Bohr-Sommerfeld Quantization

The Bohr-Sommerfeld quantization condition is the earliest successful rule for selecting discrete energy levels in quantum systems, derived from the requirement that the action variable of a periodic classical orbit be an integer multiple of Planck's constant. For a system integrable in the sense of the Liouville-Arnold theorem, each action variable $ must satisfy the quantization condition = n_i \hbar$, where $ is a non-negative integer. This condition transforms the continuous spectrum of classical action variables into the discrete spectrum of quantum energy levels, and it was the bridge that carried the old quantum theory from the planetary model of the atom to the quantum mechanics of Schrödinger and Heisenberg.

The condition is not merely a historical curiosity. It is the leading-order term of the semiclassical expansion, and in modern quantum chaos it survives as the quantization rule for systems whose classical limit is integrable. The breakdown of the Bohr-Sommerfeld condition in chaotic systems — where invariant tori are destroyed and action variables lose their meaning — is one of the central puzzles of quantum chaos, and the KAM theorem governs the boundary between the regimes where the condition holds and where it fails.