Phase-Amplitude Representation of Continuum States

TL;DR

This paper introduces a phase-amplitude numerical method for solving 1D Schrödinger equations for continuum states, accurately normalizing wavefunctions and revealing unique oscillations in electron density during potential transitions.

Contribution

The paper presents a novel phase-amplitude approach with new basis polynomials for accurate continuum state solutions and boundary enforcement, applicable to Coulomb-screened potentials.

Findings

Accurate approximation of Coulomb wavefunctions with low error

Enforcement of derivative continuity of any order

Discovery of oscillations in electron density during potential transition

Abstract

A numerical method of solving the one-dimensional Schrodinger equation for the regular and irregular continuum states using the phase-amplitude representation is presented. Our solution acquires the correct Dirac-delta normalization by wisely enforcing the amplitude and phase boundary values. Our numerical test involving point-wise relative errors with the known Coulomb functions shows that the present method approximates both the regular and irregular wavefunctions with similar, excellent accuracy. This is done by using new basis polynomials that, among other advantages, can elegantly enforce the derivative continuity of any order. The current phase-amplitude method is implemented here to study the continuum states of Coulomb-screened potentials. We discovered that, during the parametric transition from a Hydrogen atom to the Yukawa potential, the electronic density at the origin exhibits surprising oscillation -- a phenomenon apparently unique to the continuum states.