Evaluate the density matrix \(ho_{m n}\) of an electron spin in the representation that makes \(\hat{\sigma}_{x}\) diagonal.

Next, show that the value of \(\left\langle\sigma_{z}ightangle\), resulting from this representation, is precisely the same as the one obtained in Section 5.3.

The representation needed here follows from the one used in Section 5.3 by carrying out a transformation with the help of the unitary operator

\[
\hat{U}=\left(\begin{array}{rr}
1 / \sqrt{ } 2 & 1 / \sqrt{ } 2 \\
-1 / \sqrt{ } 2 & 1 / \sqrt{ } 2
\end{array}ight)
\]