Milne's Equation revisited: Exact Asymptotic Solutions

TL;DR

This paper introduces new methods for solving Milne's equation, enabling accurate and efficient computation of wave functions and asymptotic solutions across various physical contexts.

Contribution

It presents a novel approach linking a third order linear differential equation to Milne's nonlinear equation, improving solution accuracy and computational efficiency.

Findings

Achieved numerically exact asymptotic solutions for long-range potentials

Developed optimization schemes for fast, accurate wave function computation

Provided a method for non-oscillatory solutions of Milne's equation

Abstract

We present novel approaches for solving Milne's equation, which was introduced in 1930 as an efficient numerical scheme for the Schr\"odinger equation. Milne's equation appears in a wide class of physical problems, ranging from astrophysics and cosmology, to quantum mechanics and quantum optics. We show how a third order linear differential equation is equivalent to Milne's non-linear equation, and can be used to accurately calculate Milne's amplitude and phase functions. We also introduce optimization schemes to achieve a convenient, fast, and accurate computation of wave functions using an economical parametrization. These new optimization procedures answer the long standing question of finding non-oscillatory solutions of Milne's equation. We apply them to long-range potentials and find numerically exact asymptotic solutions.