Consider the quantum state

$|$:$|$

$($a$)$ Is this wave$-$function normalized? If not, normalize it$.$

$($b$)$ What are the possible results of a measurement of $S^z?$ With what probability does each result occur?

$($c$)$ Plot a histogram to indicate what you expect your experimental results in $($a$)$ would look like.

$($d$)$ What are the possible results of a measurement of $S^x?$ With what probability does each result occur?

$($e$)$ Plot a histogram to indicate what you expect your experimental results in $($c$)$ would look like.