Strong-Field Paschen-Back Effect

Crank a magnet past one tesla around a hydrogen discharge and something strange happens to the Zeeman pattern: it gets simpler. The baroque anomalous multiplets that defied classical theory for three decades collapse back toward the plain three-line pattern Lorentz predicted in 1897. Paschen and Back found this collapse at Tübingen in 1912 — the cleanest proof that fine structure is a coupling that can be broken: pay enough magnetic energy and $\hat{\mathbf{L}}$ and $\hat{\mathbf{S}}$ stop talking to each other. Zeeman Effect built the Hamiltonian and drew the regime map; Weak-Field (Anomalous) Zeeman Effect worked the low-field side, where each $j$ -level fans as $g_J\mu_B B\,m_j$ . This article owns the other side — the strong-field limit where $m_\ell$ and $m_s$ are the good labels and the spectrum is the grid $\mu_B B(m_\ell+2m_s)$ with fine structure demoted to a small $\beta\,m_\ell m_s$ residue — and the honest bridge between the two: the exact $n=2$ diagonalization, small matrices labelled by the one quantum number that survives at every field, $m_j$ . The same six levels morph from Landé fan to Paschen-Back grid, every eigenvalue machine-verified at 0.1, 0.78, 5, and 30 tesla. The practical surprise, inherited from the crossover $B_{\text{int}}=0.78$ T: for hydrogen the strong-field regime is the easy one — an ordinary lab solenoid reaches it, while sodium holds out to 37 T.

Key takeaways

Strong field means $\mu_B B\gg\langle\hat H_{SO}\rangle$ : the Zeeman term dominates the fine structure, and the good basis flips from coupled $|n\,\ell\,j\,m_j\rangle$ to uncoupled $|n\,\ell\,m_\ell\,m_s\rangle$ — the exact inverse of the weak-field logic.
Leading spectrum: $E=E_n+\mu_B B(m_\ell+2m_s)$ — an integer grid. Fine structure survives as the first-order residue $\beta\,m_\ell m_s$ (hydrogen $2p$ : $\beta=3.02\times10^{-5}$ eV) plus an $m$ -independent shift.
Only the $\hat L_z\hat S_z$ piece of $\hat{\mathbf{L}}\cdot\hat{\mathbf{S}}$ survives on the diagonal: $\langle\hat{\mathbf{L}}\cdot\hat{\mathbf{S}}\rangle=\hbar^2 m_\ell m_s$ .
$\hat J_z$ commutes with the whole Hamiltonian at every field, so $m_j$ never dies: the $n=2$ problem splits into $1\times1$ and $2\times2$ blocks solved in closed form — and same- $m_j$ branches never cross (their gap grows monotonically from $\Delta E_{fs}$ : level repulsion).
For hydrogen $n=2$ the crossover is $B_{\text{int}}=0.78$ T — laboratory magnets make hydrogen strong-field easy; sodium's $3p$ needs 37 T.
Key idea: when the Zeeman energy dwarfs the spin-orbit energy, $\hat{\mathbf{L}}$ and $\hat{\mathbf{S}}$ decouple and precess separately about $\mathbf{B}$ ; the good quantum numbers switch from $(j,m_j)$ to $(m_\ell,m_s)$ , and the spectrum becomes the regular grid $E=\mu_B B(m_\ell+2m_s)$ plus a small fine-structure shift.

Where this lives today

Strong-field decoupling is now an important experimental and theoretical tool. Ultracold-atom experiments deliberately work above the relevant crossover so spin flips can be driven with clean, well-separated resonances — the hyperfine-floor Paschen-Back regime (Back-Goudsmit) is standard territory for Feshbach-resonance control at hundreds of gauss. In astrophysics it is a magnetometer at the extreme end: white-dwarf photospheres at $10^2$ – $10^5$ T sit deep in field-dominated territory, where hydrogen lines are identifiable only through eigenvalue-vs- $B$ fans like the one computed above, extended by the diamagnetic term this chapter dropped. And the 5-30 T window of modern magnets makes the hydrogen fan a routine calibration exhibit — the same $2\times2$ blocks, measured to parts per million.

The physical picture

Picture two gyroscopes — orbital $\hat{\mathbf{L}}$ and spin $\hat{\mathbf{S}}$ — tied by an elastic band, the coupling of Spin-Orbit Coupling. In a weak field the band wins: the pair lock into a composite top $\hat{\mathbf{J}}$ precessing slowly about $\mathbf{B}$ , and only the moment's projection onto $\hat{\mathbf{J}}$ survives the fast internal whirling — the Landé physics of Weak-Field (Anomalous) Zeeman Effect. Crank the field and the magnetic torque overwhelms the band: each gyroscope precesses independently about $\mathbf{B}$ , faster than the band can synchronize them. The resultant $\hat{\mathbf{J}}$ wanders — $j$ is dead — but how much each vector leans along the field, $m_\ell$ and $m_s$ , is rock solid.

Figure

Weak field locks L and S together; strong field tears them apart

The energy-level diagram records this decoupling. The orbital moment earns $\mu_B B\,m_\ell$ , the spin moment $2\mu_B B\,m_s$ (the doubled wage of Magnetic Moment and the Gyromagnetic Ratio), so levels land on the integer grid $m_\ell+2m_s$ — for a $p$ shell $\{2,1,0,-1,-2\}$ , with two states sharing grid point 0. Far plainer than the anomalous fan — exactly the simplification Paschen and Back saw — with the residual fine structure nudging each rung by $\beta\,m_\ell m_s$ . The whole article is one picture: six $2p$ eigenvalues versus $B$ , leaving the fine-structure levels with Landé slopes, straightening into five grid rays — and in between, same- $m_j$ branches bending away from each other, never crossing: the level repulsion of Second-Order Energy Correction.

Figure

The fan: six 2p levels from Landé slopes to the Paschen-Back grid,

m_j

conserved throughout

The referee is the ratio $\mu_B B/\Delta E_{fs}$ that Zeeman Effect turned into a field scale, $B_{\text{int}}=\Delta E_{fs}/\mu_B$ : hydrogen $n=2$ , 0.78 T. A 5 T magnet puts hydrogen at ratio 6.4, deep in Paschen-Back territory, while leaving sodium's $3p$ doublet ( $B_{\text{int}}=37$ T, same owner) firmly anomalous. Strong field for hydrogen is easy.

The basis flip

Start from the Hamiltonian assembled by Zeeman Effect, pecking order reversed: Coulomb structure $\hat H_0$ , fine structure (dominated here by spin-orbit, written with the radial $\xi(r)$ of Spin-Orbit Coupling because its expectation is about to become our perturbation coefficient), and the field term:

\hat H \;=\; \hat H_0 \;+\; \xi(r)\,\hat{\mathbf{L}}\cdot\hat{\mathbf{S}} \;+\; \frac{\mu_B B}{\hbar}\big(\hat L_z+2\hat S_z\big) \qquad (1)

where: $\xi(r)=\dfrac{1}{2m^2c^2}\dfrac1r\dfrac{dV}{dr}$ carries the Thomas-halved rest-frame magnetism, and the Zeeman term uses $g_s=2$ . Read the structure: two perturbations compete to be diagonalized first; the loser is treated in first order, in the winner's eigenbasis. The strong-field (Paschen-Back) regime is reached when the Zeeman energy greatly exceeds the fine-structure energy:

\mu_B B \;\gg\; \langle \hat H_{SO}\rangle \qquad\Longrightarrow\qquad \text{Paschen-Back regime} \qquad (2)

where: $\langle\hat H_{SO}\rangle$ is the fine-structure scale of the specific manifold — for hydrogen $n=2$ the $4.53\times10^{-5}$ eV gap, putting the boundary at tenths of a tesla. Which basis diagonalizes the winner? A commutator question, and Good Quantum Numbers and Symmetry-Adapted Bases supplies the doctrine: a label is good while its operator commutes with the dominant Hamiltonian. The field term commutes with $\hat L_z$ and $\hat S_z$ separately — but not with $\hat{\mathbf{L}}\cdot\hat{\mathbf{S}}$ , and $\hat{\mathbf{L}}\cdot\hat{\mathbf{S}}$ does not commute with $\hat S_z$ (all three statements machine-verified as exact matrices on $\ell=1\otimes s=\tfrac12$ ). So once the field dominates, the uncoupled labels $(m_\ell,m_s)$ become good and the coupled pair $(j,m_j)$ loses its mandate — the exact inverse of the weak-field logic in Weak-Field (Anomalous) Zeeman Effect, where spin-orbit dominates and the field's $\hat S_z$ is the troublemaker. One line survives the reversal: $\hat J_z=\hat L_z+\hat S_z$ commutes with both competitors.

Derivation AStrong-field energies from the basis flip

We derive the Paschen-Back energy $E=E_n+\mu_B B(m_\ell+2m_s)+\beta\,m_\ell m_s$ by diagonalizing the dominant Zeeman term exactly, then feeding the demoted spin-orbit term through first-order perturbation theory in the uncoupled basis. Three checkpoints: the field term must be exactly diagonal on $|n\,\ell\,m_\ell\,m_s\rangle$ , the dot product must split into a diagonal core plus ladder terms, and the ladder terms must contribute nothing at first order. The watch-out is the last step — "the off-diagonal pieces vanish" is a diagonal-expectation statement, not a claim that they are zero operators; they return at second order and in the exact treatment below.

Step 1. Diagonalize the winner. On the uncoupled states, $\hat L_z$ and $\hat S_z$ act by eigenvalue: $\hat H_Z|n\,\ell\,m_\ell\,m_s\rangle=\mu_B B(m_\ell+2m_s)|n\,\ell\,m_\ell\,m_s\rangle$ . This is exact — the strong-field zeroth order costs no perturbation theory at all, just the right basis.

Step 2. Split the loser. Using ladder operators, $\hat{\mathbf{L}}\cdot\hat{\mathbf{S}}=\hat L_z\hat S_z+\tfrac12(\hat L_+\hat S_-+\hat L_-\hat S_+)$ — an operator identity, verified as exact matrices.

Step 3. Take the diagonal. The ladder piece maps $|m_\ell,m_s\rangle\to|m_\ell\mp1,m_s\pm1\rangle$ : orthogonal states, zero diagonal expectation (verified: the matrix $\hat L_+\hat S_-+\hat L_-\hat S_+$ has an identically zero diagonal). Only $\hat L_z\hat S_z$ survives:

\langle \hat{\mathbf{L}}\cdot\hat{\mathbf{S}}\rangle \;=\; \hbar^2\, m_\ell m_s

Why the ladder terms are harmless here but not in weak field: they connect states whose Zeeman energies differ by $\mu_B B$ — non-degenerate at strong field, so second order only, suppressed by $\beta/\mu_B B$ . In weak field the same terms connect fine-structure-degenerate states, which is why the coupled basis had to be built there.

Step 4. Assemble. $E^{(1)}_{SO}=\langle\xi(r)\rangle\,\hbar^2 m_\ell m_s\equiv\beta\,m_\ell m_s$ , with $\beta=\hbar^2\langle\xi\rangle_{n\ell}$ — the spectroscopic $\zeta_{n\ell}$ of Spin-Orbit Coupling, consumed not recomputed: its $2p$ doublet gap $4.53\times10^{-5}$ eV equals $\tfrac32\zeta_{2p}$ , so $\beta_{2p}=\zeta_{2p}=3.02\times10^{-5}$ eV. The relativistic-kinetic piece (and Darwin for $s$ -states) is diagonal here too, adding an $m$ -independent shift — for $2p$ , $-2.64\times10^{-5}$ eV. Total:

E \;=\; E_n \;+\; \mu_B B\,(m_\ell+2m_s) \;+\; \beta\, m_\ell m_s \;+\; E_{\text{kin}}^{(n\ell)}

These results can be confirmed as exact matrices: the commutators, the ladder-diagonal audit, $\beta_{2p}=(2/3)\cdot4.53\times10^{-5}\text{ eV}=3.02\times10^{-5}$ eV, $E_{\text{kin}}(2p)=-2.64\times10^{-5}$ eV, and the identity that the uncoupled-basis diagonal of the full fine-structure operator equals $E_{\text{kin}}+\beta m_\ell m_s$ on all six $2p$ states.

One useful scale check is that at $B=10\,\mathrm{T}$ the Zeeman spacing, $\mu_BB=5.79\times10^{-4}\,\mathrm{eV}$ , is much larger than the spin--orbit scale $\beta$ , so the spectrum is already close to the Paschen--Back limit. However, the states $(m_\ell,m_s)=(-1,+\tfrac12)$ and $(+1,-\tfrac12)$ remain degenerate to first order because they have both the same Zeeman energy and the same diagonal spin--orbit correction $\beta m_\ell m_s=-\beta/2$ . Their residual splitting appears only through second-order mixing (or exact diagonalization), giving about $1.6\times10^{-6}\,\mathrm{eV}$ at $10\,\mathrm{T}$ .

The exact n = 2 treatment

Between the clean limits lies the intermediate regime, and hydrogen $n=2$ is small enough to solve honestly. The key observation is that rotational symmetry about the magnetic-field axis is preserved for every field strength. Both competitors commute with $\hat J_z$ — fine structure because it is rotationally invariant, the field term because $\hat L_z+2\hat S_z$ is built from commuting $z$ -projections — so the full Hamiltonian block-diagonalizes by $m_j=m_\ell+m_s$ at every $B$ :

\big[\hat H_{fs}+\hat H_Z,\;\hat J_z\big] \;=\; 0 \qquad\text{at all } B \qquad (3)

where: $\hat H_{fs}$ is the full $n=2$ fine-structure operator, whose coupled-basis eigenvalues are the $j$ -only energies of Fine Structure of Hydrogen: $-5.66\times10^{-5}$ eV ( $j=\tfrac12$ ), $-1.13\times10^{-5}$ eV ( $j=\tfrac32$ ). The magnetic field reduces rotational symmetry to rotations about the field axis, so the conserved quantum number is $m_j=m_\ell+m_s$ . The chapter's census: weak field $(j,m_j)$ , strong field $(m_\ell,m_s)$ , all fields $m_j$ . Within $2p$ (the $2s$ pair decouples — $\hat S_z$ cannot change $\ell$ — and shifts trivially as $\pm\mu_B B$ atop its own fine-structure energy), the blocks are: $m_j=\pm\tfrac32$ , one state each, exactly linear; $m_j=\pm\tfrac12$ , a $2\times2$ each, mixing $|2p_{3/2},m_j\rangle$ with $|2p_{1/2},m_j\rangle$ .

Go deeper · Derivation B · 2 minThe m_j blocks and the closed-form fan

Derivation BThe m_j blocks and the closed-form fan

We prove the block structure and diagonalize the $|m_j|=\tfrac12$ blocks symbolically, obtaining closed-form eigenvalues whose two asymptotes are the Landé fan and the strong-field grid. Strategy: build the $2\times2$ in the coupled basis, where fine structure is diagonal and the field needs exactly one off-diagonal element — the same $-\tfrac{\sqrt2}{3}\hbar$ element of $\hat S_z$ that Weak-Field (Anomalous) Zeeman Effect identified as its breakdown mechanism. Watch-out: check the closed form against both limits, because a sign slip in the off-diagonal element leaves eigenvalues intact (it enters squared) but silently corrupts the eigenvectors.

Step 1. Blocks exist. $[\hat H(B),\hat J_z]=0$ is verified as exact $6\times6$ matrices at arbitrary field — so the Hamiltonian never connects different $m_j$ and the $6\times6$ problem falls apart into $1+1+2+2$ .

Step 2. Edge states. $m_j=\pm\tfrac32$ is pure $(m_\ell,m_s)=(\pm1,\pm\tfrac12)$ , equal to pure $|2p_{3/2},\pm\tfrac32\rangle$ : coupled and uncoupled descriptions coincide, and the energy is exactly linear, $E=E_{3/2}\pm2\mu_B B$ — the two straight rails of the fan at every field.

Step 3. The $2\times2$ . In the basis $\{|{\tfrac32},m_j\rangle,|{\tfrac12},m_j\rangle\}$ , measuring energies from the $2p_{1/2}$ fine-structure level and writing $\gamma\equiv\mu_B B$ , $\Delta\equiv\Delta E_{fs}$ : diagonal entries $\Delta+\tfrac43 m_j\gamma$ and $\tfrac23 m_j\gamma$ (the Landé slopes $g_{3/2}=\tfrac43$ , $g_{1/2}=\tfrac23$ , quoted from their owner), off-diagonal $\gamma\langle{\tfrac32},m_j|\hat S_z|{\tfrac12},m_j\rangle/\hbar=-\tfrac{\sqrt2}{3}\gamma$ (Clebsch-Gordan exact, Condon-Shortley sign).

Step 4. Diagonalize:

E_\pm(m_j) \;=\; \frac{\Delta}{2}+m_j\gamma \;\pm\; \sqrt{\left(\frac{\Delta}{2}+\frac{m_j\gamma}{3}\right)^{2}+\frac{2\gamma^{2}}{9}}

The two eigenvalues of each $2\times2$ block are given by the Breit--Rabi--type expression above (valid for either sign of $m_j$ ).

Step 5. Asymptotes. For $\gamma\ll\Delta$ : $E_+\to\Delta+\tfrac43 m_j\gamma$ , $E_-\to\tfrac23 m_j\gamma$ — the anomalous fan recovered (numerically exact at 1 mT). For $\gamma\gg\Delta$ : eigenvalues approach $\mu_B B(m_\ell+2m_s)+E_{\text{kin}}+\beta\,m_\ell m_s$ and eigenvectors rotate into the uncoupled states — at 3000 T the upper $m_j=+\tfrac12$ eigenvector overlaps $|m_\ell{=}0,\uparrow\rangle$ to better than $10^{-3}$ (verified). The closed form is the bridge between the articles flanking this one. Why the coupled basis still scaffolds a strong-field result: fine structure is diagonal there, so all field dependence sits in one real off-diagonal number. Basis choice is a tool, not an allegiance. *These results hold as exact matrices: the block commutator at arbitrary $B$ ; the CG element $-\sqrt{2}/3$ ; symbolic eigenvalues matching the closed form for $m_j=\pm\tfrac12$ ; the radicand identity $(\Delta/2+\gamma/6)^2+2\gamma^2/9$ at $m_j=+\tfrac12$ ; the weak asymptote confirmed at 1 mT; and the strong asymptote with eigenvector overlap at 3000 T.*

The closed-form expression agrees with exact numerical diagonalization of the full $6\times6$ Hamiltonian. The table lists energies (in units of $10^{-5}\,\mathrm{eV}$ relative to the unperturbed energy $E_n$ ) at representative magnetic fields.

$B$	levels ( $10^{-5}$ eV)	regime reading
0.1 T	$-5.87,\,-5.48,\,-2.29,\,-1.50,\,-0.73,\,+0.03$	two Landé fans about $-5.66$ and $-1.13$ ; foreign- $j$ weight 0.3%
0.78 T	$-10.16,\,-8.26,\,-4.83,\,-3.04,\,+2.55,\,+7.90$	$B=B_{\text{int}}$ : neither label honest — foreign- $j$ weight 9.2%
5 T	$-59.0,\,-31.7,\,-4.30,\,-3.99,\,+26.4,\,+56.8$	grid emerging: $\{-2,-1,0,0,+1,+2\}\times\mu_B B$ visible
30 T	$-348.4,\,-176.3,\,-4.18,\,-4.12,\,+171.0,\,+346.2$	deep Paschen-Back: rays $\mu_B B(m_\ell+2m_s)+$ residues

Watch the two middle levels: born in different fine-structure multiplets, they settle onto the shared grid point 0, separated at 30 T by only $0.05\times10^{-5}$ eV — a residue that shrinks like $1/B$ . And watch what never happens: same- $m_j$ branches do not cross. Their gap $2\sqrt{(\Delta/2+m_j\gamma/3)^2+2\gamma^2/9}$ starts at $\Delta E_{fs}$ and never closes at any finite field — for $m_j=+1/2$ it grows monotonically, while for $m_j=-1/2$ it first dips to $\tfrac{2\sqrt2}{3}\,\Delta E_{fs}$ near $\mu_B B=\Delta E_{fs}/3$ (an avoided crossing, held open by the off-diagonal coupling) before growing without bound: the no-crossing doctrine of Second-Order Energy Correction in its most textbook habitat.

Beyond the electron: the same divorce, one floor down

The identical drama replays at the hyperfine scale, where the nuclear spin $\hat{\mathbf{I}}$ couples to $\hat{\mathbf{J}}$ a thousand times more weakly than $\hat{\mathbf{L}}$ to $\hat{\mathbf{S}}$ : millitesla fields then play the strong-field role, $\hat{\mathbf{I}}$ and $\hat{\mathbf{J}}$ decouple, $m_I,m_J$ replace $F,m_F$ — the Back-Goudsmit effect. The quantitative machinery — hyperfine constant, $F$ -ladder, and the true Breit-Rabi formula for the hydrogen ground state — belongs to Hyperfine Structure; nothing of it is computed here. Carry the structural lesson: "strong field" is always relative to the coupling being broken, and the same $m_j$ -block logic solves every floor of the tower.

Limiting cases

case	result	meaning
$B\to0$	fine-structure multiplets, $2j+1$ degenerate	the field-free ladder of Fine Structure of Hydrogen restored
$\mu_B B\ll\Delta E_{fs}$	$E=E_j+g_J\mu_B B\,m_j$	Landé fan — owned by Weak-Field (Anomalous) Zeeman Effect; foreign- $j$ mixing $\sim(\mu_B B/\Delta E_{fs})^2$
$\mu_B B\gg\Delta E_{fs}$	$E=\mu_B B(m_\ell+2m_s)+\beta m_\ell m_s+E_{\text{kin}}$	this article: uncoupled grid plus fine-structure residue
all $B$ , $m_j=\pm(\ell+\tfrac12)$	exactly linear branches	edge states are coupled and uncoupled at once — no mixing to do
all $B$ , same $m_j$	gap $\ge\Delta E_{fs}$ , monotonically growing	level repulsion: the fan never tangles
$S=0$	grid $=\mu_B B\,m_\ell$ : normal triplet at any field	no spin, no divorce — the regime-proof case of Zeeman Effect
hyperfine analogue	$\hat{\mathbf{I}}$ – $\hat{\mathbf{J}}$ decoupling at $\sim$ mT	Back-Goudsmit regime — owner Hyperfine Structure

Common trapThe strong-field basis is not optional — and the Landé factor does not degrade gracefully

Beyond the crossover there is no meaningful Landé $g_J$ factor to correct: once $j$ is no longer conserved, the spectrum must be obtained from the $m_j$ block diagonalization (or equivalently the uncoupled basis), not from $g_J\mu_BBm_j$ . A second common mistake is to replace the Zeeman energy by $\mu_BB(m_\ell+m_s)$ ; the electron spin contributes $2\mu_BBm_s$ , so the correct strong-field grid is $\mu_BB(m_\ell+2m_s)$ . Finally, $\beta m_\ell m_s$ is only the diagonal first-order expectation value of $\mathbf L\!\cdot\!\mathbf S$ ; the off-diagonal ladder terms are precisely those retained in the exact $2\times2$ treatment.

Worked examples

Example 1 — Hydrogen $n=2$ levels at $10\,\mathrm{T}$ . Construct the strong-field spectrum relative to the unperturbed energy $E_2$ , including the Zeeman grid, first-order fine-structure corrections, and the $2s$ states.

Step 1. Regime check: $\mu_B B=5.79\times10^{-4}$ eV vs $\Delta E_{fs}=4.53\times10^{-5}$ eV — ratio 12.8, safely strong-field; the first-order formula's worst error against the exact diagonalization is $8\times10^{-7}$ eV, under 2% of the fine structure.

Step 2. Grid plus residue, $E=\mu_B B(m_\ell+2m_s)+E_{\text{kin}}+\beta m_\ell m_s$ with $E_{\text{kin}}=-2.64\times10^{-5}$ eV, $\beta=3.02\times10^{-5}$ eV (in $10^{-4}$ eV): $(1,\uparrow)$ $+11.464$ · $(0,\uparrow)$ $+5.524$ · $(-1,\uparrow)$ $-0.415$ · $(1,\downarrow)$ $-0.415$ · $(0,\downarrow)$ $-6.052$ · $(-1,\downarrow)$ $-11.690$ .

Step 3. The $2s$ pair rides the grid at $m_\ell=0$ : $E=\pm\mu_B B+E_{fs}(2s_{1/2})=+5.222$ and $-6.354$ ( $10^{-4}$ eV) — offset from the $2p$ rungs because the $s$ -state fine structure differs.

Step 4. Interpret the spectrum. Five rungs spaced by $\mu_B B$ ; the residue pushes $(1,\uparrow)$ and $(-1,\downarrow)$ outward by $+1.5\times10^{-5}$ eV and pulls the $m_\ell m_s<0$ states inward. The grid-0 pair stays degenerate at this order (both carry $m_\ell m_s=-\tfrac12$ ); the exact diagonalization splits it by $1.6\times10^{-6}$ eV. Answer: rungs at $+11.46$ , $+5.52$ , $-0.42$ (doubled), $-6.05$ , $-11.69$ in units of $10^{-4}$ eV; $2s$ pair at $+5.22$ and $-6.35$ .

Example 2 — One manifold at three magnetic fields. Using the exact fan table, describe hydrogen $2p$ at 0.1 T, 0.78 T, and 30 T: which labels are honest, and what does the spectrum look like?

Step 1. 0.1 T (ratio 0.13): levels $-5.87,-5.48$ and $-2.29,-1.50,-0.73,+0.03$ in $10^{-5}$ eV — two clean Landé fans about the fine-structure levels, slopes $\tfrac23 m_j$ and $\tfrac43 m_j$ ; foreign- $j$ weight only 0.3%. $(j,m_j)$ is honest vocabulary — Weak-Field (Anomalous) Zeeman Effect territory (its 10%-mixing boundary is 0.17 T).

Step 2. 0.78 T (ratio 1.0, the crossover itself): $-10.16,-8.26,-4.83,-3.04,+2.55,+7.90$ . Neither fan nor grid: the foreign- $j$ weight in the mixed branches is 9.2%, so neither $(j,m_j)$ nor $(m_\ell,m_s)$ describes the states to better than one part in ten. Only $m_j$ is honest. The spectrum must come from the $2\times2$ blocks — there is no shortcut formula to quote.

Step 3. 30 T (ratio 38): $-348.4,-176.3,-4.18,-4.12,+171.0,+346.2$ — five rays $\mu_B B(m_\ell+2m_s)$ dressed by residues; uncoupled labels honest (mixed branches within 3% of the $\tfrac13$ -weight asymptote). Answer: At $0.1\,\mathrm{T}$ the spectrum is well described by the Landé picture; near $0.78\,\mathrm{T}$ neither weak- nor strong-field labels are fully adequate and the exact $2\times2$ block solution is required; by $30\,\mathrm{T}$ the Paschen--Back grid is established. Throughout the crossover, $m_j$ remains the only exact quantum number.

Origins · 1 minPaschen-Back Effect and how this was discovered

Paschen-Back Effect

1912–1926 · Tübingen, Bonn

The effect is named after two experimentalists who found order where theory expected chaos. Friedrich Paschen — Tübingen's master spectroscopist, famous for the infrared hydrogen series bearing his name — and his young collaborator Ernst Back pushed Zeeman spectroscopy to fields no one had tamed, several tesla from custom electromagnets. In 1912-13 they reported something backwards: the anomalous multiplets that had embarrassed Lorentz's theory since 1897 simplified as the field grew, collapsing toward the plain normal triplet — the pattern "became normal again" precisely when the magnetic energy overwhelmed the multiplet's own splitting. Pre-quantum physics could name the phenomenon (Back's thesis called it the "transformation" of anomalous patterns) but not explain it: that waited for 1925-26, when electron spin revealed the effect as a divorce of two angular momenta whose coupling the field outbids. The regime thus carries a rare distinction — named, mapped, and calibrated a decade before the quantum numbers it decouples were proposed. Paschen's precision fields later anchored early fine-structure-constant measurements; Back refined the strong-field patterns at Tübingen and Bonn for the rest of his career.

History

1897

— Preston photographs anomalous quartets and sextets classical theory cannot count

1912

— Paschen and Back observe anomalous multiplets collapsing toward a normal-like triplet at high field

1913

— Systematic Tübingen study maps the "transformation" across elements and fields

1921

— Landé's empirical g-factors organize the weak-field patterns the strong field erases

1925

— Uhlenbeck and Goudsmit propose electron spin; the decoupling picture becomes possible

1926

— Heisenberg and Jordan derive the weak-to-strong interpolation from matrix mechanics

1929

— Back and Goudsmit demonstrate the hyperfine analogue in bismuth: nuclear spin decoupled

Going further

Griffiths & Schroeter, Introduction to Quantum Mechanics, 3rd ed., §7.4 — weak, strong, and intermediate Zeeman side by side.
Sakurai & Napolitano, Modern Quantum Mechanics, 2nd ed., §5.3 — the same competition via degenerate perturbation theory.
Cohen-Tannoudji, Diu & Laloë, Quantum Mechanics, Vol. II, Complement $D_{XII}$ — the full interpolation worked honestly.
Shankar, Principles of Quantum Mechanics, 2nd ed., Ch. 17 · Townsend, A Modern Approach to Quantum Mechanics, 2nd ed., Ch. 11.

The strong-field energy $E_n+\mu_B B(m_\ell+g_s m_s)+\beta m_\ell m_s$ and the regime condition $\mu_B B\gg\langle H_{SO}\rangle$ are standard results. The $\langle\mathbf{L}\cdot\mathbf{S}\rangle=\hbar^2 m_\ell m_s$ reduction and the weak-to-strong interpolation are likewise well established. Constants used throughout: $\mu_B\approx5.79\times10^{-5}$ eV/T, $g_s\approx2.0023$ , $\alpha\approx1/137$ (CODATA). The "few-tesla" crossover sharpens to $B_{\text{int}}=0.78$ T for hydrogen $n=2$ . This article covers the fine-structure half of the Breit-Rabi-like interpolation — the exact $n=2$ $|m_j|=\tfrac12$ blocks and their closed form $E_\pm=\Delta/2+m_j\gamma\pm\sqrt{(\Delta/2+m_j\gamma/3)^2+2\gamma^2/9}$ — while the true hyperfine Breit-Rabi formula belongs to Hyperfine Structure.

Builds on: Spin-Orbit Coupling · Zeeman Effect · Degenerate Perturbation Theory · Good Quantum Numbers and Symmetry-Adapted Bases · Fine Structure of Hydrogen Leads to: Lamb Shift · First-Order Energy and State Corrections

Why does j die in a strong field but m_j survive?

Different symmetries. $j$ belongs to $\hat J^2$ , which commutes with the rotationally invariant fine structure but not with the field term (the culprit is $\hat S_z$ , which mixes different $j$ at fixed $m_j$ ). $m_j$ belongs to $\hat J_z$ , which commutes with both. The field kills the vector's length quantum number while sparing its projection — and the surviving $m_j$ is what block-diagonalizes the intermediate regime.

Is the Paschen-Back formula exact or perturbative?

Layered. The grid $\mu_B B(m_\ell+2m_s)$ is exact in the uncoupled basis. The residue $\beta m_\ell m_s$ is first-order in the demoted fine structure, good to order $\Delta E_{fs}^2/\mu_B B$ (worst error at 10 T: $8\times10^{-7}$ eV, under 2% of the fine structure). The $m_j$ -block diagonalization is exact at every field — the fallback when the ratio is near one.

Where exactly did the ladder terms of L·S go?

Nowhere — they were denied a first-order voice, not deleted. $\tfrac12(\hat L_+\hat S_-+\hat L_-\hat S_+)$ connects states differing in Zeeman energy by $\mu_B B$ , so it enters only at second order, suppressed by $\beta/\mu_B B$ . In the exact $2\times2$ blocks those same elements are the $-\tfrac{\sqrt2}{3}\gamma$ entries — kept, not dropped — which is why the exact levels bend where the first-order rays run straight.

Why do the two grid-0 states stay degenerate even with the fine-structure correction?

Coincidence of both labels: the pair shares $m_\ell+2m_s=0$ and $m_\ell m_s=−\tfrac12$ , so grid term and residue agree for both. They sit in different $m_j$ blocks whose second-order repulsions differ only slightly — exact splitting $1.6\times10^{-6}$ eV at 10 T. A near-degeneracy of dynamics, not symmetry, invisible at first order.

Does the strong field bring back the normal Zeeman triplet?

Almost — exactly what Paschen and Back saw. Electric-dipole transitions obey $\Delta m_\ell=0,\pm1$ with $\Delta m_s=0$ (spin does not talk to the photon's electric field), so photon energies cluster into three groups spaced by $\mu_B B$ — a normal-looking triplet, each component carrying a fine substructure from the $\beta m_\ell m_s$ residues of the two levels. The full transition machinery belongs to the time-dependent chapter.

Is there a strong-field regime for the Landé g-factor itself?

No — beyond the crossover there is no $g_J$ at all: it is defined as the slope of a $j$ -level's fan, and $j$ stops being conserved. What survives is the effective moment, now trivially $-(m_\ell+2m_s)\mu_B$ per state. The anatomy of $g_J$ — special cases, vector-model reading, zeros — is Landé g-Factor's territory.

How does this connect to the hyperfine Breit-Rabi formula everyone quotes?

Same play, smaller stage: replace $\hat{\mathbf{L}}\cdot\hat{\mathbf{S}}$ by the hyperfine $A\,\hat{\mathbf{I}}\cdot\hat{\mathbf{J}}$ and the analysis repeats with millitesla crossovers — the Back-Goudsmit effect. For the hydrogen ground state the $2\times2$ blocks yield the celebrated Breit-Rabi formula, which lives with its owner, Hyperfine Structure. The $E_\pm$ closed form derived here is its fine-structure sibling — same square-root anatomy, different coupling.