Perturbation Theory for Particle in a Box

TL;DR

This paper tests a recently developed strong-coupling perturbation theory on a particle in a box, demonstrating rapid convergence to the exact quantum energy by resumming series at infinite coupling.

Contribution

It applies strong-coupling perturbation theory to a particle in a box, showing its effectiveness in a well-understood quantum system.

Findings

Strong-coupling series converge rapidly to the exact energy.

Resummation at infinite coupling yields accurate energy estimates.

Method demonstrates potential for hard-wall quantum systems.

Abstract

Recently developed strong-coupling theory open up the possibility of treating quantum-mechanical systems with hard-wall potentials via perturbation theory. To test the power of this theory we study here the exactly solvable quantum mechanics of a point particle in a one-dimensional box. Introducing an auxiliary harmonic mass term $m$, the ground-state energy $E^{(0)$ can be expanded perturbatively in powers of $1/md$, where $d$ is the box size. The removal of the infrared cutoff $m$ requires the resummation of the series at infinitely strong coupling. We show that strong-coupling theory yields a fast-convergent sequence of approximations to the well-known quantum-mechanical energy $E^{(0)= \pi ^2/2d^2$.