The presence of non-analyticities and singularities in the wavefunction and the role of invisible delta potentials

TL;DR

This paper critically analyzes the claim that square-integrable divergent wavefunctions in quantum wells are physically valid, showing instead that they imply non-physical delta potentials with infinite potential energy.

Contribution

It clarifies the nature of divergences in wavefunctions, demonstrating they correspond to non-physical delta potentials with infinite energy, thus challenging prior assumptions.

Findings

Divergent wavefunctions imply delta potentials with infinite energy

Such divergences are not physically meaningful

The correct differential equation reveals the non-physical nature of these solutions

Abstract

This article examines the suggestion made in Ref. [EPL, 115 (2016) 60001] that a solution to a particle in an infinite spherical well model, if it is square-integrable, is a physically valid solution, even if at the precise location of the singularity there is no underlying physical cause, therefore, the divergence would have to be a nonlocal phenomenon caused by confining walls at a distance. In this work we examine this claim more carefully. By identifying the correct differential equation for a divergent square-integrable solution and rewriting it in the form of the Schroedinger equation, we infer that the divergent wavefunction would be caused by the potential V(r)=-r delta(r), which is a kind of attractive delta potential. Because of its peculiar form and the fact that it leads to a divergent potential energy <V> = - infinity, the potential V(r) and the divergent wavefunction associated with it are not physically meaningful.