Quantum mechanics on topological networks

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Abstract
A quantum mechanical model of electrons on a network which has been used to obtain the line spectra of conjugated organic molecules and the band structure for various solid state systems is shown to be intimately connected with the homology and cohomology sequences of the algebraic topology of 1‐complexes. Thus, the remarkable qualitative agreement of these models to experiment implies that those calculated physical quantities are structural quantities. Quantum mechanical conservation laws analogous to Kirchhoff’s electrical conservation laws are derived by variational methods applied to the system’s Lagrangian. The topology is shown to enforce energy conservation on the network.

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