Use of Lambert’s theorem for then-dimensional Coulomb problem

Abstract

We present the analytical solution in closed form for the semiclassical limit of the quantum-mechanical Coulomb Green's 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 $\ensuremath{\nu}\ensuremath{\ge}5$, the semiclassical Green's function is found to be an excellent approximation to the quantum-mechanical Green's function.