Physical Meaning of Neumann and Robin Boundary Conditions for the Schrödinger Equation
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866913590449012736 |
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| author | Tumulka, Roderich |
| author_facet | Tumulka, Roderich |
| contents | The non-relativistic Schrödinger equation on a domain $Ω\subset \mathbb{R}^d$ with boundary is often considered with homogeneous Dirichlet boundary conditions ($ψ(x)=0$ for $x$ on the boundary), homogeneous Neumann boundary conditions ($\partial_n ψ(x)=0$ for $x$ on the boundary and $\partial_n$ the normal derivative), or Robin boundary conditions ($\partial_nψ(x)=αψ(x)$ for $x$ on the boundary and $α$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_15835 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Physical Meaning of Neumann and Robin Boundary Conditions for the Schrödinger Equation Tumulka, Roderich Quantum Physics Analysis of PDEs The non-relativistic Schrödinger equation on a domain $Ω\subset \mathbb{R}^d$ with boundary is often considered with homogeneous Dirichlet boundary conditions ($ψ(x)=0$ for $x$ on the boundary), homogeneous Neumann boundary conditions ($\partial_n ψ(x)=0$ for $x$ on the boundary and $\partial_n$ the normal derivative), or Robin boundary conditions ($\partial_nψ(x)=αψ(x)$ for $x$ on the boundary and $α$ 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. |
| title | Physical Meaning of Neumann and Robin Boundary Conditions for the Schrödinger Equation |
| topic | Quantum Physics Analysis of PDEs |
| url | https://arxiv.org/abs/2309.15835 |