Recently I started to study the formulation of quantum mechanics in the phase space. So I was introduced to the concept of Wigner function and Weyl transform. I learned that if F is an operator, then I can represent it by an integral as follows: \begin{equation} F = \int_{-\infty}^{+\infty}\frac{dpdq}{2\pi\hslash}f(p,q)\Delta(p,q) \end{equation} Where $f(p,q)$ is the Wigner transform given by: \begin{equation} f(p,q) = \int_{-\infty}^{+\infty}e^{\frac{i}{\hslash}qu}\langle p+\frac{u}{2}|F|p-\frac{u}{2}\rangle du \end{equation} and $\Delta(p,q)$: \begin{equation} \Delta(p,q) = \int_{-\infty}^{-\infty} e^{\frac{i}{\hslash}pv}|q+\frac{v}{2}\rangle \langle q-\frac{v}{2}|dv \end{equation} all of the above expressions were derived using the completeness relations as follows: \begin{equation} F = \int_{-\infty}^{+\infty}dp'dp''dq'dq''|q''\rangle\langle q''|p''\rangle \langle p''|F|p'\rangle\langle p'|q'\rangle \langle q'|\end{equation} and the following variable change was also taken \begin{equation} 2p =p'+p'', 2q = q'+q'', u = p''-p', v = q''-q' \end{equation}
Could someone show me what these calculations would look like for a numerical example of some observable operator. I tried to calculate for one of the pauli matrices, but I was stuck in the middle of the calculations. My learning becomes more consistent when I see practical examples, if anyone can help me with this problem, I will be very grateful.