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arXiv · 1612.07849

Milne's Equation revisited: Exact Asymptotic Solutions

D. Shu·I. Simbotin·R. Côté
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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.

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D. Shu, I. Simbotin, R. Côté. 2016-12-23. Milne's Equation revisited: Exact Asymptotic Solutions. https://arxiv.org/abs/1612.07849

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