Understanding Basis Choices in Quantum Mechanics

[Verdict]

Homework Statement

Alright, so this is not exactly a guided homework question. It is a rather intricate problem consisting of many steps, but one of these steps comes down to working out the hyperfine interaction between two spin 1 particles, from a hamiltonian point of view. From what I have been given, in this particular setup it can be simplified to the equation below:

Homework Equations

H = A S_{z}⊗I_{z}

where S and Z are angular momentum operators corresponding to the Z axis.

The Attempt at a Solution

Alright, so my problem is how I go about knowing which column of each matrix correspons to what. Does the first column correspond to the spin being 0, 1 or -1, basically. I have illustrated my question with the picture below, working out a specific case, where I indicate the spin of the first particle by m_S and the spin of the second particle by m_I. The reason for why the ordering is important to me is because I want to perform an experiment in which I have to be able to distinguish between the second particle being spin 1, 0 or -1, and the only way I can think of doing so in my specific setup is if I know which values of the hamiltonian correspond to which combination of (spin particle 1, spin particle 2)

Somehow I seem to remember that this choice of basis is arbitrary, which means that no specific one corresponds to spin 1, 0, or -1. This would be problematic, as I need a clear way to distinguish them from one another.

Edit: I understand that my question is a bit vague. It basically boils down to if the numbers I put under the matrix are set, or if they can be chosen arbitrarily.

 

[DrClaude]

I don't understand where you are going with this. By if you have hyperfine interaction, then $m_S$ and $m_I$ are no longer good quantum numbers and cannot be used to describe eigenstates of the system.

 

[Verdict]

Hmm. Well, I suppose I should have added more context, as looking back at it it is indeed not clear at all what I am trying to say. In the end my question boiled down to which eigenvectors corresponded to what columns of the pauli matrices for a spin 1 particle, which is easily answered. But thank you for giving my question some thought, and I apologize for wasting your time!