Improved power-series method for confined one-dimensional quantum-mechanical problems

TL;DR

This paper introduces two enhancements to the power series method for solving confined one-dimensional quantum problems, incorporating a variational step to improve convergence and discussing potential extensions to more complex systems.

Contribution

The paper presents a novel variational enhancement to the power series method, improving convergence for quantum-confined systems and suggesting broader applicability.

Findings

The variational step accelerates convergence compared to traditional methods.

The improved method outperforms the standard power series approach in an exactly-solvable model.

Potential for extending the approach to more complex quantum problems.

Abstract

We propose two improvements to the well-known power series method for confined one-dimensional quantum-mechanical problems. They consist of the addition of a variational step were the energy plays the role of a variational parameter. We compare the rate of convergence of the three methods on an exactly-solvable model. We also outline possible generalizations of the approaches to more complex problems.