Molecular quantum mechanics is usually understood in the context of a fast electron/slow nucleus approximation in which the fast electrons instantaneously adjust to slow motions of the nuclei. Recent experiments have shown that elementary steps in photosynthesis rely on non-adiabatic quantum mechanics. The non-adiabatic eigenfunctions that underlie this extraordinarily efficient process have remarkable nodal structures. These nodal structures were revealed by exactly factorizing the eigenfunctions into a vibrational marginal probability amplitude and an electronic conditional probability amplitude. Instead of the one-dimensional nodal curves familiar from adiabatic quantum mechanics, the two-dimensional non-adiabatic eigenfunctions from a model of photosynthetic energy transfer have zero-dimensional nodal points. Nodal points also occur for two-dimensional eigenfunctions from models that exhibit Berry’s geometric phase. We have made progress towards understanding the local geometry of these nodes by regarding the electronic probability amplitude as a vector field over an oriented surface of vibrational coordinates. For closed curves on the vibrational coordinate surface, proofs of the Poincaré-Hopf theorem establish that the electronic index around the curve equals the sum of the electronic indices of the nodes enclosed by the curve. The electronic indices of nodes are also related to the local radial exponent of the vibrational marginal probability amplitude and to Berry’s geometric phase. These results suggest that differential geometry and topology might help us to understand how photosynthesis generated the fossil fuels we burn and the oxygen we breathe.
This talk will be a friendly introduction to "categories", a concept that's everywhere in mathematics but that's often not introduced until the graduate curricula. The fact that we don't talk about it before then is just cultural, as the definitions are very basic and don't require that much math to formulate. I will give a lot of different examples, pooling from all kinds of classes you may or may not have taken. But worry not, even if you've taken very few math classes, there should be at least some examples that make sense. My hope is that, after this talk, if you hear the word "category" whispered by one of your peers, or casually dropped as a side remark by a professor, you'll feel like you know what that means.