The Infinite Square Well Is Subtle

TL;DR

This paper clarifies the mathematical formulation of the infinite square well potential by using distribution functions, revealing that it is not simply the limit of a finite well and resolving previous ambiguities.

Contribution

It introduces a precise potential model using Heaviside and delta functions, clarifying the nature of the infinite square well.

Findings

The potential is better described with distribution functions.

The infinite square well is not the limit of a finite well.

Ambiguities in the traditional model are resolved.

Abstract

We show that it needs a more delicate potential to confine particles inside a well. The original model containing a vague notation of infinity in the potential energy is ambiguous. Using the Heaviside step function and the Dirac delta-function, we give a precise form for the confining potential. Although such form appears unusual, the ambiguities are resolved. This form also shows that the infinite square well is not the limit of a finite square well.