# Off-Diagonal Hamiltonian elements

• cowrebellion
In summary, the conversation discusses the physical meaning of off-diagonal Hamiltonian elements in perturbation theory. It is explained that these elements represent the "mixing" between pure energy eigenstates, and can result in coupling between degenerate states. This can be seen in examples such as spin-orbit coupling. These off-diagonal elements can also correspond to scattering elements in the Hamiltonian.
cowrebellion
Hello,

I just have a quick question about Quantum Mechanics. It's probably a bit basic but I'm trying to get my head around the off-diagonal Hamiltonian elements of a perturbation. We can assume the unperturbed Hamiltonian to be degenerate.

If I have a Hamiltonian
$$H=H_{0}+H'$$
where the perturbation contains off-diagonal elements what physical meaning does that have?

edit: When i say the off-diagonal elements I mean the off-diagonal elements of:
<nr|H'|ns> where r,s represent the degeneracy. Sorry!

I know not having the off-diagonal matrix elements in the perturbation remove the degeneracy (at least to first order in perturbation theory).

If they remain what is the physical meaning? I think it has to do with the various states interating in some way but I'm not sure. Any help on this would be great!

Last edited:
Actually I've just been thinking about it and I think it just has to do with interaction energies between states. Is this correct?

Yes.

You can think of it as the "mixing" between your pure energy eigenstates. If there's coupling between degenerate states, as you show in your Hamiltonian, this means that there will be a coupling between these degenerate states.

A good physical example is spin-orbit coupling where the degenerate spin levels are coupled via Rashba Hamiltonian for instance...

Other off-diagonal elements may correspond to scattering elements in the Hamiltonian, but of course the mixing need not be between different levels, it could be between degenerate levels (most commonly spin) also, just as in your example.

Thanks for the reply I think It's all slowly falling into place.

I've not heard of the Rashba Hamiltonian but I'll try find it in a textbook somewhere.

Thanks again for the help!

## 1. What are Off-Diagonal Hamiltonian elements?

Off-Diagonal Hamiltonian elements are the elements of a Hamiltonian matrix that are not on the main diagonal. They represent the interactions between different states or variables in a system.

## 2. How do Off-Diagonal Hamiltonian elements affect a system?

Off-Diagonal Hamiltonian elements play a crucial role in determining the behavior of a system. They represent the coupling between different states or variables, which can result in energy transfer, transitions between states, and other dynamic processes.

## 3. How are Off-Diagonal Hamiltonian elements calculated?

Off-Diagonal Hamiltonian elements are calculated using mathematical formulas that take into account the energy levels and coupling strengths of the system. These calculations can be complex and require advanced mathematical techniques.

## 4. Can Off-Diagonal Hamiltonian elements be zero?

Yes, Off-Diagonal Hamiltonian elements can be zero in some systems. This means that there is no interaction or coupling between the corresponding states or variables. In other systems, these elements may have non-zero values, indicating the presence of interactions.

## 5. What is the significance of Off-Diagonal Hamiltonian elements in quantum mechanics?

In quantum mechanics, Off-Diagonal Hamiltonian elements play a crucial role in describing the behavior of quantum systems. They are used to calculate transition probabilities, determine energy levels, and understand the dynamics of quantum systems.

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