The Hamiltonian for a quantum system in the Schrödinger picture is

$$H_{0}+\lambda V(t),$$

where $H_{0}$ is independent of time and the parameter $\lambda$ is small. Define the interaction picture corresponding to this Hamiltonian and derive a time evolution equation for interaction picture states.

Let $|a\rangle$ and $|b\rangle$ be eigenstates of $H_{0}$ with distinct eigenvalues $E_{a}$ and $E_{b}$ respectively. Show that if the system is initially in state $|a\rangle$ then the probability of measuring it to be in state $|b\rangle$ after a time $t$ is

$$\frac{\lambda^{2}}{\hbar^{2}}\left|\int_{0}^{t} d t^{\prime}\left\langle b\left|V\left(t^{\prime}\right)\right| a\right\rangle e^{i\left(E_{b}-E_{a}\right) t^{\prime} / \hbar}\right|^{2}+O\left(\lambda^{3}\right)$$

Deduce that if $V(t)=e^{-\mu t / \hbar} W$, where $W$ is a time-independent operator and $\mu$ is a positive constant, then the probability for such a transition to have occurred after a very long time is approximately

$$\frac{\lambda^{2}}{\mu^{2}+\left(E_{b}-E_{a}\right)^{2}}|\langle b|W| a\rangle|^{2}$$