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

• URL: http://arxiv.org/abs/2309.15835v2
• Date: Mon, 2 Oct 2023 16:10:34 GMT
• Title: Physical Meaning of Neumann and Robin Boundary Conditions for the Schr\"odinger Equation
• Authors: Roderich Tumulka
• Abstract summary: Physically, the Dirichlet condition applies if outside of $Omega$ the potential is much higher than inside (potential well'') Our answer is, when the potential is much lower (at the appropriate level) in a thin layer before a potential well, or when a negative delta potential of the appropriate strength is added close to the potential well.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The non-relativistic Schr\"odinger equation on a domain $\Omega\subset \mathbb{R}^n$ with boundary is often considered with homogeneous Dirichlet boundary conditions ($\psi(x)=0$ for $x$ on the boundary) or 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 outside of $\Omega$ the potential is much higher than inside (``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 before a potential well, or when a negative delta potential of the appropriate strength is added close to the potential well.

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