Use of Lambert's Theorem for the n-Dimensional Coulomb Problem

TL;DR

This paper derives a closed-form semiclassical solution for the Coulomb Green function in n dimensions using Lambert's theorem, simplifying the problem to a one-dimensional form and demonstrating high accuracy for moderate quantum numbers.

Contribution

It introduces a novel projection method based on Lambert's theorem to analytically approximate the Coulomb Green function in multiple dimensions.

Findings

Semiclassical Green function closely matches quantum results for nu >= 5.

Method extends to classically forbidden regions.

Solution is suitable for efficient numerical evaluation.

Abstract

We present the analytical solution in closed form for the semiclassical limit of the quantum mechanical Coulomb Green function in position space in n dimensions. We utilize a projection method which has its roots in Lambert's theorem and which allows us to treat the system as an essentially one dimensional problem. The semiclassical result assumes a simple analytical form and is well suited for a numerical evaluation. The method can also be extended to classically forbidden space regions. Already for moderately large principal quantum numbers nu >= 5, the semiclassical Green function is found to be an excellent approximation to the quantum mechanical Green function.