Consider a two-level quantum system with energy eigenstates \(|0angle\) and \(|1angle\), with energies 0 and \(\varepsilon=\hbar \omega\), respectively, and two different pure states: \(\left|\psi_{1}\rightangle=\left((\sqrt{3} / 2)|0angle+(1 / 2)|1angle e^{-i \omega t}\right)\) and \(\left|\psi_{2}\rightangle=|0angle\). Let \(\hat{ho}_{1}=\left|\psi_{1}\rightangle\left\langle\psi_{1}\right|\) be a density matrix that describes a pure state, and let \(\hat{ho}_{2}=(2 / 3)\left|\psi_{1}\rightangle\left\langle\psi_{1}|+(1 / 3)| \psi_{2}\rightangle\left\langle\psi_{2}\right|\) be a density matrix that describes a mixed state. Express both as \(2 \times 2\) matrices. Calculate the eigenvalues of the two density matrices and show that \(\operatorname{Tr} \hat{ho}_{1}=\operatorname{Tr} \hat{ho}_{2}=1, \operatorname{Tr} \hat{ho}_{1}^{2}=1\), and \(\operatorname{Tr} \hat{ho}_{2}^{2}<1\). Use the eigenvalues to show that \(S\left[\hat{ho}_{1}\right]=0\) and that \(S\left[\hat{ho}_{2}\right]>0\).