Switching to a Matrix Hamiltonian - Conceptual Issues

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

I hope there'll be some interest in this,
 

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  • #2
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For me the grammar in some parts of your post is hard to understand, you may want to rephrase your question.

If [itex]|\psi_i\rangle[/itex] form a basis then the matrix elements [itex]h_{i j}[/itex] of the Hamiltonian in that basis are given by [itex]h_{i j} = \langle\psi_i|H|\psi_j\rangle[/itex]. I doubt this answers your question, but it seems related.
 
  • #3
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How do you discretize the Hamiltonian?

Is it exact?

What is the real space representation?

What is analytical and what is purely numerical?
 
  • #4
Hans de Vries
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How do you discretize the Hamiltonian?
Working on a discrete lattice you might typically use:

[tex]\frac\tau2~\Big\{~-1~,~~0~~,~+1~\Big\}[/tex] for first order differentiation and

[tex]\tau^2\Big\{~~~1~,~-2~,~~~1~~~\Big\}[/tex] for second order differentiation.

The rest should stay the same. Tau determines the time steps.


Regards, Hans
 

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