Expectation of $S_x$ in General Spin ${1\over 2}$ State

Let $\chi =\left (\matrix{\alpha_+ \cr \alpha_-}\right)$, be some arbitrary spin ${1\over 2}$ state. Then the expectation value of the operator

$$\begin{eqnarray*} \langle S_x \rangle & = & \langle \chi \vert S_x \vert \chi \... ...( \matrix{\alpha^*_+ \alpha_- + \alpha^*_- \alpha_+ }\right). \end{eqnarray*}$$