Existence of Schrodinger Evolution with Absorbing Boundary Condition
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arXiv
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| Format: | Preprint |
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2019
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| _version_ | 1866909772817629184 |
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| author | Frolov, Lawrence Teufel, Stefan Tumulka, Roderich |
| author_facet | Frolov, Lawrence Teufel, Stefan Tumulka, Roderich |
| contents | Consider a non-relativistic quantum particle with wave function inside a region $Ω\subset \mathbb{R}^3$, and suppose that detectors are placed along the boundary $\partial Ω$. The question how to compute the probability distribution of the time at which the detector surface registers the particle boils down to finding a reasonable mathematical definition of an ideal detecting surface; a particularly convincing definition, called the \emph{absorbing boundary rule}, involves a time evolution for the particle's wave function $ψ$ expressed by a Schrödinger equation in $Ω$ together with an ``absorbing'' boundary condition on $\partial Ω$ first considered by Werner in 1987, viz., $\partial ψ/\partial n=iκψ$ with $κ>0$ and $\partial/\partial n$ the normal derivative. We provide here a discussion of the rigorous mathematical foundation of this rule. First, for the viability of the rule it plays a crucial role that these two equations together uniquely define the time evolution of $ψ$; we point out here how, under some technical assumptions on the regularity (i.e., smoothness) of the detecting surface, the Lumer-Phillips theorem implies that the time evolution is well defined and given by a contraction semigroup. Second, we show that the collapse required for the $N$-particle version of the problem is well defined. We also prove that the joint distribution of the detection times and places, according to the absorbing boundary rule, is governed by a positive-operator-valued measure. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1912_12057 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Existence of Schrodinger Evolution with Absorbing Boundary Condition Frolov, Lawrence Teufel, Stefan Tumulka, Roderich Mathematical Physics Quantum Physics Consider a non-relativistic quantum particle with wave function inside a region $Ω\subset \mathbb{R}^3$, and suppose that detectors are placed along the boundary $\partial Ω$. The question how to compute the probability distribution of the time at which the detector surface registers the particle boils down to finding a reasonable mathematical definition of an ideal detecting surface; a particularly convincing definition, called the \emph{absorbing boundary rule}, involves a time evolution for the particle's wave function $ψ$ expressed by a Schrödinger equation in $Ω$ together with an ``absorbing'' boundary condition on $\partial Ω$ first considered by Werner in 1987, viz., $\partial ψ/\partial n=iκψ$ with $κ>0$ and $\partial/\partial n$ the normal derivative. We provide here a discussion of the rigorous mathematical foundation of this rule. First, for the viability of the rule it plays a crucial role that these two equations together uniquely define the time evolution of $ψ$; we point out here how, under some technical assumptions on the regularity (i.e., smoothness) of the detecting surface, the Lumer-Phillips theorem implies that the time evolution is well defined and given by a contraction semigroup. Second, we show that the collapse required for the $N$-particle version of the problem is well defined. We also prove that the joint distribution of the detection times and places, according to the absorbing boundary rule, is governed by a positive-operator-valued measure. |
| title | Existence of Schrodinger Evolution with Absorbing Boundary Condition |
| topic | Mathematical Physics Quantum Physics |
| url | https://arxiv.org/abs/1912.12057 |