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Viewed perpendicular to the magnetic field, the normal Zeeman effect has a $\pi$ component with linear polarization parallel to the magnetic field and two $\sigma$ components polarized perpendicular to it. Along the magnetic field, $\pi$ vanishes and the $\sigma$ pair has opposite circular polarizations. Pieter Zeeman described these geometry-dependent patterns in his lecture.^[ref]

The geometry can be derived from electric dipole radiation. Let the observation unit vector be $\mathbf{n}=(\sin \theta ,0,\cos \theta )$, with $\theta$ measured from the magnetic field. Radiation retains only the transverse part of the oscillating electric dipole moment $\mathbf{d}$:For $\pi$, take $\mathbf{d}\parallel \hat{\mathbf{z}}$; its angular light intensity factor is ${\sin }^2\theta$, so it vanishes on axis. For ${\sigma }^{\pm }$, use normalized complex dipoles $(\hat{\mathbf{x}}\pm i\hat{\mathbf{y}})/\sqrt{2}$. Their angular factors are $(1+{\cos }^2\theta )/2$. Projection gives circles on axis, lines at $\theta =\pi /2$, and generally elliptical polarization between. These factors assume equal dipole magnitudes; actual line light intensities also depend on initial quantum state populations, transition strengths and collection geometry. No absolute handedness name is needed: it depends on the viewing and phase conventions.

A transverse polarizer can separate $\pi$ from $\sigma$ even before the spectral line components are fully resolved. Along the magnetic field, a quarter-wave plate followed by a polarizer converts the two opposite circular polarizations into distinguishable linear settings. This explains why polarization analysis carried information beyond a small increase in line width.

The normal Zeeman effect is the simple three-frequency pattern obtained in the orbital limit. Assume negligible net spin, negligible hyperfine structure and quadratic shifts, and an allowed electric-dipole interaction between levels with orbital splitting factor one. A level with $J=0$ has only $m_J=0$ and no first-order shift. For the other levels, $g_J=1$ gives $\Delta E={\mu }_B{m}_{J}B$. Define $q=m_u-m_l$, rather than the final-minus-initial change of the emitting atom. The electric-dipole selection rule allows $q=0,\pm 1$, henceMany allowed transitions can share one frequency; a triplet need not mean exactly three transitions. The cleanest illustrative example is $J_u=1\to J_l=0$, with three upper quantum states and one unshifted lower quantum state. The $q=0$ component is called $\pi$, and $q=\pm 1$ components ${\sigma }^{\pm }$ with this stated convention. During emission the atom changes its magnetic quantum number by $-q$.

For an original numerical example choose $B=0.50\mathrm{T}$ and a zero-field vacuum wavelength ${\lambda }_0=500\mathrm{nm}$. Using ${\mu }_B\simeq 9.27401\times 10^{-24}\mathrm{J}{\mathrm{T}}^{-1}$ and $h=6.62607015\times 10^{-34}\mathrm{J}\mathrm{s}$ gives $\delta {\nu }_B\simeq 7.00\mathrm{GHz}$. The two outer components are separated by $2\delta {\nu }_B\simeq 14.0\mathrm{GHz}$. Differentiating the vacuum relation $\lambda =c/\nu$, with $c$ the speed of light, givesEach outer component lies about $0.00584\mathrm{nm}$ from the centre. Positive $q$ means higher frequency and shorter wavelength. Equal spacing in frequency is only approximately equal spacing in wavelength. These are calculated values for a chosen model, not historical measurements.

Use the leading nonrelativistic magnetic coupling for an electronic level with total angular momentum operator $\mathbf{J}$, and choose the $z$ axis along a uniform external magnetic field $\mathbf{B}=B\hat{\mathbf{z}}$, $B\ge 0$. Write its magnetic quantum number as $m_J$, so $J_z|J,m_J⟩=\hslash m_J|J,m_J⟩$. There are $2J+1$ possibilities from $-J$ to $J$. With Landé g-factor $g_J$, the first-order shift isHere $e>0$ is the magnitude of the electron electric charge, $m_e$ its mass, $h$ the Planck constant, and ${\mu }_B$ the Bohr magneton. Both $g_J$ and $m_J$ are dimensionless; with $B$ measured in tesla and ${\mu }_B$ in joules per tesla, $\Delta E$ is in joules. The photon frequency shift $\Delta {\nu }_{ul}$ is in hertz, not angular frequency; indices $u,l$ denote upper and lower levels. NIST states this weak-field convention and its orbital/spin assumptions explicitly.^[ref]

The sign follows directly from magnetic dipole potential energy. Within this electronic $J$ multiplet, the projected magnetic dipole moment is $\mathbf{\mu }=-g_J{\mu }_B\mathbf{J}/\hslash$. Therefore $H_Z=-\mathbf{\mu }\cdot \mathbf{B}$ has the displayed positive coefficient. Subtracting the shifts of the two levels and using photon energy $h{\nu }_{ul}=E_u-E_l$ gives the transition formula. For the orbital case, ${\mu }_B/h=e/(4\pi m_e)$ gives the same leading frequency spacing as the classical charged oscillator. This explains the historical sensitivity to charge divided by mass; the quantum state derivation here is modern.

“Weak” compares interaction energies: mixing with neighbouring fine structure levels must be small, so $J$ remains a useful quantum number. The formula keeps the term linear in $B$ and neglects quadratic shifts. It also omits hyperfine structure; if that is resolved and controls the coupling, the appropriate total electronic-plus-nuclear angular momentum and splitting factor must replace $J,g_J$. Weak compared with fine structure does not automatically mean weak compared with hyperfine structure.

In August 1896, Pieter Zeeman observed broadened sodium spectral lines at Leiden, initially without resolved components. His lecture describes a slit and concave diffraction grating, then compares the broadening with the sodium doublet separation: about one thirtieth, roughly $0.2$ ångström. This supplied a spectral reference scale, not an absolute magnetic field calibration. Later cadmium photographs displayed resolved transverse triplets and axial doublets.^[ref]

The fuller report specifies a concave diffraction grating of ten-foot radius with 14,438 lines per inch, an adjustable slit and a micrometer eyepiece with vertical crosswire. Salt in a flame between electromagnet poles supplied the sodium spectral lines. Repeated switching broadened and restored them. To test changes in flame temperature or vapour density, Pieter Zeeman examined photon absorption through a heated porcelain tube with water-cooled glass windows. The arc lamp was displaced from the magnet and its light redirected by a mirror to avoid disturbing the arc. Rotating a narrower tube made the absorption spectral lines equally broad at different heights; the magnetic field still broadened their whole length.^[ref]

For the axial test, pierced electromagnet poles let the light travel along the magnetic field. A quarter-wave plate and polarizer selected one broadened edge; rotating the analyser by $90^{\circ }$ or reversing the magnet current exchanged the visible edge. The crosswire made its displacement detectable. This tested electromagnetic polarization as well as line width. For its rough estimate the report compared each side's widening with the sodium doublet separation, obtaining about one fortieth, and stated an approximate magnetic field intensity in historical units. These passages give no absolute field-calibration procedure; the inferred charge-to-mass ratio was explicitly a first approximation.^[ref]

Hendrik Antoon Lorentz's charged-oscillator theory connected the observed splitting to moving electric charge. In modern language, a line-position measurement needs a wavelength or frequency calibration and an independent magnetic field calibration before it determines a magnetic dipole moment. Resolving the individual components adds the transition information used in the worked example below.

An equal incoherent mixture within a fixed total-$J$ multiplet has density operator $\rho =P_J/(2J+1)$, where $P_J$ is its orthogonal projection. If the applied-field Hamiltonian operator is proportional to $L_z+2S_z$ and no spin-orbit interaction is retained, its energy eigenstates are the uncoupled states $|m_L,m_S⟩$. The Born rule gives their populations by the formula above, or equivalently by squared Clebsch-Gordan decomposition coefficients. Equal populations of coupled states need not give equal populations of magnetic energy eigenvalues.

For $L=1,S=1/2,J=3/2$, the nonextremal coupled states have weights $2/3$ and $1/3$ in their two uncoupled components. The five dimensionless magnetic energies $-2,-1,0,1,2$ then have populations $1/4,1/6,1/6,1/6,1/4$; the zero energy contains two independent states, each with population $1/12$. In a different, weak-field approximation with a dominant spin-orbit interaction, projecting into the isolated multiplet gives the Landé g-factor spectrum instead. The full operator and its projection must not be interchanged.

For two visible, approximately massless leptons, both measured four-momenta enterThis is a reconstructed invariant mass, including longitudinal information, rather than the lower bound obtained from transverse mass. As an original example, two opposite-azimuth leptons with $p_{T+}=p_{T-}=45\mathrm{GeV}$ and equal rapidities have $m_{\ell \ell }=90\mathrm{GeV}$. Changing their rapidity difference to one raises it to about $101.486\mathrm{GeV}$, even though their transverse momenta are unchanged.

The ideal transverse mass bound concerns the actual event's parent invariant mass, not a perfectly sharp universal number. A finite decay width spreads parent invariant masses; particle jet recoil changes daughter directions; invariant-mass resolution, momentum resolution and imperfect reconstruction can smear an observed endpoint. Recoil alone does not invalidate the ideal inequality when both daughter transverse momenta are correct. Additional invisible particles, radiated photons and a massless-daughter approximation can invalidate the simple two-body reconstruction. Likewise, a Z boson pair-mass peak requires control of identification, detection efficiency, instrumental response and backgrounds. A peak position and an endpoint are different summaries of selected event populations.

The 1983 UA2 experiment Z boson paper reports eight candidate electron pairs, including radiative decays, at $\sqrt{s}=550\mathrm{GeV}$. Four events entered its invariant mass determination. The published result was $91.9\pm 1.3\pm 1.4\mathrm{GeV}/c^2$, the second uncertainty systematic. These are that paper's reported numbers, not inputs fitted to the illustration below.^[ref]

The 1983 UA1 experiment paper used proton–antiproton collisions at $\sqrt{s}=540\mathrm{GeV}$. A high-transverse momentum electron candidate required a track matched to an electromagnetic particle shower, with little energy behind it in the hadronic particle calorimeter. Test beams checked electron and pion responses; ordinary events and particle jet samples checked the imbalance resolution. Two searches, beginning with isolated electrons and with missing transverse energy, converged on the same candidates. Conversion pairs, hadrons and heavy-flavor decays were examined as backgrounds.^[ref]