On the functional form of the infinite square well model

TL;DR

This paper refines the infinite square well quantum model by deriving a precise functional form of the potential energy, eliminating ambiguities, and enabling direct calculation of eigenstates and eigenvalues without boundary conditions.

Contribution

It provides a clear functional form of the potential energy in the infinite square well, resolving previous ambiguities and simplifying the derivation of eigenstates and eigenvalues.

Findings

Derived a precise functional form of V(x)

Confirmed Ehrenfest's theorem directly

Eliminated boundary condition ambiguities

Abstract

The original model of the infinite square well contains a vague notation infinity and therefore results some ambiguities. We investigate to obtain a functional form for the potential energy V(x). This is done by substituting back the original energy eigenstates and eigenvalues into the Schrodinger equation. We then obtain a precise functional form of the V(x). From this reformed model, we show that energy eigenstates and eigenvalues can directly be obtained without the need of imposing boundary condition, Ehrenfest's theorem can directly be confirmed, and ambiguities in the original model can be resolved.