Suppose $a=(a_1,a_2,a_3)$ is a normalized vector in $\R^3$. Compute the eigen-values of $S_a\colon=\tfrac12(a_1\s_1+a_2\s_2+a_3\s_3)$. Compute the eigen-vectors of $S_a$ for $$ a=(\cos\vp\sin\theta,\sin\vp\sin\theta,\cos\theta)\in S^2~. $$

We already know that $\exp(-itS_a)\in\SU(2)$ and that $\pm1/2$ are the eigen values of $S_a$ , cf. exam. So let's compute the eigen-vectors: $x=x_1e_1+x_2e_2$ is a normalized eigen-vector of $S_a$ for the eigen-value $1/2$ iff $$ |x_1|^2+|x_2|^2=1 \quad\mbox{and}\quad (a_3-1)x_1+(a_1-ia_2)x_2=0~. $$ which implies $$ \frac{x_2}{x_1}=\frac{1-a_3}{a_1-ia_2} =\frac{1-\cos\theta}{\cos\vp\sin\theta-i\sin\vp\sin\theta} =\frac{e^{i\vp}\sin(\theta/2)}{\cos(\theta/2)}~. $$ Thus the eigen-vector of $S_a$ for the eigen-value $1/2$ is given by $$ x=e^{-i\vp/2}\cos(\theta/2)e_1+e^{i\vp/2}\sin(\theta/2)e_2 $$ Analogously the eigen-vector of $S_a$ for the eigen-value $-1/2$ is given by: $$ y=-e^{-i\vp/2}\sin(\theta/2)e_1+e^{i\vp/2}\cos(\theta/2)e_2~. $$

bloch sphere
Bloch sphere: Every unit vector $a$ represents a state $x$ satisfying $S_ax=\tfrac12x$, i.e. the spin in the direction $a$ has the definite value $+1/2$. The unit vector $-a$ represents a state $y$ with $S_ay=-y$, i.e. the spin in the direction $a$ has the definite value $-1/2$.