How do i find the eigenvalues of this tough Hamiltonian?

In summary, the conversation discusses a Hamiltonian with G matrices in different spaces and the difficulty in finding eigenvalues due to the 6 indices. The possibility of constructing a 6D tensor or a matrix with the i' indices going down and the i indices going across is mentioned. The idea of using tensor contraction as a possible solution is also suggested.
  • #1
baouba
41
0
I have this Hamiltonian --> (http://imgur.com/a/lpxCz)

Where each G is a matrix.

I want to find the eigenvalues but I'm getting hung up on the fact that there are 6 indices. Each G matrix lives in a different space so I can't just multiply the G matrices together. If I built this Hamiltonain explicitly on the computer it would be a 6D tensor right? So how do I change it into a matrix so I can find the eigenvalues?

One thing I've been told it that I could build a matrix with the i' indices going down and the i indices going across. Neglecting the sum over the alphas, if I did this would the first row of my matrix be (G1,1 G1,1G1,1,G1,1 G1,1 G1,2,G1,1 G1,1 G1,3,...,G1,1 G1,1 G1,10,G1,1 G1,2 G1,1,...,G1,1 G1,2 G1,10,...,G1,1 G1,10 G1,10,G1,2 G1,1G1,1,...G1,10 G1,10 G1,10)
where each element is the product of the elements of the G matrices. I don't see you can take the product of the elements of the G matrices if they aren't defined in the same space. Would constructing a matrix like this really give the eigenvalues of H?
 
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  • #2
The other thing I'm thinking might work is to use some sort of tensor contraction. Is there a way to do this? I'm not sure how to approach this problem, any help would be greatly appreciated!
 

1. How do I find the eigenvalues of a Hamiltonian?

Finding the eigenvalues of a Hamiltonian involves solving the eigenvalue equation, which is a mathematical equation that describes the relationship between the Hamiltonian operator and its eigenvalues. This can be done through various methods such as diagonalization, perturbation theory, or numerical methods.

2. What is the importance of eigenvalues in a Hamiltonian?

The eigenvalues of a Hamiltonian represent the possible energy levels of a quantum mechanical system. They are crucial in understanding the behavior and properties of the system, as they determine the probabilities of different energy states and the evolution of the system over time.

3. Can all Hamiltonians be solved analytically?

No, not all Hamiltonians can be solved analytically. Some Hamiltonians are too complex to be solved using traditional methods and require numerical techniques or approximations to find the eigenvalues.

4. How do I interpret the eigenvalues of a Hamiltonian?

The eigenvalues of a Hamiltonian can be interpreted as the possible energy values that a system can have. The lowest eigenvalue represents the ground state energy, while the higher eigenvalues correspond to excited states. The difference between the eigenvalues also gives insight into the energy transitions that can occur in the system.

5. Are there any applications of finding the eigenvalues of a Hamiltonian?

Yes, there are many applications of finding the eigenvalues of a Hamiltonian, especially in quantum mechanics and physics. Some examples include predicting the energy levels of atoms and molecules, understanding the behavior of particles in a potential well, and analyzing the stability of a system.

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