Adjacency Graphs and Long-Range Interactions of Atoms in Quasi-Degenerate States: Applied Graph Theory

TL;DR

This paper explores how adjacency graphs can reveal hidden symmetries and analyze energy level evolutions, including level crossings, in quantum systems with long-range interactions, with applications to hydrogen atom interactions.

Contribution

It introduces the use of adjacency matrices and graphs to analyze symmetries and level crossings in quantum systems, extending the understanding of the no-crossing theorem in higher dimensions.

Findings

Adjacency graphs help identify hidden symmetries in quantum systems.

Level crossings can occur in higher-dimensional irreducible matrices.

Application to hydrogen 2S-2S interactions informs optical measurement techniques.

Abstract

We analyze, in general terms, the evolution of energy levels in quantum mechanics, as a function of a coupling parameter, and demonstrate the possibility of level crossings in systems described by irreducible matrices. In long-range interactions, the coupling parameter is the interatomic distance. We demonstrate the utility of adjacency matrices and adjacency graphs in the analysis of "hidden" symmetries of a problem; these allow us to break reducible matrices into irreducible subcomponents. A possible breakdown of the no-crossing theorem for higher-dimensional irreducible matrices is indicated, and an application to the 2S-2S interaction in hydrogen is briefly described. The analysis of interatomic interactions in this system is important for further progress on optical measurements of the 2S hyperfine splitting.