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It's probably very clear and well-established for those who rigorously studied Quantum Mechanics but I don't think what I am going to ask is easily 'google'-able but if it is so - please send me to the correct source before spending time.

But don't recommend the whole Landau-Lifgarbagez QM text please.

The thing is, I am routinely performing numerical simulations that involve a discretized single particle, one-band effective mass Hamiltonian almost everyday. I discretizefree space (note that I am using an effective mass, so that's okay even for an electron moving in a solid

which is okay.

Which is not okay is that if I don't use an effective mass approach and decide to go to an ATOMISTIC hamiltonianwhich could be derived from first principlesthe lattice I am going to work on will bediscrete by itself!The point is, everything is made up of atoms and I will have to work on a discrete latticeanyway.

And this Hamiltonian will beexactif I am not mistaken. Now the question:

How do you start from an analytical Hamiltonian and obtain an exact matrix representation??

I kind of know everything (numerical discretization, real lattice discretization, eigenspace discretization,) boils down to the concept of basis functionsbut I just can't connect the dots.

I hope there'll be some interest in this,

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# Switching to a Matrix Hamiltonian - Conceptual Issues

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