Variational approach to the Schr\"odinger equation with a delta-function potential

TL;DR

This paper develops a variational method to accurately compute eigenvalues of the one-dimensional Schrödinger equation with a delta-function potential, emphasizing the importance of basis functions that incorporate the delta's effects.

Contribution

It demonstrates that the Rayleigh-Ritz variational method can be effectively applied to delta-function potentials with appropriate basis functions.

Findings

Accurate eigenvalues obtained for the delta potential problem.

Validation of the variational approach for singular potentials.

Basis sets that include delta effects improve accuracy.

Abstract

We obtain accurate eigenvalues of the one-dimensional Schr\"odinger equation with a Hamiltonian of the form $H_{g}=H+g\delta (x)$, where $\delta (x)$ is the Dirac delta function. We show that the well known Rayleigh-Ritz variational method is a suitable approach provided that the basis set takes into account the effect of the Dirac delta on the wavefunction.