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

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

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

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 higherdimensional 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