$ H =\begin{pmatrix} \alpha & \sqrt{\frac{3}{4}}\beta \\ \sqrt{\frac{3}{4}}\beta & \alpha+ \beta \end{pmatrix} $
1) Find the spectrum.
I have found this to be
$ E_{1} = -\frac{3}{4} \beta^{2} + \alpha $
$ E_{2} = \alpha +\beta - \frac{3}{4}\beta^{2} $
by solving $ det(H-EI)=0$
Not sure How to find the corresponding eigenvectors?
Also The second part, I am not sure how to start it.
Operator $ Z = \zeta\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} $
At Time t = 0 , Z is measured with outcome $\zeta$
find $ | \psi(t) > $ for t>=0 and find the probabilities of measuring the possible outcomes for energy and Z as a function of time.