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Rigidity, separability, and cusp conditions of a wave function
Quantum Physics · Submitted 11 Aug 2022
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Through these concepts and their mathematical implications, we introduce and formulate cusp conditions and cusp functions as fundamental properties of an arbitrary $N$-body quantum system with $N\ge 2$, greatly expanding their relevance beyond the Coulombic systems.
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The abstract, sentence by sentence
We introduce in quantum mechanics a concept of \textit{rigidity} and a concept of a \textit{pinned point} of a wave function. The concept of a pinned point is a generalization of a familiar concept in the description of a vibrating string, while the concept of rigidity is introduced to describe the sensitivity of a wave function to changes in energy, potential, and/or external perturbation. Through these concepts and their mathematical implications, we introduce and formulate cusp conditions and cusp functions as fundamental properties of an arbitrary $N$-body quantum system with $N\ge 2$, greatly expanding their relevance beyond the Coulombic systems. The theory provides rigorous constraints on an arbitrary $N$-body quantum system, specifically on its short-range pair correlation that is essential to a better understanding of strongly correlated systems. More broadly, the theory and the derivations presented here are part of a reconstruction of the mathematical and conceptual foundation of an $N$-body quantum theory, incorporating previously hidden properties and insights revealed through the concepts of rigidity and pinned points. It includes general analytic properties of a 2-body wave function versus energy and their relations to cusp conditions and cusp functions. It includes a rigorous derivation and an understanding, in terms of an emergent length scale, of the 2-particle separability in an $(N>2)$-body quantum system and its relations to cusp conditions. It also includes a classification of quantum systems, both 2-body and $N$-body, based on the universal behaviors in their short-range correlation.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We introduce in quantum mechanics a concept of \textit{rigidity} and a concept of a \textit{pinned point} of a wave function. | contribution 1.00 | 0.94 | 0.08 | asserted 0.99 |
| s1 | The concept of a pinned point is a generalization of a familiar concept in the description of a vibrating string, while the concept of rigidity is introduced to describe the sensitivity of a wave function to changes in energy, potential, and/or external perturbation. | contribution 0.66 | 0.56 | 0.06 | asserted 1.00 |
| s2 | Through these concepts and their mathematical implications, we introduce and formulate cusp conditions and cusp functions as fundamental properties of an arbitrary $N$-body quantum system with $N\ge 2$, greatly expanding their relevance beyond the Coulombic systems. | contribution 0.98 | 0.94 | 0.48 | asserted 1.00 |
| s3 | The theory provides rigorous constraints on an arbitrary $N$-body quantum system, specifically on its short-range pair correlation that is essential to a better understanding of strongly correlated systems. | interpretation 0.46 | 0.85 | 0.27 | asserted 1.00 |
| s4 | More broadly, the theory and the derivations presented here are part of a reconstruction of the mathematical and conceptual foundation of an $N$-body quantum theory, incorporating previously hidden properties and insights revealed through the concepts of rigidity and pinned points. | contribution 0.61 | 0.88 | 0.19 | asserted 1.00 |
| s5 | It includes general analytic properties of a 2-body wave function versus energy and their relations to cusp conditions and cusp functions. | contribution 0.93 | 0.85 | 0.08 | asserted 0.97 |
| s6 | It includes a rigorous derivation and an understanding, in terms of an emergent length scale, of the 2-particle separability in an $(N>2)$-body quantum system and its relations to cusp conditions. | contribution 0.84 | 0.94 | 0.13 | asserted 1.00 |
| s7 | It also includes a classification of quantum systems, both 2-body and $N$-body, based on the universal behaviors in their short-range correlation. | contribution 0.91 | 0.88 | 0.14 | asserted 1.00 |
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The paper
Original abstract
We introduce in quantum mechanics a concept of \textit{rigidity} and a concept of a \textit{pinned point} of a wave function. The concept of a pinned point is a generalization of a familiar concept in the description of a vibrating string, while the concept of rigidity is introduced to describe the sensitivity of a wave function to changes in energy, potential, and/or external perturbation. Through these concepts and their mathematical implications, we introduce and formulate cusp conditions and cusp functions as fundamental properties of an arbitrary $N$-body quantum system with $N\ge 2$, greatly expanding their relevance beyond the Coulombic systems. The theory provides rigorous constraints on an arbitrary $N$-body quantum system, specifically on its short-range pair correlation that is essential to a better understanding of strongly correlated systems. More broadly, the theory and the derivations presented here are part of a reconstruction of the mathematical and conceptual foundation of an $N$-body quantum theory, incorporating previously hidden properties and insights revealed through the concepts of rigidity and pinned points. It includes general analytic properties of a 2-body wave function versus energy and their relations to cusp conditions and cusp functions. It includes a rigorous derivation and an understanding, in terms of an emergent length scale, of the 2-particle separability in an $(N>2)$-body quantum system and its relations to cusp conditions. It also includes a classification of quantum systems, both 2-body and $N$-body, based on the universal behaviors in their short-range correlation.
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How the literature uses it
1 arXiv paper cite this in 10 sentences
Papers citing it · 7 sentences without a confident label
Builds on it 1 paper
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We mentioned that while the formulation of Ref. [ref] should be sufficient for systems of particles that only repel each other, we needed at least one more piece of the foundation, built around the concept of arrangement, for systems of particles of which some or all can attract and bind.
Arrangement and chemical complexity in an $N$-body quantum system · Introduction · 0.91
What it cites · 6 sentences without a confident label
Cites as background 1 paper
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A recent development on interaction is that of a multichannel quantum-defect theory for anisotropic potentials (MQDTA) [ref] .
Multichannel quantum-defect theory for anisotropic interactions · Introduction · 0.90
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