(a) A quantum system has Hamiltonian $H=H_{0}+V(t)$. Let $\{|n\rangle\}_{n \in \mathbb{N}_{0}}$ be an orthonormal basis of $H_{0}$ eigenstates, with corresponding energies $E_{n}=\hbar \omega_{n}$. For $t<0$, $V(t)=0$ and the system is in state $|0\rangle$. Calculate the probability that it is found to be in state $|1\rangle$ at time $t>0$, correct to lowest non-trivial order in $V$.

(b) Now suppose $\{|0\rangle,|1\rangle\}$ form a basis of the Hilbert space, with respect to which

$$\left(\begin{array}{cc} \langle 0|H| 0\rangle & \langle 0|H| 1\rangle \\ \langle 1|H| 0\rangle & \langle 1|H| 1\rangle \end{array}\right)=\left(\begin{array}{cc} \hbar \omega_{0} & \hbar v \Theta(t) e^{i \omega t} \\ \hbar v \Theta(t) e^{-i \omega t} & \hbar \omega_{1} \end{array}\right)$$

where $\Theta(t)$ is the Heaviside step function and $v$ is a real constant. Calculate the exact probability that the system is in state $|1\rangle$ at time $t$. For which frequency $\omega$ is this probability maximized?