$7\mathrm{.}3$ $-$ and $/2-$Pulses $($a$)$ For zero detuning, and the initial conditions c $(0)1$ and c$(0)-0,$ solve Eq $(1)$ to find both ex$($t$)$ and cy$($t$).($b$)$ Prove that a $-$pulse gives the operationc $(1)+22+1))($c$)$ What is the overall effect of two z$-$pulses acting on $1)($d$)$ Show that a $/2-$puise gives $(1)(11)-12)/2($e$)$ What is the overall effect of two $/2-$pulses acting on $17$ When state $(2)$ experiences a phase shift of $,$ between the two puises, show that the probabilities of ending up in states $(1)$ and $(2)$ are sin$^2(0/2)$ and cos$"(0/2),$ respectively. $($f$)$ Calculate the effect of the three$-$pulse sequences$/2-$s$-/2,$ with a phase shift of between the second and third pulses. $($The operators can be written as $2$x $2$ unitary matrices, although this is not really necessary for this simple case.$)$