I'm trying to get into graphene, in detail, I try to derive the elec. energy dispersion. Sadly, I am not that familiar with condensed matter QM by now, so I got some basic questions and I hope to find some answers here. I considered the two different sublattices A and B. Now i defined some position states $|A_{j}\rangle$ $\in$ $H_{A}$ and $|B_{j}\rangle$ $\in$ $H_{B}$ I'd like them to be orthonormal, so $\langle A_{i}|A_{j}\rangle=\delta_{i,j}$ and $\langle B_{i}|B_{j}\rangle=\delta_{i,j}$ . Completeness: $\sum_{j} | A_{j}\rangle\langle A_{j}|=1$ and $\sum_{j} | B_{j}\rangle\langle B_{j}|=1$ Siml. for the corresponding k-states $|k_{A_{j}}\rangle$ and $|k_{B_{j}}\rangle$ except some constants mainly set by discrete Fourier transformation, they shall be orthogonal and complete. Now before going to create a Hamiltonian, I'd like to calc $\langle A_{j} |k_{A_{i}}\rangle$ . In case of a continous |x> this would simply be proportional to $e^{ip_{i}*x}$ . But in this case, my position states are discrete as well aren't they? I am unsure how to solve this, since the known proof for this proportionality depends on contnious x. In fact the whole p-operator is not working in its canonical definition anymore. How do I solve this problem and calculate my scalar product? Thank you very much!

Going on, I created some tight binding hamiltonian for next neighbour jumping: $H=t*\sum_{j}( \sum_{i=1 to 3} |A_{j}\rangle \langle A_{j}+e_i| +h.c.)$ with $e_i$ being the three translations from an A lattice atom to a net neighbour in B. So $\langle A_{j}+e_i|$ coressponds to a state from B. I guess it would be a good idea to express the x-states in k-states and replace them in the hamiltonian and throw them at my k states to calculate expectation values. But I really don't get how to do it exactly, especially since I got this problem with the scalar product and the continuity described above and since I am not sure how to express this translation getting from an A state to a b state mathematically, so I can calculate a scalar product with a coresponding k state.

I would be very thankful to anyone helping me to understand this topic, Best Regards Stefan


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