Solution
= Solution
In the <interaction picture>, first-order <time-dependent perturbation theory> gives the transition amplitude
$$
c_j^{(1)}(T)
=-\frac{i}{\hbar}\langle j|V|i\rangle
\int_0^T e^{i\omega_{ji}t}\,dt
=-\frac{\langle j|V|i\rangle}{\hbar\omega_{ji}}
\left(e^{i\omega_{ji}T}-1\right).
$$
Taking its squared modulus and using $|e^{i\theta}-1|^2=4\sin^2(\theta/2)$ gives
$$
\boxed{P_{ij}\simeq
4|\langle i|V|j\rangle|^2
\frac{\sin^2(\omega_{ij}T/2)}{(\hbar\omega_{ij})^2}.}
$$