Rigidity, separability, and cusp conditions of a wave function
- URL: http://arxiv.org/abs/2208.06035v1
- Date: Thu, 11 Aug 2022 20:41:22 GMT
- Title: Rigidity, separability, and cusp conditions of a wave function
- Authors: Bo Gao
- Abstract summary: We introduce in quantum mechanics a concept of textitrigidity and a textitpinned point of a wave function.
The concept of rigidity is introduced to describe the sensitivity of a wave function to changes in energy, potential, and/or external perturbations.
We formulate cusp conditions and cusp functions as fundamental properties of an arbitrary $N$-body quantum system with $Nge 2$.
- Score: 4.152510794431946
- License: http://creativecommons.org/licenses/by-nc-nd/4.0/
- 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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