(a) Show that the state
$\chi = \frac{1}{\sqrt{2}}(\alpha_1\beta_2 - \beta_1\alpha_2)$
is a singlet spin-state. Also, show that it is normalized.
(b) Show that the states
$\chi_1 = \alpha_1\alpha_2$
$\chi_0 = \frac{1}{\sqrt{2}}(\alpha_1\beta_2 + \beta_1\alpha_2)$
$\chi_{-1} = \beta_1\beta_2$
form the three components of a triplet spin-state with $m = -1, 0, 1$, respectively. Also, show that they are normalized.
(c) Apply the global $S_+$ to the three components of triplet states, $\chi_1$, $\chi_0$ and $\chi_{-1}$. Compare your results with
$S_+|SM_s\rangle = \sqrt{S(S+1) - M(M+1)}|SM_s + 1\rangle$