Dirac's notation? (QM)

Question

I have a question regarding Dirac's notation in quantum physics.

As far as I understand: $\langle a|b\rangle=(a1^*,a2^*)*(b1,b2)^T$

But what does $\langle1/2,1/2|J|1/2,-1/2\rangle$ mean?

Answer 1

If $J=\begin{pmatrix}a & b \\ c & d\end{pmatrix}$ then $\langle x,y\mid J\mid z,t\rangle=axz+bxt+cyz+dyt$.