Exact quantization of nonsolvable potentials: The role of the quantum phase

Abstract

Semiclassical quantization is exact only for the so-called solvable potentials, such as the harmonic oscillator. In the nonsolvable case, the semiclassical phase, given by a series in ({Dirac_h}/2{pi}), yields more or less approximate results and eventually diverges due to the asymptotic nature of the expansion. A quantum phase is derived to bypass these shortcomings. It achieves exact quantization of nonsolvable potentials and allows us to obtain the quantum wave function while locally approaching the best predivergent semiclassical expansion. An iterative procedure allowing us to implement practical calculations with a modest computational cost is also given. The theory is illustrated on two examples for which the limitations of the semiclassical approach were recently highlighted: cold atomic collisions and anharmonic oscillators in the nonperturbative regime.