the Zeeman effect
/ ZAY-mahn /
Place an atom in an external magnetic field and its spectral lines split into several closely-spaced lines. The reason is intuitive: an atom carries a magnetic moment from its electrons' orbital and spin angular momentum, and a magnet in a field has an energy that depends on its orientation, so states that were degenerate in orientation now spread out in energy. Pieter Zeeman saw this in 1896 and won a Nobel Prize; the field's own strength is written on the atom's spectrum.
The interaction is H = (mu_B/hbar)(L + g_s S)·B, with mu_B the Bohr magneton and g_s ≈ 2 the electron spin g-factor. In the weak-field regime, where the field is small compared to fine structure, J stays a good quantum number and each level shifts by Delta E = g_J mu_B m_J B, splitting a level into its 2J+1 equally-spaced m_J sublevels. The Landé g-factor g_J = 1 + [j(j+1) + s(s+1) - l(l+1)]/[2 j(j+1)] encodes how orbital and spin moments project onto J. In the strong-field limit (Paschen-Back regime) the field overwhelms spin-orbit coupling, L and S decouple, and the pattern simplifies again.
The Zeeman effect is a foundational diagnostic: it measures magnetic fields in sunspots and distant stars, underlies magnetic resonance and atomic clocks, and provides the m_F-substructure that laser cooling and trapping exploit. The historically decisive point is honesty about the name: the 'normal' Zeeman effect — the simple, classically-predicted triplet — occurs only when the total spin S = 0. The generic case, with spin present, gives a richer 'anomalous' pattern that classical physics could not explain and that was one of the strongest early clues that the electron carries spin.
A ^2S_(1/2) level (j = 1/2, l = 0, s = 1/2) has g_J = 2, so in a field B it splits into m_J = +1/2 and -1/2 sublevels separated by 2 mu_B B; a ^2P_(1/2) level has g_J = 2/3, a different spacing. That the two levels split by different amounts is the hallmark of the anomalous Zeeman effect.
Different g_J values for the upper and lower levels make the split pattern 'anomalous'.
The Landé formula assumes weak fields where J is still good; when the field rivals fine structure you leave the linear regime, and at very strong fields the Paschen-Back pattern (labelled by m_l and m_s) takes over.