Please post your questions, comments and results related to the 6x6 matrix describing 2 electrons and two sites. Feel free to edit the matrix in here, as well as some eigenvalues and their eigenvectors.
The best initial focus for this is, I believe, to complete the eigenvalue spectrum; that is to find all 6 eigenvalues and to think about their arrangement as a function of energy from lowest to highest. To get an explicit result from which we can begin to understand the essential physics, let's look for a result valid to 1st order in I/U, where I/U is less than one, and also the specific case I = 1 eV and U = 4 eV.
For each eigenvector there is a story. Why does it have the energy it has? How does it acquire that; what is the nature of its eigenvector? All of the eigenvectors have a story. Some of them have really interesting stories. Find them and tell their stories here.
PS. To give you a starting point to think (and dream and wonder) about this, I believe eigenvectors think mostly about one thing: energy. Some of them are pretty vain and they are focused on their own eigenvalue. (To be fair, some of them think also about relationships, e.g., orthogonality.)
For example, think about the triplets. What are they? What are they thinking? What do they have in common? What are that good at doing?
ReplyDeleteI meant: "what are they good at doing?"
DeleteMaybe ask yourself this: how do those eigenstates feel about U?
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