quantum mechanicsquantum-spin
Could you explain how to derive the Pauli matrices?
$$\sigma_1 = \sigma_x =
\begin{pmatrix}
0&1 \\ 1&0
\end{pmatrix}\,, \qquad
\sigma_2 = \sigma_y =
\begin{pmatrix}
0&-i\\ i&0
\end{pmatrix}\,, \qquad
\sigma_3 = \sigma_z =
\begin{pmatrix}
1&0\\0&-1
\end{pmatrix}
$$
Maybe you can also link to an easy to follow tutorial ?
As Qmechanic states: Most calculations in modern physics do not actually depend on the explicit realization of the Pauli matrices. At the end the physical quantities depend on the bilinear functions like the vector and axial currents.
However, it's entirely possible to transform the standard Pauli matrix representation into a spatially symmetric and entirely real valued representation that produces the exact same physics but is much easier to interpret as the complex asymmetric representation.
This representation uses 4x4 real valued matrices instead of 2x2 complex ones and the slightly larger group structure of $SO(4)\cong Spin(3)\otimes Spin(3)$ allows us to make the representation symmetric in the x, y and z-coordinates.
The 4 complex components of the bispinor field become 8 real values and the spatial symmetry of the representation becomes apparent if we visualize the linear relations between these 8 parameters for each of the matrices used, as shown in the images below. The red numbers represent the 4 right chiral parameters and the black numbers represent the 4 left chiral parameters.
For the detailed meaning of this all you can have a look here:
The real symmetric representation of the Dirac equation
Short overviews are given here: Physics-quest site and here: blog post
I think it is easier to compute direct products when you write the matrices in component form; basically, you just have to multiply each element of the first matrix by the whole second matrix: $$ \mathbf{A}\otimes\mathbf{B} = \begin{bmatrix} A_{11} \mathbf{B} & \cdots & A_{1n} \mathbf{B} \\ \vdots & \ddots & \vdots \\ A_{n1} \mathbf{B} & \cdots & A_{nn} \mathbf{B}\end{bmatrix} $$ In your case, using the Pauli matrices $\boldsymbol{\sigma}_2$ and $\boldsymbol{\eta}_1$, we get: $$ \boldsymbol{\sigma}_2 \otimes \boldsymbol{\eta}_1 = \begin{bmatrix} 0 & -i\boldsymbol{\eta}_1 \\ i\boldsymbol{\eta}_1 & 0 \end{bmatrix} = \begin{bmatrix} 0 & 0 & 0 & -i \\ 0 & 0 & -i & 0 \\ 0 & i & 0 & 0 \\ i & 0 & 0 & 0\end{bmatrix} $$
Best Answer
That certainly depends on what exactly you mean. I take your question as "how do you see that the (non-relativistic) electron spin (or more generally, Spin-1/2) is described by the Pauli matrices?"
Well, to start, we know that measuring the electron spin can only result in one of two values. From this we see that we need matrices of at least dimension 2. The simplest choice is then of course exactly dimension 2.
Moreover the spin is an angular momentum, and thus described by three operators obeying the angular momentum algebra: $[L_i, L_j] = \mathrm{i} \hbar\epsilon_{ijk} L_k$. This together with matrix dimension 2 basically restricts the choice to sets of three matrices which are equivalent to $\hbar/2$ times the Pauli matrices (the freedom of choice of those matrices corresponds to the freedom to use three arbitrary orthogonal directions as $x$, $y$ and $z$ direction).
So now why choose from those equivalent choices exactly the Pauli matrices? Well, there's always one measurement direction which is represented by a diagonal matrix; this makes calculations much easier. Of course it makes sense to choose the matrices in a way that this direction is one of the coordinate directions. By convention, the $z$ direction is chosen. This ultimately fixes the matrices.