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:
a) Determine the eigenvalues $\lambda_i$ and the normalized eigenvectors $f_i$ of $H$.
b) Compute the time evolution operator $U(t)=e^{-iHt}$ of the system, according to $f(H)=\sum_{i=1}^{2}f(\lambda_i)\left . \left | e_i \right \rangle\right . \left \langle \left . e_i \right | \right .$ for $f$ and analytic function for all $\lambda_i$. Compute the time evolved state $\psi(t)=U(t)\psi(t=0)$.
I do not understand what to do for problem b). I need help for this one. For a), I found out that the eigenvalues $\lambda_i$ are $\pm 1$, and the normalized eigenvectors $f_i$ are $\frac{1}{\sqrt 2}\begin{pmatrix} i\\ 1 \end{pmatrix}$ and $\frac{1}{\sqrt 2}\begin{pmatrix} -i\\ 1 \end{pmatrix}$