This is an extension to the previous post.
We will study the time-evolution of a finite dimensional quantum system. To this end, let us consider a quantum mechanical system with the Hilbert space $\mathbb{C}^2$. We denote by $\left . \left | 0 \right \rangle\right .$ and $\left . \left | 1 \right \rangle\right .$ the standard basis elements $(1,0)^T$ and $(0,1)^T$. Let the Hamiltonian of the system in this basis be given by $$ H=\begin{pmatrix} 0 &-i \\ -i &0 \end{pmatrix} $$ and assume that for $t=0$ the state of the system is just given by $\psi(t=0)=\left . \left | 0 \right \rangle\right .$. In the following, we also assume natural units in which $\hbar=1$.
Problems:
i) Compute the expectation value of a $Z$-measurement at time $t$: $\left \langle \sigma_z \right \rangle_{\psi(t)}=\left \langle \psi(t)\mid \sigma_z\psi(t) \right \rangle$, where $$ \sigma_z=\begin{pmatrix} 1 &0 \\ 0 &-1 \end{pmatrix} $$
ii) Instead of evolving the quantum states in time, we can alternatively evolve the observables according to $\sigma_z(t)=e^{iHt}\sigma_ze^{-iHt}$, called the Heisenberg evolution of $\sigma_z$. At which time should we perform our measurement in order to maximize the expectation value of $\sigma_z$?
In the previous post, I've concluded that $$ \psi(t)=\begin{pmatrix} \frac{1}{2}e^{t}+\frac{1}{2}e^{-t}\\ -\frac{1}{2}e^{t}+\frac{1}{2}e^{-t} \end{pmatrix} $$ I am not really sure how to calculate $\left \langle \psi(t)\mid \sigma_z\psi(t) \right \rangle$. Can you help me with this part?