When is Bohr-Sommerfeld quantization a good estimate and why is it sometimes (almost) exact?

Question

Bohr Sommerfeld quantization, where the allowed energies of quantum systems are determined from closed orbits with constructively interfering de Broglie waves (or closed loop action integrals given by multiples of the reduced Planck constant) is exact or near exact in some cases, such as the Coulomb potential or harmonic oscillator, but not always. Why might this be the case?

Answer 1

I'm pretty sure what you're referring to is old quantum theory, which comes from Hamilton-Jacobi Theory, from my understanding it has to with Legendre transformations. But in some coordinates it might easier to solve for the equations of motion, however when we do that we can lose information. The Sommerfeld approximation comes from the idea that the phase space can be squeeze and stretch causing an approximation, you can get a change in x in the position space and a change in momentum in momentum. The area must remain constant in time, hence deltap * Delta\x = constant.