Semiclassical quantization ($I/\hslash \gg 1$)

Semiclassical quantization uses classical trajectories and actions to approximate quantum states in a regime where relevant actions are large compared with $\hslash$. Phase consistency on a periodic orbit leads to Bohr-Sommerfeld quantization. Soliton collective coordinates can be quantized this way, while fluctuations supply additional corrections to the leading classical mass.

Single-valued semiclassical phase on a closed classical orbit quantizes its action, with Maslov index $\mu$ accounting for turning-point phase corrections. A cyclic angle has no oscillator turning points: its wavefunctions are $e^{i\ell q}$, with signed $\ell \in \mathbb{Z}$ for a $2\pi$ period and canonical momentum $\hslash \ell$. There is no half-integer shift for that free angular coordinate.