$21.$ a$.$ Evaluate

$\backslash$Sigma d$(\backslash$beta $)$m mm m $=$ j for any j $($integer or half$-$integer$)$; then check your answer for j $=1.$ b$.$ Prove, for any j$,$

j m$=-$J $1$ m$^2($B$)^2=(+1)$ sin$^2+$ m$^2(3$cos$^2-1).$

$[$Hint: This can be proved in many ways. You may, for instance, examine the rotational properties of J$2$ using the spherical $($irreduci$-$ ble$)$ tensor language.$]$