.3 — the competition between the Zeeman interaction and fine structure, and the change of good basis across the crossover field. Cambridge, 2021.
[^ct-stark]: **Cohen-Tannoudji, Diu & Laloë**, _Quantum Mechanics_ Vol. II, Complement D_XII — the Stark effect, the vanishing linear shift of nondegenerate states, and the static polarizability of the hydrogen ground state. Wiley, 1977.
[^codata]: **CODATA 2018 recommended values**, National Institute of Standards and Technology — Bohr magneton $\mu_B = 5.7883818\times10^{-5}\ \text{eV/T}$, Bohr radius $a_0 = 5.29177\times10^{-11}\ \text{m}$. [physics.nist.gov/cuu/Constants](https://physics.nist.gov/cuu/Constants/).