Matrix operation in dirac matrices If we define $\alpha_i$  and $\beta$ as Dirac matrices which satisfy all of the conditions of spin $1/2$ particles , and $p$ is the momentum of the particle, then how can we get the matrix form
\begin{equation}
\alpha_i p_i= \begin{pmatrix} p_z & p_x-ip_y  \\ p_x+ip_y & -p_z  \end{pmatrix}.
\end{equation}

It's just a matrix manipulation. Let $\sigma_i$ pauli matrices. 
\begin{equation}
\alpha_i p_i= \begin{pmatrix} 0& \sigma_i \\ \sigma_i & 0  \end{pmatrix} p_i .
\end{equation}
$ \alpha_i p_i= \begin{pmatrix} 0& p_1 \sigma_1 \\ p_1\sigma_1 & 0  \end{pmatrix}   +  \begin{pmatrix} 0& p_2 \sigma_2 \\ p_i\sigma_2 & 0  \end{pmatrix}  +  \begin{pmatrix} 0& p_3 \sigma_3 \\ p_3\sigma_3 & 0  \end{pmatrix} $ 
But $ \sigma_1 p_1 = \begin{pmatrix} 0& 1 \\\ 1 & 0  \end{pmatrix}p_1=\begin{pmatrix} 0& p_1 \\\ p_1 & 0  \end{pmatrix}$ ,
$ \sigma_2 p_2= \begin{pmatrix} 0& -i \\\ -i & 0  \end{pmatrix}p_2=\begin{pmatrix} 0& -ip_2 \\\ ip_2 & 0  \end{pmatrix}$ 
$ \sigma_3 p_3= \begin{pmatrix} 1& 0 \\\ 0 & -1  \end{pmatrix}p_3= \begin{pmatrix} p_3& 0 \\\ 0 & -p_3 \end{pmatrix}$
Now adding these we  get ($1\rightarrow x $, $2\rightarrow y $ ,$3\rightarrow z $)
,
\begin{equation}
\alpha_i p_i= \begin{pmatrix} p_z & p_x-ip_y  \\ p_x+ip_y & -p_z  \end{pmatrix}  .
\end{equation}
