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I am working on the matrix form of the Hamiltonian in the second quantization. I haven't taken any course on second quantization and I'm learning it on my own. I'm a little bit confused about the actual form of a system-bath Hamiltonian. Can anyone please introduce some references with numerical examples on this topic? like how the Hamiltonian looks like for a system-bath with given values of frequency of the modes.

I'm talking about a simple aggregate Hamiltonian in the one-exciton basis. The total Hamiltonian is the summation of three terms: the electronic part, the vibrational part and the coupling term that describes the interaction between the electronic excitation and the vibrational modes. Like the one in this paper:

The Hamiltonian is in the form of equation (A5) in the paper (page 10). The Hamiltonian should take the matrix form and that is where I'm not quite sure how to get it right. In particular the vibrational term which is ${H_{vib}} = \sum\limits_{n = 1}^N {\sum\limits_{j = 1}^M {{\omega _{nj}}} } a_{nj}^\dagger {a_{nj}}$. I understand that for this term the Hamiltonian is diagonal, but I'm not quite sure what values the number operator takes.

I already have the values of frequency (omega) in table 1 (page 5). I think the number operator is one for all the non-zero(the diagonal)terms of ${H_{vib}}$ . Another thing that confuses me is the two summations, where N is the number of monomers and M the number of modes. This makes it kind of confusing for me to write down the H matrix for example for a dimer(N=2) and trimer(N=3), while I have some arbitrary number of modes,let's say M=6 or 4. ${H_{vib}}$ has to take a square form, but with these numbers I'm not quite sure how to write it down.

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Could you give an example ? Are you talking about Leggett-Caldeira like calculation ? If so, I can discuss that more in an answer. – Adam Jan 26 '14 at 20:57
ok, I see. In order to have a self-contained question (as it should be), could you rewrite your question, including the typical Hamiltonian you are interested in ? And don't really understand what you want. Is it the Hamiltonian in a matrix form ? – Adam Jan 26 '14 at 22:26

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