Particle in a box with a delta-function potential: strong and weak coupling limits

TL;DR

This paper analyzes a quantum particle in a box with a delta-function potential, revealing solvable limits in weak and strong coupling, and discusses spectral degeneracies and energy states.

Contribution

It provides an exact analysis of the particle in a box with delta potential in both coupling limits, highlighting spectral degeneracies and energy state behaviors.

Findings

Spectra are identical in weak and strong coupling limits.

A single negative-energy state exists for attractive potentials.

Degeneracy occurs when 1/|λ|=0.

Abstract

A particle in a one-dimensional delta-function potential and particle in a box are two well-known pedagogical examples; their combination, particle in a box with a delta-function potential V_\lambda(x)=\lambda\delta(x-x_0), too, has been recently explored. We point out that it provides a unique example that is solvable in the weak (\lambda->0) and the strong (1/\lambda->0) coupling limits. In either limit, the attractive and repulsive potentials lead to identical spectra, with the possible exception of a single negative-energy state that is present when the potential is attractive. We numerically obtain the spectra near the strong-coupling limit and discuss the consequences of the degeneracy that arises when 1/|\lambda|=0.