Simple one-dimensional quantum-mechanical model for a particle attached to a surface

TL;DR

This paper introduces a simple one-dimensional quantum model for a particle attached to a surface, solving the Schrödinger equation with Weber functions, analyzing eigenvalues, eigenfunctions, and zero-point energy relevant to surface adsorption.

Contribution

It provides an exact analytical solution for a surface-adsorbed particle model and discusses applications to hydrogen on palladium surfaces, including variational methods.

Findings

Eigenvalues and eigenfunctions characterized analytically.

Zero-point energy calculated for H on Pd(100).

Asymptotic behavior of eigenvalues derived.

Abstract

We present a simple one-dimensional quantum-mechanical model for a particle attached to a surface. We solve the Schr\"odinger equation in terms of Weber functions and discuss the behavior of the eigenvalues and eigenfunctions. We derive the virial theorem and other exact relationships as well as the asymptotic behaviour of the eigenvalues. We calculate the zero-point energy for model parameters corresponding to H adsorbed on Pd(100) and also outline the application of the Rayleigh-Ritz variational method.