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

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.