Infinite potential well with a sinusoidal bottom

TL;DR

This paper introduces a matrix method to solve quantum wave equations, enabling solutions for a broader class of potentials, exemplified by an exact solution for a sinusoidal bottom in an infinite well.

Contribution

The authors develop a tridiagonal matrix approach that generalizes solvable potentials beyond traditional methods, demonstrated through an exact sinusoidal bottom solution.

Findings

Broader class of solvable potentials identified

Exact solution for infinite well with sinusoidal bottom provided

Matrix approach simplifies solving complex wave equations

Abstract

We construct a tridiagonal matrix representation of the wave operator that maps the wave equation into a three-term recursion relation for the expansion coefficients of the wavefunction. Finding a solution of the recursion relation is equivalent to solving the original problem. Consequently, a larger class of solvable potentials is obtained. The usual diagonal representation constraint results in a reduction to the conventional class of solvable potentials. To exhibit the power of this approach, we give an exact solution for the infinite potential well with sinusoidal bottom.