= Solution
\b[<Stimulated emission> adds a <photon> to an already occupied radiation mode; <spontaneous emission> can occur when that mode initially contains no <photons>.] In the <electric-dipole approximation> the atom-field interaction is $H_{\rm int}=-\mathbf d\cdot\mathbf E$. Quantizing the radiation field makes the emission <matrix element> into a mode with occupation $n$ proportional to
$$
\langle n+1|a^\dagger|n\rangle=\sqrt{n+1}.
$$
This <photon occupation enhancement of emission> makes the emission rate into that mode proportional to $n+1$: the part proportional to $n$ is stimulated and the remaining part is spontaneous. Absorption is proportional to $|\langle n-1|a|n\rangle|^2=n$. The <photon> energy is the atomic energy difference, up to any specified recoil or line broadening.
The stimulated contribution preferentially populates the stimulating mode, including its propagation direction and <polarization>; it provides <optical gain>. Spontaneous <photons> populate the available modes according to the transition's dipole <matrix elements> and radiation environment. They are not universally isotropic. An excited bare atom with the field vacuum is not an <eigenstate> of the fully coupled <Hamiltonian>, which resolves the apparent paradox of a nominally stationary excited state radiating.<Topics in Quantum Mechanics by David Tong>[§4.3.2, why the electromagnetic field must be included to describe spontaneous decay]{ref}
\Image[/past-exam-of-the-advanced-quantum-physics-course-of-the-university-of-cambridge/2016/paper-1-emission.png]
{title=Original two-level transition diagrams and the $n$ versus $n+1$ occupation factors for <photon> absorption and emission}
For lower and upper levels 1 and 2, with radiation spectral <energy density> $u_\nu$ per unit ordinary frequency, the rates are
$$
R_{\rm abs}=N_1B_{12}u_\nu,\qquad
R_{\rm stim}=N_2B_{21}u_\nu,\qquad
R_{\rm sp}=N_2A_{21}.
$$
Thermal detailed balance with the <Planck spectrum> gives
$$
\boxed{g_1B_{12}=g_2B_{21},\qquad
\frac{A_{21}}{B_{21}}=\frac{8\pi h\nu^3}{c^3}.}
$$
The statistical weights $g_i$ here are level degeneracies, not <Landé g-factors>. A definition of $B$ multiplying intensity or angular-frequency density would change the normalization factors.
For a specified free-space electric-dipole transition, summing radiation modes gives
$$
A_{21}=\frac{\omega_{21}^3|\mathbf d_{12}|^2}
{3\pi\epsilon_0\hbar c^3},
\qquad \omega_{21}=2\pi\nu.
$$
Appropriate sums over final magnetic substates and averages over initial substates are needed for an unresolved degenerate level. The radiative lifetime is $1/A_{21}$ only when this is the sole decay channel. In a <laser>, net stimulated gain requires $N_2/g_2>N_1/g_1$, as well as sufficient gain to overcome cavity loss; <population inversion> alone does not specify the lasing threshold.
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