Physical Meaning of Neumann and Robin Boundary Conditions for the Schr\"odinger Equation

Abstract

The non-relativistic Schr\"odinger equation on a domain $\Omega\subset \mathbb{R}^d$ with boundary is often considered with homogeneous Dirichlet boundary conditions ($\psi(x)=0$ for $x$ on the boundary), homogeneous Neumann boundary conditions ($\partial_n \psi(x)=0$ for $x$ on the boundary and $\partial_n$ the normal derivative), or Robin boundary conditions ($\partial_n\psi(x)=\alpha\psi(x)$ 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.