Pedagogical applications of the one-dimensional Schr\"odinger's equation to proximity effect systems: Comparison of Dirichlet and Neumann boundary conditions

TL;DR

This paper investigates how Dirichlet and Neumann boundary conditions affect the solutions of the one-dimensional Schrödinger equation in proximity effect systems, revealing significant differences in the lowest eigenstates and unusual behaviors.

Contribution

It compares the impact of boundary conditions on proximity effect systems modeled by Schrödinger's equation, highlighting their dramatic influence on eigenstates and system behavior.

Findings

Boundary conditions significantly alter the lowest eigenstate.

Neumann and Dirichlet conditions lead to different system behaviors.

Unusual solution behaviors are observed in certain potential wells.

Abstract

Proximity effect systems in superconducting films can be modeled by a one-dimensional Schr\"odinger equation. Several systems are studied using Dirichlet and Neumann boundary conditions. It is observed that the two boundary conditions have a dramatic effect on the lowest eigenstate allowed in these systems, and points to unusual behavior for solutions of Schr\"odinger's equation in certain potential wells and proximity effect systems.