The semi-classical quantization rule generalizes the standing-wave idea. Instead of quantizing one circular orbit, it quantizes the action accumulated by each periodic coordinate: \(\oint p_i\,dq_i=n_i h\). The integral measures how much classical phase is accumulated over a closed cycle.
Use this as a method. First identify the periodic motion; then write the conjugate momentum; then integrate over a full cycle; finally solve the condition for the allowed energy, radius or orbit. The rule is not the final quantum theory, but it explains why boundary conditions and action become central.
The Sommerfeld-Wilson-Ishiwara rule quantizes the action accumulated over one complete cycle of a periodic coordinate. It is the natural generalization of "an integer number of wavelengths must fit" when the momentum is not constant.
The index \(i\) allows more than one coordinate to be quantized, which is essential for orbital problems beyond a single circular motion.
If momentum is constant, the standing-wave count is simply length divided by wavelength. Using de Broglie's relation gives
When \(p\) changes along the path, the product \(p\,dq\) must be integrated. Action is the classical quantity that naturally measures accumulated phase, so quantizing action is an early way to impose wave self-consistency.
The rule should be used as a calculation sequence:
This recipe is why the next examples focus on \(p(x)=\sqrt{2m[E-V(x)]}\): the potential determines the momentum along the classical path.
The rule is useful but not final. It gives good intuition for bound periodic systems, yet it misses effects that require full wave mechanics: zero-point shifts, degeneracy structure, spin and operator-based selection rules.
This page is designed to support short guided exercises on: Action integrals, allowed orbits and the logic behind early quantization rules.
Use these anchors to design compact exercises. The exercise should be answerable from this page plus standard algebra, while longer derivations, full worked examples and broader context should point back to the original book.