Switching to a Matrix Hamiltonian - Conceptual Issues

• sokrates
In summary, the conversation discusses the challenges of switching from an effective mass Hamiltonian to an atomistic Hamiltonian in numerical simulations. The main question is how to obtain an exact matrix representation from an analytical Hamiltonian. The concept of basis functions is mentioned as a key factor in discretizing the Hamiltonian. The conversation also touches on the use of discrete lattices and the use of first and second order differentiation in time steps.

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,

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.

How do you discretize the Hamiltonian?

Is it exact?

What is the real space representation?

What is analytical and what is purely numerical?

sokrates said:
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

What is a matrix Hamiltonian?

A matrix Hamiltonian is a mathematical representation of the energy of a quantum system, typically used in quantum mechanics. It is represented by a matrix, with elements corresponding to the energy of different states of the system.

Why would someone switch to a matrix Hamiltonian?

Switching to a matrix Hamiltonian can make calculations of energy levels and transitions in a quantum system more efficient and accurate. It is particularly useful for systems with many energy states.

What are some conceptual issues that may arise when switching to a matrix Hamiltonian?

One conceptual issue is understanding the mathematical representation of a matrix Hamiltonian and how it relates to the physical system. Another issue is ensuring the matrix is properly constructed and accurately represents the energy states of the system.

Are there any limitations to using a matrix Hamiltonian?

Yes, one limitation is that it is typically used for systems with discrete energy levels, and may not be suitable for systems with continuous energy spectra. Additionally, the size of the matrix can become computationally intensive for larger systems.

How is a matrix Hamiltonian used in practical applications?

A matrix Hamiltonian is used to predict the energy levels and transitions of quantum systems, which can then be used in applications such as designing new materials or developing quantum technologies. It is also used in theoretical research to better understand the behavior of quantum systems.