For an N-electron system, the z component of the total spin angular momentum operator is
\hat{S}_{z,total} = \sum_{k} \hat{S}_{z,k}
If we define the spin eigenstates such that \hat{S}_{z,k}\alpha(k) = \frac{1}{2}\hbar\alpha(k) and \hat{S}_{z,k}\beta(k) = -\frac{1}{2}\hbar\beta(k)
then find the eigenvalues of \hat{S}_{z,total} for the two spin-orbit eigenstates specified below. Note
that k labels the electron, and the spatial orbital in which the electron resides is also
indicated in the Slater determinants provided.
(a) \psi = \frac{1}{\sqrt{2}}\begin{vmatrix} 1s\alpha(1) & 1s\beta(1) \\ 1s\alpha(2) & 1s\beta(2) \end{vmatrix} Evaluate \hat{S}_{z,total}\psi.
(b) \psi = \frac{1}{\sqrt{6}}\begin{vmatrix} 1s\alpha(1) & 1s\beta(1) & 2s\alpha(1) \\ 1s\alpha(2) & 1s\beta(2) & 2s\alpha(2) \\ 1s\alpha(3) & 1s\beta(3) & 2s\alpha(3) \end{vmatrix} Evaluate \hat{S}_{z,total}\psi.
