1) The Hamiltonian operator of a two-state system is given as:
$H = -i\kappa |1\rangle \langle 2| + i\kappa |2\rangle \langle 1|$
where $|1\rangle$ and $|2\rangle$ are orthonormal basis kets, and $\kappa$ is a real constant with the dimension of energy. At
$t = 0$, the state of the system is:
$|\alpha\rangle = \frac{1}{\sqrt{3}}|1\rangle + \sqrt{\frac{2}{3}}|2\rangle$.
(a) Find $|\alpha, t\rangle$.
(b) When the energy of the system is measured at time $t$, what are the possible outcomes, and with what
probabilities will they be found?
