Quantum Mechanics - Pauli Spin Matrices

[Tangent87]

The Pauli Spin matrices:

$$\sigma_1=\left[<br /> \begin{array}{ c c }<br /> 0 & 1 \\<br /> 1 & 0<br /> \end{array} \right],\sigma_2=\left[<br /> \begin{array}{ c c }<br /> 0 & -i \\<br /> i & 0<br /> \end{array} \right],\sigma_3=\left[<br /> \begin{array}{ c c }<br /> 1 & 0 \\<br /> 0 & -1<br /> \end{array} \right]$$

are used to represent the angular momentum operators for spin 1/2 in 3 dimensions for a single particle. My question is how are they derived? Particularly, what properties of the matrices correspond to general features of angular momentum and which are specific to spin half?

If we had two spin-half particles (A and B say) with basis states $$|\uparrow>_{A}|\uparrow>_{B},|\downarrow>_{A}|\uparrow>_{B},|\uparrow>_{A}|\downarrow>_{B},|\downarrow>_{A}|\downarrow>_{B}$$ then is it possible to find matrix representations of the spin matrices for each of A and B in the three axes? I'm guessing they would be 4x4 matrices since we have four basis states.

I know that's a lot to ask but the wikipedia page on Pauli matrices hardly mentions their relevance to quantum mechanics until the end of the page.

 

[arkajad]

You want an irreducible unitary representation of the group SU(2). So you want antihermitian matrices of trace 0. Multiplying by i, you want Hermitian traceless matrices satisfying the commutation relations of the generators of the rotation group. Pauli's matrices is one solution in 2 complex dimension. Any other solution is unitarily equivalent.

For two spins you take tensor product of the two representations. Then you may like to restrict it to antisymmetric tensors.