Switching to a Matrix Hamiltonian - Conceptual Issues

[sokrates]

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

I hope there'll be some interest in this,

 

[isabelle]

For me the grammar in some parts of your post is hard to understand, you may want to rephrase your question.

If $|\psi_i\rangle$ form a basis then the matrix elements $h_{i j}$ of the Hamiltonian in that basis are given by $h_{i j} = \langle\psi_i|H|\psi_j\rangle$. I doubt this answers your question, but it seems related.

 

[sokrates]

How do you discretize the Hamiltonian?

Is it exact?

What is the real space representation?

What is analytical and what is purely numerical?

 

[Hans de Vries]

How do you discretize the Hamiltonian?

Working on a discrete lattice you might typically use:

$$\frac\tau2~\Big\{~-1~,~~0~~,~+1~\Big\}$$ for first order differentiation and

$$\tau^2\Big\{~~~1~,~-2~,~~~1~~~\Big\}$$ for second order differentiation.

The rest should stay the same. Tau determines the time steps.Regards, Hans