Closed form solution for a double quantum well using Gr\"obner basis

TL;DR

This paper derives analytical expressions for the spectrum, eigenfunctions, and dipole matrix elements of a double quantum well with varying potential heights and effective masses, using Gr"obner basis to solve the coupled polynomial equations.

Contribution

It introduces a novel application of Gr"obner basis to analytically solve for eigenvalues and eigenfunctions in complex double quantum well systems.

Findings

Analytical expressions for spectrum and eigenfunctions are obtained.

The method handles different potential heights and effective masses.

Eigenvalue equations are solved without explicit transcendental solutions.

Abstract

Analytical expressions for spectrum, eigenfunctions and dipole matrix elements of a square double quantum well (DQW) are presented for a general case when the potential in different regions of the DQW has different heights and effective masses are different. This was achieved by Gr\"obner basis algorithm which allows to disentangle the resulting coupled polynomials without explicitly solving the transcendental eigenvalue equation.