From quartic anharmonic oscillator to double well potential

TL;DR

This paper demonstrates a method to accurately approximate eigenfunctions and eigenvalues of double-well potentials by combining solutions from quartic anharmonic oscillators, simplifying the analysis of these quantum systems.

Contribution

It introduces a novel approach that constructs double-well eigenfunctions from quartic oscillator solutions, providing highly accurate results.

Findings

Accurate eigenfunctions for double-well potentials obtained

Eigenvalues closely approximated using the new method

Method simplifies analysis of complex quantum potentials

Abstract

It is already known that the quantum quartic single-well anharmonic oscillator $V_{ao}(x)=x^2+g^2 x^4$ and double-well anharmonic oscillator $V_{dw}(x)= x^2(1 - gx)^2$ are essentially one-parametric, their eigenstates depend on a combination $(g^2 \hbar)$. Hence, these problems are reduced to study the potentials $V_{ao}=u^2+u^4$ and $V_{dw}=u^2(1-u)^2$, respectively. It is shown that by taking uniformly-accurate approximation for anharmonic oscillator eigenfunction $\Psi_{ao}(u)$, obtained recently, see JPA 54 (2021) 295204 [1] and Arxiv 2102.04623 [2], and then forming the function $\Psi_{dw}(u)=\Psi_{ao}(u) \pm \Psi_{ao}(u-1)$ allows to get the highly accurate approximation for both the eigenfunctions of the double-well potential and its eigenvalues.