Physical Meaning of Neumann and Robin Boundary Conditions for the Schrödinger Equation

The non-relativistic Schrödinger equation on a domain $\mathrm{\Omega }\subset {\mathbb{R}}^d$ with boundary is often considered with homogeneous Dirichlet boundary conditions ($\psi (x)=0$ for $x$ on the boundary), homogeneous Neumann boundary conditions (${\mathrm{\partial }}_n\psi (x)=0$ for $x$ on the boundary and ${\mathrm{\partial }}_n$ the normal derivative), or Robin boundary conditions (${\mathrm{\partial }}_n\psi (x)=\alpha \psi (x)$ for $x$ on the boundary and $\alpha$ a real parameter). Physically, the Dirichlet condition applies if the potential is much higher outside than inside the domain (``potential well’'). We ask, when does the Neumann or Robin condition apply physically? Our answer is, when the potential is much lower (at the appropriate level) in a thin layer along the surface of a potential well, or when a negative delta potential of the appropriate strength is added at a surface close to the surface of the potential well.