BohrSommerfeldQuantisierung

Bohr-Sommerfeld quantization is an early quantum theory development that extends Bohr’s atomic model by allowing non-circular orbits and treating quantization as a condition on classical motion. Proposed by Niels Bohr and Arnold Sommerfeld in the early 20th century, it applies to systems whose motion is integrable and expresses quantization in terms of action variables rather than wavefunctions.

The central idea is that for each degree of freedom with coordinate q_i and conjugate momentum p_i, the action integral ∮ p_i dq_i over one period must be an integer multiple of Planck’s constant: ∮ p_i dq_i = n_i h, where n_i are integers. In modern notation, and with turning-point corrections, this condition is often written as ∮ p_i dq_i = 2πħ(n_i + μ_i/4), where μ_i are Maslov indices depending on the system. These quantization rules yield discrete energy levels and a set of allowed orbital motions for a given potential.

In the hydrogen-like atom, the method reproduces the existence of discrete energy levels that depend on a principal quantum number n, which itself arises from a combination of radial and angular quantization (often described as n = n_r + l + 1). The theory also implies discrete angular momenta and predicts elliptical orbits in the old quantum framework, contrasting with the purely circular orbits of Bohr’s original model.

Limitations of Bohr-Sommerfeld theory are significant. It could not account for many spectroscopic features, spin, fine structure, or multi-electron atoms, and it was superseded by full quantum mechanics in the mid-1920s. Nevertheless, it influenced the development of semiclassical methods, such as EBK quantization and the WKB approximation, and remains a historical stepping stone in quantum theory.