= Solution
Define $\hbar\omega_0=E_2-E_1$ and use <radiation energy density> $u(\omega)$ per unit <angular frequency>, excluding the zero-point contribution. The <Angular-frequency Einstein coefficients> give rates per atom: $B_{12}u(\omega_0)$ for absorption, $B_{21}u(\omega_0)$ for <stimulated emission>, and $A_{21}$ for <spontaneous emission>. Stimulated emission adds a photon to the stimulating mode; spontaneous emission persists with no incident photons.
\Image[/past-exam-of-the-advanced-quantum-physics-course-of-the-university-of-cambridge/2013/paper-1-aqp2013-radiative-transitions.png]
{title=Absorption, stimulated emission and spontaneous emission between two nondegenerate levels}
{source=paper-1-aqp2013-radiative-transitions.py}
In thermal equilibrium the <Boltzmann distribution> gives $N_2/N_1=e^{-\hbar\omega_0/(k_{\rm B}T)}$. Balancing up and down rates yields
$$
N_1B_{12}u=N_2(A_{21}+B_{21}u),\qquad
u=\frac{A_{21}}{B_{12}e^{\hbar\omega_0/(k_{\rm B}T)}-B_{21}}.
$$
Matching the <Planck law> for every temperature fixes
$$
\boxed{B_{12}=B_{21},\qquad\frac{A_{21}}{B_{21}}=\frac{\hbar\omega_0^3}{\pi^2c^3}.}
$$
No degeneracy factor is needed for these nondegenerate levels.<Topics in Quantum Mechanics by David Tong>[§4.3.2, printed pp.123–127: thermal balance and Einstein coefficients]{ref}
For plates of large area $A$ and spacing $d$, the surviving low-frequency <transverse electromagnetic mode> has one polarization, $\mathbf E\parallel\hat{\mathbf z}$, and in-plane <wavevector>. Periodic in-plane counting gives
$$
dN=\frac{A}{(2\pi)^2}\,2\pi k\,dk,\qquad
\frac1{Ad}\frac{dN}{d\omega}=\frac{\omega}{2\pi c^2d}.
$$
Multiplying this <single-polarization two-dimensional photon density of states> by $\hbar\omega$ and the <Bose-Einstein distribution> produces the <thermal radiation between parallel conducting plates>:
$$
\boxed{u_{\rm plates}(\omega)=\frac{\hbar\omega^2}{2\pi c^2d}\frac1{e^{\hbar\omega/(k_{\rm B}T)}-1}.}
$$
The same rate balance therefore gives
$$
\left(\frac AB\right)_{\rm plates}=\frac{\hbar\omega_0^2}{2\pi c^2d},\qquad
\boxed{\frac{(A/B)_{\rm plates}}{(A/B)_{\rm free}}=\frac{\pi c}{2\omega_0d}\gg1.}
$$
These $B$ coefficients refer to the radiation polarization in their respective environments: isotropic free-space $B$ must not automatically be equated to the coefficient for a specified $z$-polarized field.
For <hydrogen>, take the quantization axis along $z$, neglect <spin-orbit coupling>, and assume perfect plates, a wide in-plane region and an unmodified atomic wavefunction: $a_{\rm B}\ll d\ll c/\omega_0$, with negligible surface-induced mixing or shifts. The <electric-dipole selection rules for hydrogen> require $\Delta l=\pm1$; the allowed $z$ field has $\Delta m_\ell=0$. Hence the <polarization-selective spontaneous emission between conducting plates> gives
$$
\boxed{m_\ell=0:\quad A_{\rm plates}\sim A_{\rm free}\frac{c}{\omega_0d},\qquad
\tau_{\rm plates}\sim\tau_{\rm free}\frac{\omega_0d}{c}.}
$$
The <thin-cavity enhancement of a normal electric dipole> applies: the $2p_0\to1s$ dipole points along $z$ and couples to the enhanced mode density. For $m_\ell=\pm1$, the transition dipoles lie in the $x$–$y$ plane and couple to none of these low-frequency modes: \b[the electric-dipole decay rate is zero in the ideal model], so this channel's lifetime is infinite. Real finite, imperfect plates and additional decay or mixing channels limit the lifetime. The order-one orientation factor for the allowed transition does not change the requested lifetime scaling.
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