Book: The Correspondence Principle and its Application to the Quantum Theory of Spectral Lines

Overview
Bohr presents a systematic account of how the correspondence principle links classical electrodynamics to quantum transitions, using it as a tool to analyze the structure, frequencies, intensities, and polarization of spectral lines. Building on the Bohr–Sommerfeld model of quantized orbits, he shows that many empirical regularities of atomic spectra, Ritz’s combination principle, selection rules, multiplet patterns, and field-induced splittings, emerge by demanding agreement with classical radiation theory in the limit of large quantum numbers.

The correspondence principle
The principle states that for high quantum numbers the behavior of a quantized system approaches that of the corresponding classical system, not only for energies and mean motions but also for emitted radiation. Classically, a bound electron’s motion can be decomposed into Fourier components with discrete harmonics; quantum-mechanically, radiation arises from transitions between stationary states. Bohr identifies “corresponding” quantities: the frequency of a quantum transition between neighboring states matches the orbital frequency, and more generally transitions with quantum change Δn = k correspond to the k-th classical harmonic. This mapping becomes exact in the large-quantum-number limit and guides the construction of rules for finite quantum numbers.

Frequencies and the combination principle
Quantum transition frequencies are differences of state “terms,” reproducing Ritz’s combination principle. The correspondence principle connects this with classical motion by equating transition frequencies to the harmonics present in the classical trajectory of the electron. For hydrogen-like systems, this accounts for the Balmer and Lyman regularities and ties the line positions to the action-quantized orbits, while indicating how similar reasoning extends to more complex spectra.

Intensities from classical amplitudes
Classically, the power radiated in a given harmonic is proportional to the square of the corresponding Fourier amplitude of the electron’s acceleration. Bohr argues that, in the high-quantum limit, the relative intensities of quantum lines approach these classical values. This leads to prescriptions for transition probabilities: lines associated with strong classical harmonics are intense, while classically weak or absent harmonics produce weak or forbidden lines. The argument anticipates the later Einstein-coefficient formalism by tying emission probabilities to dipole amplitudes constrained by classical electrodynamics.

Selection rules and angular momentum
Applying the dipole radiation scheme to the quantized orbits gives definite selection rules. Because electric dipole radiation couples to the displacement of the charge, only transitions changing the orbital character contribute substantially. Bohr identifies the dominant rule Δl = ±1 for electric dipole transitions, with Δl = 0 strongly suppressed. In the presence of a preferred axis (a magnetic field), the space quantization of angular momentum components introduces the magnetic quantum number m, and the correspondence with the classical precessional motion yields Δm = 0, ±1, matching observed polarization patterns. These rules rationalize the observed absence of many classically conceivable lines and the structure of series.

Polarization and spatial structure of radiation
The vector character of the classical dipole field determines the polarization of spectral components. Lines with Δm = 0 correspond to radiation polarized parallel to the field direction (π components), while Δm = ±1 correspond to circularly polarized components (σ) viewed along the field, or to orthogonal linear polarizations viewed perpendicular to it. Bohr shows how these correspondences reproduce the characteristic polarization of multiplets and field-split lines.

External fields and multiplets
When weak magnetic or electric fields perturb the orbits, the classical motion acquires additional frequencies through slow precession and oscillation. The correspondence principle translates these into line splittings: the normal Zeeman triplet and the Stark components follow from the allowed Δm changes and the perturbative shifts of term values. In alkali and related spectra, coupling between orbital and core-induced motions produces multiplet structures whose patterns and relative intensities can be sketched by the same reasoning.

Significance
The analysis turns the correspondence principle into a working method: start from a classical model of the electron’s motion, identify its harmonics, amplitudes, and polarization, then impose quantization on the actions and map classical harmonics to quantum transitions. The approach unifies disparate spectral regularities, fixes many ambiguities of the old quantum theory, and sets the stage for later, more exact quantum mechanics, where selection rules and transition probabilities emerge from operator matrix elements that realize Bohr’s correspondences algebraically.