The $S_x$ operator in matrix form is given by
$\begin{align*}S_x = \sigma_x \frac{\hbar}{2} = \frac{\hbar}{2}\left(\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix} \rig...$
An eigenspinor of $S_x$ with eigenvalue $-\hbar/2$ must satisfy

$\begin{align}S_x \xi = \frac{-\hbar}{2} \xi \label{eqn83:1}\end{align}$
We can convert all of the answers to vector notation by using the identifications
$\begin{align*}| \uparrow\rangle &\to \left(\begin{matrix} 1 \\ 0 \\ \end{matrix}\right) \\| \downarrow\rangle &\to \l...$
We can then check whether the condition in equation (1) is satisfied. We find that it is satisfied only for answer (C). Therefore, answer (C) is correct. Physically, this means that the eigenspinor in answer (C) represents a particle with definite $x$ -component of its spin vector equal to $-\hbar/2$ .

Solution to 2001 Problem 83