Question 2:
Consider a spin-\frac{1}{2} particle in the general state:
$|\psi\rangle = \alpha |+\rangle + \beta |-\rangle$
(1)
with $|\alpha|^2 + |\beta|^2 = 1$.
(a) Using the representations of the Pauli operators X, Y, and Z:
$X = \begin{pmatrix} 0 & 1\\ 1 & 0 \end{pmatrix}$, $Y = \begin{pmatrix} 0 & -i\\ i & 0 \end{pmatrix}$, $Z = \begin{pmatrix} 1 & 0\\ 0 & -1 \end{pmatrix}$,
(2)
compute $\langle X \rangle$, $\langle Y \rangle$, and $\langle Z \rangle$.
(b) Compute also $\langle X^2 \rangle$, $\langle Y^2 \rangle$, and $\langle Z^2 \rangle$.
(c) Consider the operator $S^2 = \frac{\hbar^2}{4}(X^2 + Y^2 + Z^2)$ corresponding to the square of the spin. Determine the eigenvectors and eigenvalues of this operator.
