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**Switching to a Matrix Hamiltonian -- Conceptual Issues**

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 discretize

**free 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 hamiltonian

**which could be derived from first principles**the lattice I am going to work on will be

**discrete by itself!**The point is, everything is made up of atoms and I will have to work on a discrete lattice

*anyway*.

And this Hamiltonian will be

*exact*if 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 t

*o the concept of basis functions*but I just can't connect the dots.

I hope there'll be some interest in this,