Spin Coupling, Matrix Building

[sol66]

So I understand that the tensor product of two vector spaces constitutes a new vector space. In the case of spin coupling you have some arbitrary vector |x,z> where x represents the spin state of particle 1 and z represents the spin state of particle 2; bot spin states being general 2 element vectors.

This is where I question my understanding, from what I can see ... the tensor product of individual eigenspaces containing vector x and z creates a new eigenspace that holds the general vector |x,z>. Also it seems that you can construct a matrix out of your eigenstates using the basis from your eigenspaces holding the spinstates x and z. So is my understanding correct?

If I am right, my other question is ... how do you construct such a matrix?

 

[DrClaude]

You need to list out all possible two-particle states, such as
$$
\begin{pmatrix}
| x, z \rangle \\
| x, -z \rangle \\
| -x, z \rangle \\
| -x, -z \rangle
\end{pmatrix}
$$
to use the notation in the OP, or more commonly
$$
\begin{pmatrix}
| z, z \rangle \\
| z, -z \rangle \\
| -z, z \rangle \\
| -z, -z \rangle
\end{pmatrix}
$$
As always, the matrix representation is not unique, as the choice of basis is not unique, nor is the order of the elements in a given basis.