Comider a systens whose wave function is given by

$(,)=\frac{1}{2}{Y}_{00}(,)+\frac{1}{\sqrt[\phantom{2}]{3}}{Y}_{11}(,)+\frac{1}{2}{Y}_{1,-1}(,)+\frac{1}{\sqrt[\phantom{2}]{6}}{Y}_{22}(,)$

$($a$)$ Is $(,)$ normalized?

$($b$)$ Is $(0.)$ an eigenstate of hat$(L{)}^2$ and hat$(L{)}_z?$

$($c$)$ Calculate hat$(L{)}_{+-}(,)$ and $($:$|hat(L{)}_{+-}|$:$).$

$($d$)$ If$,$ a me asurement of the $z-$component of the orbital angular momentiom is carried out, find ue probabilitis corresponding the results $0,,-,$ and $2t.$