Anharmonic oscillator and double-well potential: approximating eigenfunctions

TL;DR

This paper introduces a simple uniform approximation for the ground state eigenfunction's logarithmic derivative in anharmonic oscillators and double-well potentials, enabling a rapidly convergent perturbation theory.

Contribution

It presents a novel uniform approximation method for the eigenfunctions of anharmonic oscillators and double-well potentials, improving perturbation theory convergence.

Findings

Fast convergence of the proposed perturbation theory.

Effective approximation for ground state eigenfunctions.

Applicable to potentials with positive and negative quadratic terms.

Abstract

A simple uniform approximation of the logarithmic derivative of the ground state eigenfunction for both the quantum-mechanical anharmonic oscillator and the double-well potential given by $V= m^2 x^2+g x^4$ at arbitrary $g \geq 0$ for $m^2>0$ and $m^2<0$, respectively, is presented. It is shown that if this approximation is taken as unperturbed problem it leads to an extremely fast convergent perturbation theory.