Q$\mathrm{.}1$: Two spin$-$states are described by $|+n_1$:$|$ and $|+n_2$:$|.$ The fidelity is defined as the probability to find state $($:$|+n_1\}$ in state $($:$|+n_2\}$ when measured, i$.$e$.F_{12}=(+n_2$:$)|.$ Show that this works out to be $co{s}^2\frac{}{2}$ with $$ as the angle between the unit vectors hat$(n{)}_1$ and hat$(n{)}_2.[$See Q$.5$ of HW$8$ for relation between hat$(n)$ and $(,).]$

Q$\mathrm{.}2$: A linearly polarized light beam with intensity $I_0$ when passed through a polarizer with its horizontal pass axis leads to intensity $3\frac{I_0}{4}.$

$($a$)$ Now if the pass axis is rotated by a $45$ angle keeping plane of polarizer same, what are the possible values of the intensity?

$($b$)$ When the polarizer is rotated by angle $$ the intensity is found to be the same for clockwise and anticlockwise rotations. Find $.$