Wilson-Sommerfeld quantization rule revisited

A fresh look at the origin of the Wilson-Sommerfeld quantization rule has b een pursued to gain new insight. The rule is shown to provide states that s atisfy several well-known theorems of standard quantum mechanics. A few oth er useful results and scaling relations are also derived. They emerge to ac t as nice guiding rules of thumb in the course of rigorous computations. Ce rtain features of true excited-state densities can be understood. Goodness of approximate densities can be assessed. Compressed systems can be studied profitably. A route is also sketched that allows one to retrieve classical trajectories from near-exact energy eigenfunctions for both bound and reso nant states by exploiting this rule. Additionally, a discussion on semiclas sical perturbation theory is presented emphasizing the asymptotic behavior. Pilot calculations demonstrate the success of the present endeavor under v arious circumstances. (C) 2001 John Wiley & Sons, Inc.