Consider the spin vector \(\sigma_{S}\).

\[\sigma_{S}=\frac{\hbar}{2}\left(\sigma_{x} \hat{\mathbf{i}}+\sigma_{y} \hat{\mathbf{j}}+\sigma_{z} \hat{\mathbf{k}}ight)\]

Calculate the matrix \(\sigma_{S}^{2}\) and determine its eigenvalues.

Note that the numerical factor of the eigenvalue has the form \(m(m+1)\) with \(m=\frac{1}{2}\). We shall show in Chapter 8 that \(\sigma_{S}^{2}\) is the square of an electron's spin angular momentum.