Wave function at the origin for large-n S-wave levels

Abstract

We show, within the WKB approximation, that for large-n S-wave levels |ϕ_n+1(0)|-|ϕ_n(0)| has the same sign as the curvature V"(r) of the potential, for a large class of central potentials with definite curvature. The result is shown to be supported by known exact results for solvable models and provides an extension to high-n levels of Martin's result for |ϕ_2S(0)|-|ϕ_1S(0)|. As a byproduct, we also obtain the result (∂/∂m){(1/m)[φ_n(0)]²}>~<0 if V"(r)<~>0.