The $1/n$ expansion in quantum mechanics

The classical approximation ($l=n-1\to\infty$) for the energy $\varepsilon^{(0)}$ and the semiclassical expansion in problems of quantum mechanics are discussed. A recursive method is proposed for calculating the quantum corrections of arbitrary order to $\varepsilon^{(0)}$, this being valid for both bound and quasistationary states. The generalization of the method to states with an arbitrary number of nodes and the possibility of a more general choice of the parameter of the semiclassical expansion are considered. The method is illustrated by the example of the Yukawa and “funnel” potentials and for the Stark effect in the hydrogen atom. These examples demonstrate the rapid convergence of the $1/n$ expansion even for small quantum numbers.