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

TL;DR

This paper explores the physical interpretation of Neumann and Robin boundary conditions in the Schrödinger equation, linking them to potential configurations near the boundary such as thin layers or delta potentials.

Contribution

It provides a physical explanation for Neumann and Robin boundary conditions, relating them to low potential layers or surface delta potentials near the boundary.

Findings

Neumann conditions correspond to low potential layers near the boundary.

Robin conditions relate to surface delta potentials of specific strength.

Dirichlet conditions occur when the potential outside is much higher.

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.