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

american journal of physics

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