Zeeman Effect Revisited

The Zeeman effect is the splitting or shifting of spectral lines in an applied magnetic field. In historical order, it was one of the first signs that light emitted by atoms knows about internal angular momentum and magnetic moments. In modern language, the effect is a perturbation of atomic energy levels by magnetic coupling.

The compact named-effect reference is Zeeman Effect, Zeeman Effect as a Perturbation Example develops the generic weak-field and crossover methods, and Zeeman Effect in Atoms treats Breit–Rabi energies, polarization patterns, and spectroscopic inference. This page revisits the effect from the spin chapter’s perspective: why the normal Zeeman effect looked partly classical, why the anomalous Zeeman effect was so important, and how spin turns the anomaly into organized angular-momentum physics.

Normal Zeeman Effect

For a magnetic moment $\boldsymbol\mu$ in a magnetic field $\mathbf B$, the basic interaction is

$$H_Z = - \boldsymbol\mu\cdot\mathbf B.$$

For electron orbital angular momentum, using $e>0$ and electron charge $q=-e$,

$$\boldsymbol\mu_L = - \mu_B \frac{\mathbf L}{\hbar}, \qquad \mu_B = \frac{e\hbar}{2m_e}.$$

If the field is $\mathbf B=B\hat{\mathbf z}$, the orbital Zeeman Hamiltonian becomes

$$H_Z^{(L)} = \mu_B B \frac{L_z}{\hbar}.$$

An orbital eigenstate with $L_z=\hbar m_\ell$ is shifted by

$$\Delta E = \mu_B B m_\ell.$$

This simple result explains why a magnetic field can split a spectral line into components separated by equal energy intervals. For electric-dipole transitions in the simplest spinless picture, the allowed changes in magnetic quantum number are

$$\Delta m_\ell=0,\pm 1.$$

Those three possibilities give the familiar normal Zeeman triplet: one unshifted component and two symmetrically shifted components, with polarization depending on the observation geometry.

The normal effect was historically encouraging because it connected spectroscopy, magnetism, and angular momentum. It was also incomplete. Many atoms did not produce the simple triplet pattern.

Anomalous Zeeman Effect

The anomalous Zeeman effect was the name given to splitting patterns that did not fit the simple spinless orbital model. Alkali spectra, fine-structure multiplets, and other atomic lines showed more components, different spacings, and effective magnetic responses that could not be reduced to $\boldsymbol\mu_L$ alone.

The difficulty was not that the experiments were unreliable. The difficulty was that the old model had too few quantum numbers. A purely orbital magnetic moment gives one projection label $m_\ell$. Real atomic levels also involve electron spin, spin-orbit coupling, and total angular momentum.

In the common weak-field Russell-Saunders coupling regime, levels are described schematically by

$$\mathbf J = \mathbf L+\mathbf S, \qquad J_z\lvert J,m_J\rangle = \hbar m_J\lvert J,m_J\rangle.$$

The first-order magnetic shift is then written

$$\Delta E = g_J\mu_B B m_J,$$

where $g_J$ is the Landé factor for the level. The important point is conceptual: the magnetic splitting now depends on how orbital and spin angular momentum combine, not on orbital motion alone.

The detailed $g$-factor conventions and signs are collected in Magnetic Moments and Spin in Magnetic Field Hamiltonian. The present page uses the formula only to explain why the anomalous effect became intelligible once spin was included.

Why Anomalous Splitting Mattered

Anomalous Zeeman patterns mattered because they made the missing structure visible. They were not minor corrections to an otherwise complete picture; they showed that atomic states carried more information than the orbital quantum numbers of the old quantum theory.

Several lessons came together:

• spectral lines split according to angular-momentum projection, so magnetic fields reveal hidden level structure;
• simple orbital magnetism gives only a limited set of patterns;
• Pauli’s two-valued quantum number and the exclusion principle pointed to an additional electron label;
• Uhlenbeck and Goudsmit’s spin proposal supplied an intrinsic angular momentum and magnetic moment;
• Pauli’s spin matrices and Dirac’s relativistic equation later gave the formal setting in which the patterns no longer looked anomalous.

The word “anomalous” is therefore historical. It records the mismatch with the pre-spin model. In modern quantum mechanics, these splittings are expected consequences of spin, angular momentum coupling, and magnetic perturbation theory.

Spin and Modern Explanation

Modern atomic spectroscopy starts from a hierarchy of interactions. The Coulomb problem gives large-scale energy levels. Fine structure, including spin-orbit coupling, splits those levels. An external magnetic field then adds Zeeman shifts whose form depends on the field strength and coupling regime.

In weak fields, one often treats the magnetic field as a perturbation of already-coupled atomic levels. A state may be labeled by $L$, $S$, $J$, and $m_J$, and the shift is organized by the Landé factor. In stronger fields, the external field can compete with spin-orbit coupling. Then the appropriate labels change, and the system approaches the Paschen–Back regime rather than the weak-field Zeeman regime.

The angular-momentum algebra behind the internal $L$-$S$ coupling is developed in Spin–Orbit Coupling.

This is why there is no single universal Zeeman formula. The correct expression depends on the particle, charge convention, magnetic moment, angular-momentum coupling scheme, and field strength. The stable physical core is the magnetic coupling

$$H_Z = - \boldsymbol\mu\cdot\mathbf B,$$

with $\boldsymbol\mu$ determined by the relevant orbital, spin, nuclear, or effective degrees of freedom.

Common Misconceptions

• The normal Zeeman effect is not the generic Zeeman effect; it is a special simple pattern.
• “Anomalous” does not mean experimentally suspicious. It means anomalous relative to the old spinless orbital model.
• A magnetic field does not merely split a line; it splits or shifts energy levels, and spectral lines reflect differences between shifted levels.
• The formula $\Delta E=g_J\mu_B B m_J$ assumes a weak-field coupling regime and specific angular-momentum labels.
• Spin explains much of the anomalous effect, but real spectra can also involve fine structure, hyperfine structure, configuration mixing, and field-strength effects.
• The sign of a Zeeman shift is convention-dependent unless the magnetic moment and charge convention are stated.

Cross-Links

• Spin and Discrete Outcomes
• Magnetic Moments
• Electron Spin
• Pauli Exclusion Principle
• Pauli Matrices in Historical Context
• Space Quantization
• Stern–Gerlach Experiment
• Zeeman Effect
• Zeeman Effect in Atoms
• Zeeman Effect as a Perturbation Example
• What Spin Is and Is Not
• Magnetic Moments from Orbital Motion
• Spin in Magnetic Fields
• Spin–Orbit Coupling
• Angular Momentum Algebra
• Spin in Magnetic Field Hamiltonian
• Pauli Hamiltonian
• Pauli Equation

References

• P. Zeeman, “On the Influence of Magnetism on the Nature of the Light Emitted by a Substance,” Philosophical Magazine 43, 226-239, 1897.
• A. Landé, “Term Structure and Zeeman Effect of the Multiplets,” Zeitschrift für Physik 15, 189-205, 1923.
• W. Pauli, “Über den Zusammenhang des Abschlusses der Elektronengruppen im Atom mit der Komplexstruktur der Spektren,” Zeitschrift für Physik 31, 765-783, 1925.
• G. E. Uhlenbeck and S. Goudsmit, “Spinning Electrons and the Structure of Spectra,” Nature 117, 264-265, 1926, DOI: 10.1038/117264a0.
• W. Pauli, “Zur Quantenmechanik des magnetischen Elektrons,” Zeitschrift für Physik 43, 601-623, 1927.
• C. J. Foot, Atomic Physics, Oxford University Press, 2005.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.

Exercises

1. In the spinless orbital model, take $\mathbf B=B\hat{\mathbf z}$ and $H_Z^{(L)}=\mu_B B L_z/\hbar$. What are the shifts for $m_\ell=-1,0,+1$?

Solution

Using $L_z=\hbar m_\ell$,

$$\Delta E = \mu_B B m_\ell.$$

Thus the shifts are

$$-\mu_B B, \qquad 0, \qquad +\mu_B B.$$

They are equally spaced, which is the simplest normal-Zeeman pattern.

2. Why did the anomalous Zeeman effect point toward an additional electron degree of freedom?

Solution

A spinless orbital model supplies orbital projection labels and an orbital magnetic moment. Many observed spectra showed splitting patterns with more structure than this model could produce. The additional two-valued degree of freedom later identified as electron spin supplied intrinsic angular momentum and a spin magnetic moment, allowing the observed multiplets to be organized by total angular momentum.

3. Use

$$g_J = 1+ \frac{ J(J+1)+S(S+1)-L(L+1) }{ 2J(J+1) }$$

to find $g_J$ for $L=1$, $S=1/2$, and $J=3/2$.

Solution

The angular-momentum factors are

$$J(J+1)=\frac{15}{4}, \qquad S(S+1)=\frac{3}{4}, \qquad L(L+1)=2.$$

Therefore

$$g_J = 1+ \frac{15/4+3/4-2}{2(15/4)} = 1+ \frac{5/2}{15/2} = \frac{4}{3}.$$

4. What changes qualitatively between the weak-field Zeeman regime and the Paschen–Back regime?

Solution

In the weak-field Zeeman regime, the external field is treated as a perturbation of levels already organized by internal angular-momentum coupling, often with labels $L$, $S$, $J$, and $m_J$. In the Paschen–Back regime, the external magnetic field competes strongly with spin-orbit coupling, so $L$ and $S$ can decouple relative to the field and the useful quantum labels change.