System of two spin 1/2 particles in an external magnetic field

[ConorDMK]

Homework Statement

To save typing, I copied an image of the question and of my attempt at a solution. And the part I am having trouble with is part (iii).

Relevant Equations

Pauli spinors and matrices for x.

So what I'm not sure on, is calculating the matrix elements for part (iii) with Pauli spinors and Pauli matrices, and then finding the form of the corresponding states. As I don't see how using the hint helps.

The following is using the eigenvalues of the spin-operators.

Provided what I have is correct, I'm not sure on how to get to the form of the states.

 

[DrClaude]

Once you have adopted a basis, you must stick with it. The problem tells you that you have to use a basis $|m_1 m_2 \rangle$, which are eigenstates of $\hat{S}_z$ for the respective particles. You must then express $\hat{H}_1$ as a matrix in the same basis, not using spin up/down with respect to $\hat{S}_x$, as you are doing after.

 

[ConorDMK]

Once you have adopted a basis, you must stick with it. The problem tells you that you have to use a basis $|m_1 m_2 \rangle$, which are eigenstates of $\hat{S}_z$ for the respective particles. You must then express $\hat{H}_1$ as a matrix in the same basis, not using spin up/down with respect to $\hat{S}_x$, as you are doing after.

Oh yeah, of course, thank you! It's been a while since I've done this stuff.
So I use,

which makes sense as to why the hint says to use Pauli Matrices and Spinors.