= Solution
{c}
Expand the <wavefunction> in unperturbed <energy eigenstates>,
$$
|\psi(t)\rangle=c_1(t)e^{-iE_1t/\hbar}|1\rangle+c_2(t)e^{-iE_2t/\hbar}|2\rangle.
$$
The $c_j$ are <interaction picture> <probability amplitudes>, so $|c_j|^2$ are level populations and $|c_1|^2+|c_2|^2=1$ in this closed <two-level system>. A classical field $\mathbf E_0\cos\omega t$ supplies the <electric-dipole interaction> $-\mathbf d\cdot\mathbf E_0\cos\omega t$. Choosing the relative phase of the <energy eigenstates> makes $\Omega=-\langle1|\mathbf d\cdot\mathbf E_0|2\rangle/\hbar$ real. This is the resonant <Rabi angular frequency>.
Combining the field's two exponentials with the unperturbed <quantum phases> produces terms oscillating at $\omega-\omega_0=\delta$ and at $\omega+\omega_0$. The <rotating-wave approximation> drops the latter, leaving exactly the supplied equations with coupling $\Omega/2$. It requires $|\Omega|,|\delta|\ll\omega_0$ and a field envelope slow enough on the carrier timescale. <Spontaneous emission> and <dephasing> must be negligible during the sequence. For a ground-state <hyperfine level> transition, the analogous direct drive is a <magnetic dipole transition>; the same equations hold with the magnetic <matrix element> replacing the electric one.
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