The Zeeman and Stark Effects

Placing hydrogen in a static external field adds a perturbation that competes with the atom's internal fine structure. 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 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

where ${\mu }_B=e\hslash /2m=5.788\times 10^{-5}\text{eV/T}$ is the Bohr magneton.¹ 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 ${\alpha }^{4}m{c}^2\sim 10^{-4}\text{eV}$. For hydrogen the crossover field is about $1\text{T}$, so laboratory fields span all three regimes.

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 $|n,\ell ,j,m_j⟩$. The Zeeman term is a perturbation on the fine-structure levels, and its shift is the diagonal expectation $E_Z^1=\frac{{\mu }_{B}B}{\hslash }⟨{\hat{L}}_z+2{\hat{S}}_z⟩=\frac{{\mu }_{B}B}{\hslash }⟨{\hat{J}}_z+{\hat{S}}_z⟩$. Since $⟨{\hat{J}}_z⟩=m_j\hslash$ is immediate, only $⟨{\hat{S}}_z⟩$ 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 $\vec{J}$,

Carrying this through, $⟨{\hat{L}}_z+2{\hat{S}}_z⟩=g_J{m}_j\hslash$ with the Landé g-factor

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.

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 $|n,\ell ,m_{\ell },m_s⟩$, in which ${\hat{L}}_z+2{\hat{S}}_z$ is already diagonal. The Paschen–Back shift is

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 $m_{\ell }+2m_s$.

The intermediate regime

When ${\mu }_{B}B\sim E_{fs}$ neither basis diagonalizes the total perturbation ${\hat{H}}^{\prime }={\hat{H}}_{fs}^{\prime }+{\hat{H}}_Z^{\prime }$, and the two must be diagonalized together within each degenerate $n$ shell. Choose either basis, build the matrix of ${\hat{H}}^{\prime }$, 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 $|m_j|$ 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 $B$).

The quadratic Stark effect for the ground state

An external electric field $\vec{E}=E\hat{k}$ adds ${\hat{H}}_S^{\prime }=eE\hat{z}$ (the electron's potential energy in the field). The ground state $|100⟩$ is nondegenerate, so nondegenerate perturbation theory applies. The first-order shift vanishes by parity: $\hat{z}$ is odd, $|{\psi }_{100}{|}^2$ is even, so $⟨100|\hat{z}|100⟩=0$. 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,

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

The linear Stark effect for n = 2

The $n=2$ level is four-fold degenerate (ignoring spin): $|200⟩$, $|210⟩$, $|21,+1⟩$, $|21,-1⟩$. Degenerate perturbation theory requires the $4\times 4$ matrix of ${\hat{H}}_S^{\prime }=eE\hat{z}$ 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 $\ell$,

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 $-3e{a}_{0}E$, whose eigenvalues are $\pm 3e{a}_{0}E$ with good states $(|200⟩\mp |210⟩)/\sqrt{2}$.

The linear (rather than quadratic) response is a direct consequence of the degeneracy: because $|200⟩$ and $|210⟩$ 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.

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.

Reading the field strength against the internal scale selects the basis, and the degenerate 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 bounds a ground state without any expansion, and the WKB approximation handles slowly varying potentials semiclassically.