Define the interaction picture for a quantum mechanical system with Schrödinger picture Hamiltonian $H_{0}+V(t)$ and explain why either picture gives the same physical predictions. Derive an equation of motion for interaction picture states and use this to show that the probability of a transition from a state $|n\rangle$ at time zero to a state $|m\rangle$ at time $t$ is

$$P(t)=\frac{1}{\hbar^{2}}\left|\int_{0}^{t} e^{i\left(E_{m}-E_{n}\right) t^{\prime} / \hbar}\left\langle m\left|V\left(t^{\prime}\right)\right| n\right\rangle d t^{\prime}\right|^{2}$$

correct to second order in $V$, where the initial and final states are orthogonal eigenstates of $H_{0}$ with eigenvalues $E_{n}$ and $E_{m}$.

Consider a perturbed harmonic oscillator:

$$H_{0}=\hbar \omega\left(a^{\dagger} a+\frac{1}{2}\right), \quad V(t)=\hbar \lambda\left(a e^{i \nu t}+a^{\dagger} e^{-i \nu t}\right)$$

with $a$ and $a^{\dagger}$ annihilation and creation operators (all usual properties may be assumed). Working to order $\lambda^{2}$, find the probability for a transition from an initial state with $E_{n}=\hbar \omega\left(n+\frac{1}{2}\right)$ to a final state with $E_{m}=\hbar \omega\left(m+\frac{1}{2}\right)$ after time $t$.

Suppose $t$ becomes large and perturbation theory still applies. Explain why the rate $P(t) / t$ for each allowed transition is sharply peaked, as a function of $\nu$, around $\nu=\omega$.