Zeeman Effect

In 1896 a Leiden experimentalist widened a sodium flame's yellow lines by switching on an electromagnet, and atomic physics acquired its most versatile dial. In a magnetic field every level shifts by an amount depending on how the electron's angular momentum is oriented — the degeneracy Degeneracy and the Role of the Magnetic Quantum Number proved compulsory for every central potential finally breaks, and $m$ stops being bookkeeping and starts printing measurable spectral lines. The mechanism is one Hamiltonian, $\hat{H}_Z=(\mu_B B/\hbar)(\hat{L}_z+2\hat{S}_z)$ , but its consequences split into regimes, because the atom already carries an internal magnetic scale — the fine structure of the previous serials. The external field competes with the spin-orbit locking of $\hat{\mathbf{L}}$ to $\hat{\mathbf{S}}$ , and the winner decides which quantum numbers survive. Weak field: $j,m_j$ hold and the splitting inherits the Landé factor (the next article's territory). Strong field: $\hat{\mathbf{L}}$ and $\hat{\mathbf{S}}$ are ripped apart and $m_\ell,m_s$ take over — the Paschen-Back regime, two serials ahead. This article builds the Hamiltonian, derives the crossover scale that referees the competition, and solves the one case needing no referee: spinless states, whose universal three-line "normal" pattern Lorentz predicted within weeks of Zeeman's measurement — and which turns out to be nature's exception, not its rule. Every number below is machine-verified: the 14.0 GHz-per-tesla dial, the 0.78 T crossover that makes hydrogen's "weak field" surprisingly hard to reach, and the e/m ratio Zeeman and Lorentz extracted from line widths months before Thomson caught his cathode rays.

Key takeaways

An external field couples to the electron's total moment: $\hat{H}_Z=(\mu_B B/\hbar)(\hat{L}_z+2\hat{S}_z)$ — the spin factor 2 (from Magnetic Moment and the Gyromagnetic Ratio) is the source of every complication.
Scale: $\mu_B=5.788\times10^{-5}$ eV/T ⇒ $\mu_B B/h=14.0$ GHz at 1 T (computed) — tiny against optical frequencies, comparable to fine structure.
The regime is one ratio, $\mu_B B/\Delta E_{fs}$ : weak ⇒ perturb $|n\,\ell\,j\,m_j\rangle$ (anomalous Zeeman, next article); strong ⇒ perturb $|n\,\ell\,m_\ell\,m_s\rangle$ (Paschen-Back); comparable ⇒ diagonalize blocks.
Hydrogen $n=2$ crosses over at $B_{\text{int}}=\Delta E_{fs}/\mu_B\approx0.78$ T (computed): an ordinary lab magnet already leaves the weak-field formulas behind. Sodium's $3p$ holds out to 37 T.
Spinless ( $S=0$ ) states are regime-proof: three components at $0,\pm\mu_B B$ for any $\ell$ — the normal triplet, with π/σ polarizations fixed by $\Delta m_\ell$ .
Zeeman + Lorentz read $e/m\approx1.8\times10^{11}$ C/kg off the 1896 broadening — the electron weighed by light, months before Thomson.
Key idea: $\hat{L}_z+2\hat{S}_z=\hat{J}_z+\hat{S}_z$ is not proportional to $\hat{J}_z$ — how a level splits depends on which internal coupling survives the field, and the Zeeman chapter is the taxonomy of that competition.

The core insight

The interaction $\hat{H}_Z=(\mu_B B/\hbar)(\hat{L}_z+2\hat{S}_z)$ is fixed and simple; the physics is a competition of scales. Because $\hat{\mathbf{L}}+2\hat{\mathbf{S}}$ is not parallel to $\hat{\mathbf{J}}$ , the pattern depends on whether $\Delta E_{fs}$ or $\mu_B B$ dominates — weak, strong, and intermediate regimes, each with its own good quantum numbers.

Where this lives today

The Zeeman effect is physics' magnetic-field ruler at every scale. Solar telescopes invert the polarization of Zeeman-split Fe lines into vector magnetograms of the whole solar surface — Hale's 1908 trick with 2020s spectropolarimetry (a sunspot's 0.3 T splits lines by 4.2 GHz, computed). Optically pumped magnetometers track Larmor precession of Zeeman sublevels in alkali vapor to femtotesla sensitivity — magnetoencephalography without cryogenics — and every atomic clock's error budget is dominated by Zeeman management: clock transitions run between $m=0$ states with no linear shift while their $m\ne0$ neighbors serve as built-in gauges. And magneto-optical traps use the position-dependent Zeeman shift of a quadrupole field as the restoring force of laser cooling — the effect Zeeman found in a flame is the spring constant of nearly every ultracold-atom experiment on Earth.

The physical picture

Start with the classical cartoon the quantum calculation vindicates almost line by line. An orbiting electron is a current loop with moment proportional to its angular momentum; spin adds a second moment at twice the classical rate ( $g_s\approx2$ , the anomaly of Magnetic Moment and the Gyromagnetic Ratio). The total moment $\boldsymbol{\mu}\propto-(\hat{\mathbf{L}}+2\hat{\mathbf{S}})$ is therefore tilted away from $\hat{\mathbf{J}}=\hat{\mathbf{L}}+\hat{\mathbf{S}}$ . Were $g_s=1$ , $\boldsymbol{\mu}$ would be anti-parallel to $\hat{\mathbf{J}}$ , every level would fan as $\mu_B B\,m_j$ , and this chapter would be one article long.

Figure

The tilted moment: L + 2S is not parallel to J — the one geometric fact behind all three regimes

Because $g_s=2$ , the moment precesses. In a weak field the spin-orbit coupling is the fast dynamics: $\hat{\mathbf{L}}$ and $\hat{\mathbf{S}}$ whirl around their locked resultant $\hat{\mathbf{J}}$ , and only the projection of the tilted moment onto $\hat{\mathbf{J}}$ survives the average — the Landé factor, derived where it belongs in Projection Theorem for Vector Operators and deployed next serial. In a strong field the external precession is faster: $\hat{\mathbf{L}}$ and $\hat{\mathbf{S}}$ lock onto $\mathbf{B}$ separately and the pair divorce — Strong-Field Paschen-Back Effect handles the settlement. The referee is a single number in tesla, and computing it is this article's central skill.

Figure

The regime map: one axis (field strength), three territories, three owners

One habit for the whole chapter: the field also defines the axis. Choosing $\mathbf{B}=B\hat z$ promotes one $\hat{L}_z$ out of the democratic family of Degeneracy and the Role of the Magnetic Quantum Number — its "m is a choice of axis" doctrine is exactly the statement that until a field arrives, there is nothing for $m$ to measure.

The Zeeman Hamiltonian

The atom's magnetic moment was assembled piece by piece in Magnetic Moment and the Gyromagnetic Ratio: the orbital current loop contributes $\boldsymbol{\mu}_L=-(\mu_B/\hbar)\hat{\mathbf{L}}$ , and spin contributes $\boldsymbol{\mu}_S=-g_s(\mu_B/\hbar)\hat{\mathbf{S}}$ with $g_s=2.00232\approx2$ — we consume that dictionary here rather than re-derive it. A moment in a field has orientation energy $-\boldsymbol{\mu}\cdot\mathbf{B}$ , so the perturbation the field adds to the atomic Hamiltonian is

\hat{H}_Z \;=\; -\boldsymbol{\mu}\cdot\mathbf{B} \;=\; \frac{\mu_B}{\hbar}\,\big(\hat{\mathbf{L}}+g_s\hat{\mathbf{S}}\big)\cdot\mathbf{B} \qquad (1)

where: $\mu_B=e\hbar/2m_e=9.274\times10^{-24}$ J/T $=5.788\times10^{-5}$ eV/T (recomputed from CODATA constants) and $g_s\approx2$ . Read it physically: the field pays $\hat{\mathbf{L}}$ and $\hat{\mathbf{S}}$ different wages for alignment. With equal wages the operator would be proportional to $\hat{\mathbf{J}}$ , already diagonal in the fine-structure basis — no regimes to map. Point the field along $\hat z$ (this defines $z$ ; nothing in the isolated atom did) and set $g_s=2$ :

\hat{H}_Z \;=\; \frac{\mu_B B}{\hbar}\,\big(\hat{L}_z+2\hat{S}_z\big) \;=\; \frac{\mu_B B}{\hbar}\,\big(\hat{J}_z+\hat{S}_z\big) \qquad (2)

where: the second form uses $\hat{L}_z=\hat{J}_z-\hat{S}_z$ (verified as exact $6\times6$ matrices on $\ell=1\otimes s=\tfrac12$ ). The $\hat{J}_z$ piece is harmless — diagonal in every basis this chapter uses. All the physics lives in the leftover $\hat{S}_z$ : diagonal in the uncoupled $|m_\ell\,m_s\rangle$ basis but not within a fixed- $j$ multiplet — precisely why the weak-field regime needs projection machinery and the strong-field regime doesn't. Two scale anchors. Per tesla the quantum is small: $\mu_B B/h=14.0$ GHz at 1 T (computed) — a $3\times10^{-5}$ nudge on an optical line, resolved as a $\sim0.01$ nm split. But the field the atom makes for itself is not small: hydrogen's $n=2$ fine structure, $4.53\times10^{-5}$ eV (consumed from Fine Structure of Hydrogen), is matched by $\mu_B B$ already at $0.78$ T. External magnets compete with the atom's interior far sooner than intuition suggests. One term we deliberately drop: minimal coupling also produces a diamagnetic piece $e^2B^2(x^2+y^2)/8m_e$ . Its ratio to the linear term is $\sim10^{-6}$ at 1 T (computed; parity only near $10^6$ T — white-dwarf and neutron-star territory, condensed-matter's Landau-level physics, not this chapter's).

Derivation AThe Zeeman Hamiltonian and the identity that organizes the chapter

We assemble equation (1) from the moment dictionary, reduce it to the working form (2), and prove the operator identity $\hat{L}_z+2\hat{S}_z=\hat{J}_z+\hat{S}_z$ that every later regime argument leans on. The derivation is deliberately anticlimactic — two known moments dotted into a constant field — because the value is bookkeeping done once, correctly, with the $g_s$ weighting explicit. Watch where $g_s=2$ enters: it is an input from the moment dictionary, not something this derivation produces.

Step 1. Orbital piece. A charge $-e$ in orbit is a current loop; Magnetic Moment and the Gyromagnetic Ratio established $\boldsymbol{\mu}_L=-(\mu_B/\hbar)\hat{\mathbf{L}}$ , the minus sign recording the negative charge.

Step 2. Spin piece. The same dictionary gives $\boldsymbol{\mu}_S=-g_s(\mu_B/\hbar)\hat{\mathbf{S}}$ , $g_s=2.00232$ . We take $g_s=2$ (Dirac value) throughout; the QED correction shifts splittings at the $10^{-3}$ level, below anything tracked here.

Step 3. Orientation energy. $\hat{H}_Z=-(\boldsymbol{\mu}_L+\boldsymbol{\mu}_S)\cdot\mathbf{B}=(\mu_B/\hbar)(\hat{\mathbf{L}}+2\hat{\mathbf{S}})\cdot\mathbf{B}$ . The sign flipped: aligned moment lowers energy, and moments are anti-parallel to angular momenta, so the field rewards negative $m$ . Why B may be treated as external and classical: laboratory fields vary over millimetres, the atom over ångströms — uniform across the cloud to one part in 10⁷. Quantized-field corrections belong to QED; their famous consequence in this chapter is the Lamb shift, one serial back.

Step 4. Choose the axis. With $\mathbf{B}=B\hat z$ only $z$ -components survive: $\hat{H}_Z=(\mu_B B/\hbar)(\hat{L}_z+2\hat{S}_z)$ . Here the degenerate $m$ -family of Degeneracy and the Role of the Magnetic Quantum Number stops being conventional: the field direction is a physical reference and different $m$ finally means different energy.

Step 5. The organizing identity: $\hat{L}_z+2\hat{S}_z=(\hat{J}_z-\hat{S}_z)+2\hat{S}_z=\hat{J}_z+\hat{S}_z$ — a diagonal-everywhere piece plus the troublemaker $\hat{S}_z$ . The next article projects it, the strong-field article diagonalizes it, the intermediate regime inverts it honestly. Verified: the identity as exact 6×6 matrices on ℓ=1 ⊗ s=½; μ_B from CODATA e, ħ, m_e = 9.2740×10⁻²⁴ J/T = 5.7884×10⁻⁵ eV/T; diamagnetic-to-linear ratio 1.06×10⁻⁶ at 1 T (article verify script).

The competition: who owns which regime

The unperturbed atom already contains magnetic physics: Spin-Orbit Coupling locked $\hat{\mathbf{L}}$ to $\hat{\mathbf{S}}$ at energy $\sim\alpha^2$ Ry, and Fine Structure of Hydrogen assembled the $j$ -ladder. The field does not get a clean slate — it competes:

\frac{\mu_B B}{\Delta E_{fs}} \;\begin{cases} \ll 1 & \text{weak field: } j,m_j \text{ good — perturb the fine-structure levels}\\[2pt] \gg 1 & \text{strong field: } m_\ell,m_s \text{ good — fine structure becomes the perturbation}\\[2pt] \sim 1 & \text{intermediate: no shortcut — diagonalize each } m_j \text{ block} \end{cases} \qquad (3)

where: $\Delta E_{fs}$ is the fine-structure splitting of the manifold in question (hydrogen $n=2$ : the $2p_{3/2}$ – $2p_{1/2}$ gap, $4.53\times10^{-5}$ eV, consumed from Fine Structure of Hydrogen). This is the doctrine of Good Quantum Numbers and Symmetry-Adapted Bases in action: a label survives only while the coupling that defines it dominates. Weak limit — $\hat{\mathbf{J}}$ 's grip holds, $m_j$ labels the fan; strong limit — the field's grip holds, $(m_\ell,m_s)$ label it; in between, neither family commutes with enough of the Hamiltonian, and honesty requires matrices.

Go deeper · Derivation B · 1 minThe crossover field, and why hydrogen is weak-field-hostile

Derivation BThe crossover field, and why hydrogen is weak-field-hostile

We convert the regime criterion (3) into a number with units of tesla — the internal field scale $B_{\text{int}}$ — and evaluate it for this chapter's two workhorses, hydrogen's $n=2$ manifold and sodium's $3p$ doublet. The point is a cultural correction: "weak field" sounds generous, but for hydrogen it ends below one tesla, while for sodium it extends past any ordinary magnet. Same formula, opposite practical verdicts.

Step 1. Define the crossover where the energies tie: $B_{\text{int}}=\Delta E_{fs}/\mu_B$ . ( $O(1)$ factors like the $g_J$ -difference of the levels shift the practical boundary slightly; the tesla-scale verdicts are insensitive.)

Step 2. Hydrogen, $n=2$ . From Fine Structure of Hydrogen, $\Delta E_{fs}(2p_{3/2}\!-\!2p_{1/2})=\tfrac{13.6\,\text{eV}\,\alpha^2}{16}=4.53\times10^{-5}$ eV, so $B_{\text{int}}=0.78$ T (computed). Routine hardware passes hydrogen's crossover: the clean Landé patterns live below $\sim0.1$ T, and by 2 T the manifold is closer to Paschen-Back. Why so low: fine structure is α² · Ry ≈ 10⁻⁴ of the gross spectrum, further suppressed with n. Small internal splittings mean small internal fields to beat.

Step 3. Sodium, $3p$ . The D-doublet wavelengths 589.0/589.6 nm give $\Delta E_{fs}=2.14\times10^{-3}$ eV (computed from the wavelengths — sodium's core-penetrating $3p$ makes the hydrogenic formula inapplicable, so experiment supplies the splitting, as in Spin-Orbit Coupling). $B_{\text{int}}=37$ T (computed): every tabletop sodium experiment is deeply weak-field — why the D lines are the canonical anomalous showcase, and why Paschen and Back needed brutal magnets.

Step 4. Owners. Below $B_{\text{int}}$ : Weak-Field (Anomalous) Zeeman Effect, fan $g_J\mu_B Bm_j$ . Far above: Strong-Field Paschen-Back Effect, fan $\mu_B B(m_\ell+2m_s)$ with fine structure demoted to a residual $\propto m_\ell m_s$ . Near it: small matrices per $m_j$ sector — the same owner's intermediate machinery. Hyperfine structure replays the whole map a thousand times lower (Breit-Rabi, Hyperfine Structure). Verified: B_int(H, n=2) = 0.782 T, B_int(Na 3p) = 37.0 T, Na doublet 2.142 meV recomputed from the wavelengths (article verify script).

The one regime-proof case: the normal triplet

One family of states is immune to the taxonomy: spin singlets. With $S=0$ the troublesome $\hat{S}_z$ has nothing to act on, $\hat{H}_Z=(\mu_B B/\hbar)\hat{L}_z$ exactly, and every level fans with the same universal spacing — no projection theorem, no competition, no anomaly. This is the case Lorentz solved with a classical electron on a spring, within weeks of the 1896 measurement.

E^{(1)}_Z \;=\; \mu_B B\,m_\ell \qquad\Longrightarrow\qquad h\nu \;=\; h\nu_0+\mu_B B\,\Delta m_\ell,\quad \Delta m_\ell=0,\pm1 \qquad (4)

where: $\nu_0$ is the zero-field line and the electric-dipole selection rule $\Delta m_\ell=0,\pm1$ (stated here; derived with the transition machinery of the time-dependent chapter) collapses the many level-to-level possibilities to exactly three photon energies. However rich the two $\ell$ -manifolds — a $d\to p$ transition connects 15 sublevel pairs — the spacing $\mu_B B$ is common to upper and lower levels, so the differences take only the values $0,\pm\mu_B B$ . Three lines, always, with the two shifted ones displaced by 14.0 GHz per tesla.

Go deeper · Derivation C · 1 minNormal Zeeman triplet with its polarization fingerprint

Derivation CNormal Zeeman triplet with its polarization fingerprint

We derive the universal three-line pattern for transitions between singlet terms, with the π/σ polarization dictionary that doubles as a field-direction diagnostic. Strategy: first-order shifts on both levels ( $\hat{L}_z$ is already diagonal in the $m_\ell$ basis — the "fortunate basis" case of Degenerate Perturbation Theory), then the selection rule prunes the differences. Watch how the count three rests on two facts at once — equal spacings above and below, and $|\Delta m_\ell|\le1$ — and loses to both generalizations (unequal $g$ 's next article, unequal effective spacings in Paschen-Back).

Step 1. Shifts. For $S=0$ the eigenstates $|n\,\ell\,m_\ell\rangle$ already diagonalize $\hat{H}_Z=(\mu_B B/\hbar)\hat{L}_z$ : upper level shifts by $\mu_B B\,m'_\ell$ , lower by $\mu_B B\,m''_\ell$ . No mixing, no secular matrix.

Step 2. Photon energies. $h\nu=h\nu_0+\mu_B B(m'_\ell-m''_\ell)$ with $\Delta m_\ell\in\{0,\pm1\}$ ⇒ the set of energies is $\{h\nu_0,\,h\nu_0\pm\mu_B B\}$ — three members regardless of $\ell',\ell''$ (swept programmatically for $d\to p$ : 15 allowed pairs, exactly 3 distinct energies).

Step 3. Polarizations. $\Delta m_\ell=0$ radiates like a dipole oscillating along $\mathbf{B}$ : linear (π), invisible down the field axis — a dipole does not radiate along itself. $\Delta m_\ell=\pm1$ radiate like charge circling perpendicular to $\mathbf{B}$ : circular (σ±), the only light seen along the axis. Transverse viewing shows all three, σ then linearly polarized ⊥ $\mathbf{B}$ . Why this is a gift: polarization encodes the field's direction while splitting encodes its magnitude — the two dials of solar magnetography.

Step 4. Scale. At 1 T the σ components sit 14.0 GHz off center; at 500 nm, $\Delta\lambda=\lambda^2\mu_B B/hc=0.0117$ nm (both computed) — visible to 1890s Rowland gratings only as a broadening, exactly what Zeeman first reported. Verified: the 15-pair d→p sweep collapses to {0, ±μ_B B}; Δν = 13.996 GHz/T; Δλ(500 nm, 1 T) = 0.01167 nm; μ_B/(hc) = 0.4669 cm⁻¹/T (article verify script).

One honest label. Nature mostly refuses the "normal" pattern: it needs $S=0$ , which no one-electron atom can supply. Real singlet showcases are paired-spin two-electron atoms — the cadmium 643.8 nm line ( $^1D_2\to{}^1P_1$ ), the clean triplet standard. In hydrogen and the alkalis spin is always on stage, $g_J\ne1$ , and the pattern is anomalous — the word a fossil of pre-spin bafflement. The quantitative anomalous machinery is the next serial's.

Limiting cases

case	result	meaning
$B\to0$	degenerate $m$ -multiplets restored	the isolated-atom democracy of Degeneracy and the Role of the Magnetic Quantum Number returns; $m$ unmeasurable
$S=0$ , any $B$	three lines at $0,\pm\mu_B B$	normal triplet: regime-proof because the tilted $\hat{S}_z$ piece is absent
$\mu_B B\ll\Delta E_{fs}$	$E=g_J\mu_B Bm_j$	anomalous fan; $g_J$ machinery owned by Weak-Field (Anomalous) Zeeman Effect
$\mu_B B\gg\Delta E_{fs}$	$E=\mu_B B(m_\ell+2m_s)+\mathcal{O}(\Delta E_{fs})$	Paschen-Back: field divorces $\hat{\mathbf{L}}$ from $\hat{\mathbf{S}}$ ; owned by Strong-Field Paschen-Back Effect
$\mu_B B\sim\Delta E_{fs}$	block diagonalization per $m_j$	intermediate field, anticrossing curves; same owner
$B\sim10^6$ T	diamagnetic $B^2$ term rivals the linear term	atoms deform into field-aligned cylinders — magnetar/white-dwarf spectroscopy, beyond this chapter

Common trapThe splitting is not always the normal triplet — and 1 tesla is not always weak

Two errors dominate lost Zeeman marks. First: applying the three-line normal pattern to states that carry spin. The triplet requires $S=0$ ; anything with $s=\tfrac12$ fans with $g_J\neq1$ into four, six, or more components — the default in one-electron and alkali spectra. Second: reaching for the weak-field formula because the field "sounds small." The comparison is with the fine-structure splitting of the specific manifold, never with intuition: hydrogen $n=2$ crosses over at 0.78 T, so a superconducting solenoid is decisively strong-field there and $g_J\mu_B Bm_j$ produces confidently wrong level orders. Compute $B_{\text{int}}=\Delta E_{fs}/\mu_B$ first; let the number pick the article.

Worked examples

Example 1 — the cadmium triplet as a field gauge. The Cd 643.8 nm line ( $^1D_2\to{}^1P_1$ , true singlet-to-singlet) is observed at 1 T, transverse to $\mathbf{B}$ . Predict the pattern.

Step 1. Regime check: trick question — singlets are regime-proof. Both terms have $S=0$ , so the triplet logic applies at any laboratory field.

Step 2. Spacing: $\mu_B B=5.788\times10^{-5}$ eV $=14.0$ GHz; in wavelength, $\Delta\lambda=\lambda^2\mu_B B/hc=0.0193$ nm (computed) — comfortably resolved by a Fabry-Pérot étalon.

Step 3. Pattern: transverse viewing shows an unshifted π line polarized along $\mathbf{B}$ , flanked at $\pm0.0193$ nm by σ lines polarized perpendicular. Along the field the π line vanishes and the σ pair turn circular. Answer: a symmetric triplet, $643.8000\pm0.0193$ nm, whose π/σ signature confirms the field direction.

Example 2 — classify before you compute. An exam hands you hydrogen $n=2$ at $B=0.05$ T and sodium's $3p$ doublet at $2$ T. Which splitting formulas are even legal?

Step 1. Hydrogen at 0.05 T: $\mu_B B=2.9\times10^{-6}$ eV against $\Delta E_{fs}=4.53\times10^{-5}$ eV — ratio $0.064$ (computed). Comfortably weak-field: the Landé fan $g_J\mu_B Bm_j$ of Weak-Field (Anomalous) Zeeman Effect applies (the $j$ -mixing contamination is percent-level; the next article computes where 10% is reached).

Step 2. Sodium at 2 T: ratio $1.16\times10^{-4}/2.14\times10^{-3}=0.054$ . Also weak-field, despite the 40-times-larger field — sodium's crossover is 37 T. Same formula family.

Step 3. Swap: hydrogen at 2 T gives ratio $2.6>1$ — Paschen-Back territory, $E=\mu_B B(m_\ell+2m_s)$ plus residual spin-orbit, owned by Strong-Field Paschen-Back Effect; no Landé factor appears anywhere in it.

Step 4. The habit: divide $\mu_B B$ by the manifold's $\Delta E_{fs}$ before touching any formula. Ratios 0.064, 0.054, 2.6 → weak, weak, strong. Answer: hydrogen@0.05 T and sodium@2 T take the weak-field treatment; hydrogen@2 T requires the strong-field one — three different-looking problems, one classifying ratio.

Origins · 1 minPieter Zeeman and how this was discovered

Pieter Zeeman

1865–1943 · Leiden and Amsterdam

Zeeman measured the effect that bears his name with apparatus everyone else already had — a sodium flame, a Ruhmkorff electromagnet, a Rowland grating — plus the patience to look for a distortion others had reasoned away (Faraday tried flame spectra between magnet poles in 1862; the instruments weren't ready). In August 1896 he found the D lines widening when the magnet fired. Laboratory legend insists he was defying his supervisor's instructions and was dismissed for it — a story he politely never confirmed. Within weeks Lorentz had the classical theory: a bound "ion" circling in a field must emit a triplet with exactly the polarizations Zeeman then verified line by line. Better still, the width of the splitting handed them the charge-to-mass ratio of the vibrating particle — about $10^{11}$ C/kg, a thousand times hydrogen's electrolytic value, charge negative: they had weighed the electron months before Thomson, using nothing but light. The 1902 Nobel Prize (physics' second ever) went to both. Zeeman's later Amsterdam lab kept its great magnet on a vibration-proof concrete island — fitting, because he had also found, immediately after the triplets, lines splitting into four and six components that the classical theory forbade. That "anomalous" zoo is the next article's opening problem.

History

1862

— Faraday looks for magnetic effects on flame spectra; instruments too coarse, none found

1896

— Zeeman resolves a magnetic broadening of the sodium D lines at Leiden

1897

— Lorentz's triplet-and-polarization prediction confirmed; e/m of the "light ion" extracted

1897

— Preston photographs quartets and sextets: the anomalous patterns defy the classical theory

1902

— Nobel Prize to Zeeman and Lorentz, the second physics prize awarded

1908

— Hale reads sunspot magnetic fields of several kilogauss from Zeeman-split iron lines

1912

— Paschen and Back reach the strong-field regime where anomalous patterns collapse back toward triplets

Going further

Griffiths & Schroeter §7.4 — the Zeeman effect at the source register, weak and strong cases side by side.
Sakurai & Napolitano §5.3 — the same taxonomy with the projection theorem carrying the weak-field case.
Cohen-Tannoudji, Diu & Laloë Vol. II, Complement $D_{XII}$ — the full intermediate-field diagonalization worked honestly.
Foukal, Solar Astrophysics — Zeeman diagnostics as the working tool of solar magnetometry.

Provenance note: Hamiltonian $H_Z=\frac{\mu_B}{\hbar}(\mathbf{L}+g_s\mathbf{S})\cdot\mathbf{B}$ and the Landé formula are standard (Griffiths §7.4, Sakurai §5.3, Cohen-Tannoudji Vol II Complement D_XII). A verifier should: (1) confirm $\mu_B=9.274\times10^{-24}\,\mathrm{J/T}$ and $g_s=2.00232$ (CODATA); (2) recompute $g_J$ via the projection theorem and check the worked values $g_J(^2S_{1/2})=2$ , $g_J(^2P_{1/2})=2/3$ , $g_J(^2P_{3/2})=4/3$ ; (3) verify the strong-field shift $\mu_B B(m_\ell+2m_s)+A m_\ell m_s$ by diagonalizing $L_zS_z$ in the $|m_\ell m_s\rangle$ basis; (4) check the normal-Zeeman spacing $\mu_B B$ and the conversion $\mu_B/(hc)\approx0.4669\,\mathrm{cm^{-1}/T}$ ; (5) confirm the Breit-Rabi formula against Sakurai/standard hyperfine references. No nonstandard numbers were introduced. — Items (1)/(4) run in this article's verify script, with the crossover fields, the d→p sweep, the diamagnetic ratio, and the e/m extraction; item (3)'s diagonal $\langle L_zS_z\rangle=m_\ell m_s\hbar^2$ is verified matrix-wise here, the strong-field treatment belonging to Strong-Field Paschen-Back Effect; item (2) runs where it is owned — derivation in Projection Theorem for Vector Operators, deployment in Weak-Field (Anomalous) Zeeman Effect. Edition note: the base's Breit-Rabi formula (its Eq. 8) is a hyperfine-regime result — written with $m_j$ where the standard formula carries $m_F$ — and is not reproduced here: intermediate-field machinery is owned BY NAME by Strong-Field Paschen-Back Effect, Breit-Rabi proper by Hyperfine Structure; the base's Landé and Paschen-Back derivations likewise relocate to those owner serials per chapter ownership.

Why does the magnetic field define the z-axis instead of just perturbing the atom?

An isolated atom has no preferred direction — states of different $m$ are related by rotations and must share an energy. The field breaks that democracy physically: writing $\mathbf{B}=B\hat z$ is the statement that an axis now exists against which orientation is measurable. That is why the degeneracy can finally lift, and why every Zeeman quantum number ( $m_j$ or $m_\ell,m_s$ ) is a projection along the field.

Is the Zeeman effect a first-order perturbation result or exact?

The operator is exact (up to the dropped diamagnetic term); the treatment is perturbative. In the weak and strong limits a basis exists in which the relevant perturbation is diagonal within each degenerate manifold, so first-order theory suffices. In between, no such basis exists and one diagonalizes $\hat{H}_{fs}+\hat{H}_Z$ within each $m_j$ block — small matrices, exact answers, no series.

Where exactly does the factor g_s = 2 enter, and what if I forget it?

It enters once, in the moment dictionary: $\boldsymbol{\mu}_S=-g_s(\mu_B/\hbar)\hat{\mathbf{S}}$ . Forgetting it (taking $g_s=1$ ) makes $\hat{H}_Z\propto\hat{J}_z$ , predicting a universal triplet for every transition — precisely the classical expectation that the anomalous patterns of 1897 falsified. Every "anomalous" feature of the Zeeman chapter is the experimental signature of that single factor of 2.

Why is the term quadratic in B negligible, and when does it stop being negligible?

The diamagnetic term $e^2B^2(x^2+y^2)/8m_e$ compares to the linear term as $\sim ea_0^2B/4\hbar$ : about $10^{-6}$ at 1 T (computed), reaching parity only near $10^6$ T. In laboratory atomic physics it matters only for huge-radius states — Rydberg atoms, whose $\langle r^2\rangle\propto n^4$ amplifies it by eight orders of magnitude at $n=100$ — and in astrophysics on white dwarfs ( $10^2$ – $10^5$ T) and neutron stars, where it reorganizes atomic structure entirely.

If the normal triplet needs S = 0, why did Zeeman see clean behavior in sodium, which has spin?

He initially saw only broadening, not resolved components — 1896 resolution could not expose the anomalous structure of the D lines (which split into 4 and 6 components, as the next article computes). Early triplet confirmations came from lines that approximate a triplet at the available resolution. Preston's sharper photographs a year later revealed the quartets and sextets, and "anomalous Zeeman effect" became the standing crisis of pre-spin atomic theory.

How do astronomers measure fields when Doppler broadening swamps the splitting?

Polarization. The σ components carry opposite circular handedness, so differencing right- and left-circular spectra (Stokes V) isolates a signature proportional to $B_\parallel$ even when the splitting is far below the line width — a resolution problem converted into a signal-to-noise problem, routine for the Sun and for white dwarfs.