On matching conditions in the WKB method

The modified matching conditions for quasiclassical wave functions on both sides of a turning point for the radial Schrödinger equation have been obtained. They differ significantly from the usual Kramers condition which holds for the one-dimensional case. Namely, the ratio C₂/C₁ in the subbarrier and the classical allowed regions is not a universal constant ( , as usual), but depends on the values of the orbital angular momentum l, energy E and on the behaviour of the potential V(r) at r → 0. The comparison with exact and numerical solutions of the Schrödinger equation shows that the modified matching conditions not only make the quasiclassical approximation in the subbarrier region asymptotically exact within the n → ∞ limit, but also considerably enhances its accuracy even in the case of small quantum numbers, n 1. The power-law, funnel and short-range potentials are considered in detail.