A didactically motivated reexamination of a particle's quantum mechanics with square-well potentials

TL;DR

This paper critically reexamines the standard assumptions in quantum mechanics regarding square-well potentials, using a trapezoidal potential approach to assess the validity of common simplifications and continuity assumptions.

Contribution

It provides a didactic analysis questioning the traditional justifications for ignoring potential segments and continuity assumptions in square-well problems.

Findings

The justification for neglecting potential's vertical segments is scrutinized.

Continuity of eigenfunctions and derivatives at jump points is analyzed.

A trapezoidal potential approach offers insights into standard assumptions.

Abstract

We address two questions regarding square-well potentials from a didactic perspective. The first question concerns whether or not the justification of the standard a priori omission of the potential's vertical segments in the analysis of the eigenvalue problem is licit. The detour we follow to find out the answer considers a trapezoidal potential, includes the solution, analytical and numerical, of the corresponding eigenvalue problem and then analyzes the behavior of that solution in the limit when the slope of the trapezoidal potential's ramps becomes vertical. The second question, obviously linked to the first one, pertains whether or not eigenfunction's and its first derivative's continuity at the potential's jump points is justified as a priori assumption to kick-off the solution process, as it is standardly accepted in textbook approaches to the potential's eigenvalue problem.