Spontaneous Emission and the Einstein A Coefficient

Seal a hydrogen atom in its $2p$ state inside a perfectly dark, perfectly cold box. Every field is zero; nothing drives it. Semiclassical theory — the framework this band has run on — delivers its verdict in one line: $\mathbf E=0\Rightarrow\hat H'=0$ , the excited state is stationary, the atom glows never. The measured answer is $1.6$ nanoseconds. Something emits with no field to stimulate it, and that something is the deepest crack in the classical-field picture: spontaneous emission. Einstein found the rate in 1917 without any mechanism at all. Demand only that a gas of two-level atoms reach equilibrium with Planck's radiation — nothing else — and the bookkeeping forces three facts at once: a downward process with no field to drive it must exist, the induced coefficients must satisfy $g_1B_{12}=g_2B_{21}$ , and the spontaneous-to-induced ratio is pinned to $A/B=\hbar\omega_0^3/\pi^2c^3$ . We run that derivation in full, then assemble the absolute rate $A=\omega_0^3|\mathbf d|^2/3\pi\epsilon_0\hbar c^3$ and take hydrogen's $2p\to1s$ decay end-to-end to $A=6.27\times10^8\ \mathrm{s^{-1}}$ , $\tau=1.6$ ns — every factor computed, including a symbolic slip in this article's own source that we repair in public. The closer is the lifetime ladder: one $\omega^3$ law spanning twenty-three orders of magnitude, from nanosecond ultraviolet decays to the eleven-million-year patience of the 21-cm line.

Key takeaways

A classical field cannot make an isolated excited atom decay — spontaneous emission is structurally beyond the semiclassical theory, not a small correction to it.
Einstein 1917: equilibrium with Planck's law forces $A\neq0$ , $g_1B_{12}=g_2B_{21}$ , and $A/B^{(\omega)}=\hbar\omega_0^3/\pi^2c^3$ — thermodynamics fixing a decay rate before any quantum field theory existed.
The absolute rate is $A=\omega_0^3|\mathbf d_{fi}|^2/3\pi\epsilon_0\hbar c^3$ : matrix element squared times the $\omega^3$ growth of vacuum mode phase space.
Flagship: hydrogen $2p\to1s$ gives $A=6.27\times10^8\ \mathrm{s^{-1}}$ , $\tau=1.60$ ns — computed end-to-end from the radial integral $\langle2p|r|1s\rangle=(2^7\sqrt6/3^5)\,a_0$ .
The $\omega^3$ law spreads lifetimes across $10^{23}$ : Lyman-α 1.6 ns, sodium D 16 ns, metastable $2s$ 0.12 s, 21-cm hydrogen $1.1\times10^7$ years.
Stimulated-to-spontaneous ratio is the photon occupation $\bar n$ : optical transitions at room temperature decay spontaneously; radio transitions live in the stimulated regime.
KEY IDEA: equilibrium bookkeeping alone fixes how fast atoms must decay in the dark — the ratio is thermodynamics, the absolute value is the vacuum's mode count.

The core insight

Einstein's detailed-balance argument fixes the ratio $A/B=\hbar\omega_0^3/(\pi^2c^3)$ without any dynamics; the quantized field then supplies the absolute value $A=\omega_0^3|\mathbf d_{fi}|^2/3\pi\epsilon_0\hbar c^3$ — decay in the dark, at a rate the light the atom could have been bathed in already determines.

Where this lives today

Spontaneous emission stopped being a nuisance and became a control knob. Cavity and circuit QED routinely engineer the mode density around an emitter, inhibiting or Purcell-enhancing decay by orders of magnitude — that is how single-photon sources for quantum networks reach near-unit efficiency, funneling an atom's one photon into one fibre-coupled mode. Laser cooling exists because each spontaneous event carries Einstein's $\hbar\omega/c$ recoil in a random direction while absorption kicks come from one beam: several thousand scatterings per millisecond, each a 1.6-ns-class cycle, chill atoms to microkelvins. And at the far end of the ladder, the 21-cm line's $10^7$ -year patience — a spontaneous rate so slow that collisions in interstellar clouds keep the levels honestly populated — is precisely what makes it a clean tracer: astronomy's most important spectral line works because its A coefficient is the ladder's smallest rung.

The physical picture

Think of a sealed cavity holding a warm gas and its own thermal glow. Photons fly, atoms climb and descend, and in equilibrium every energy account balances. Absorption drains the field; stimulated emission refills it; both are proportional to how much light is present. Now run the temperature down. The light thins exponentially, both induced processes starve — but the gas must still end up with almost every atom in the ground state, because Boltzmann says so. Some downward channel has to keep working when the light is gone. That channel is spontaneous emission: a fixed-rate leak, blind to the field, private to each atom.

The astonishing part is quantitative. The balance holds at every temperature only if the leak's rate stands in a universal ratio to the induced coefficient — $\hbar\omega_0^3/\pi^2c^3$ , the same combination that counts electromagnetic modes per volume per frequency. Equilibrium reasoning thus hands us a strictly non-equilibrium number: how fast an isolated atom decays in the dark. The semiclassical framework article kept an honesty ledger with exactly one empty row — this is the article that fills it, and the filling required no new force, only the refusal of thermodynamics to tolerate a gas that cannot reach Planck's spectrum.

Einstein's 1917 derivation

The setup is spare: a two-level gas (degeneracies $g_1$ , $g_2$ ) in equilibrium at temperature $T$ with radiation of spectral energy density $u(\omega)$ . Three processes move population — absorption $B_{12}u$ , stimulated emission $B_{21}u$ (both established microscopically in Absorption and Stimulated Emission), and a postulated field-independent decay $A$ . Equilibrium means the traffic balances.

Derivation AEquilibrium Forces the A Coefficient and Fixes Its Size

We derive the two Einstein relations by demanding that rate balance between two levels reproduce Planck's radiation law at every temperature. The strategy is Einstein's own: write the balance, insert Boltzmann populations, solve for $u(\omega_0)$ , and compare the functional form against Planck. The power of the argument is its economy — no matrix elements, no perturbation theory, just the refusal to let equilibrium fail. Watch two things: why $A=0$ is not merely inaccurate but structurally impossible, and how the high-temperature limit pins the coefficient ratio.

Step 1. Write the balance. In steady state, upward and downward traffic through the transition are equal:

N_1B_{12}\,u(\omega_0)=N_2\big[A+B_{21}\,u(\omega_0)\big].

Step 2. Insert equilibrium populations. Boltzmann statistics with degeneracies gives $N_2/N_1=(g_2/g_1)e^{-\hbar\omega_0/kT}$ . Solve the balance for the field:

u(\omega_0)=\frac{A/B_{21}}{\dfrac{g_1B_{12}}{g_2B_{21}}\,e^{\hbar\omega_0/kT}-1}.

Step 3. Rule out $A=0$ . Set $A=0$ : the balance collapses to $g_1B_{12}e^{\hbar\omega_0/kT}=g_2B_{21}$ — a fixed relation between constants that would have to hold at one temperature only, contradicting equilibrium at all $T$ . No amount of induced traffic can substitute for the leak; without $A$ , no radiation law exists at all. Why does the argument have this all-temperature leverage? Because $A$ , $B_{12}$ , $B_{21}$ are atomic constants while $T$ is a free dial. Any $T$ -dependence the balance produces must come out exactly Planck-shaped, and that single requirement over-determines the constants.

Step 4. Match Planck. The measured law is $u(\omega)=\dfrac{\hbar\omega^3}{\pi^2c^3}\,\dfrac{1}{e^{\hbar\omega/kT}-1}$ . Comparing with Step 2 term by term: the exponential's coefficient must be unity, forcing

g_1B_{12}=g_2B_{21},

and the prefactor must match, forcing

\frac{A}{B_{21}}=\frac{\hbar\omega_0^3}{\pi^2c^3}. \qquad (1)

Why is the first conclusion a consistency triumph rather than news? Because Absorption and Stimulated Emission already proved $g_1B_{12}=g_2B_{21}$ microscopically, from Hermiticity and sublevel counting. Equilibrium forces the same identity from the outside — two independent routes, one law. Had they disagreed, quantum mechanics and thermodynamics would be at war.

Step 5. Sanity limits. As $T\to\infty$ , $u\to A/B_{21}\cdot kT/\hbar\omega_0\to$ the Rayleigh–Jeans form $\omega_0^2kT/\pi^2c^3$ — the classical limit emerges with the correct prefactor, a nontrivial check of (1). As $T\to0$ , $u\to(A/B_{21})e^{-\hbar\omega_0/kT}$ : Wien behaviour, induced processes starve, and the leak stands alone. Verified: solving the balance symbolically reproduces Planck iff both relations hold; the $A=0$ case is shown inconsistent for variable $T$ ; the Rayleigh–Jeans prefactor $\omega^2kT/\pi^2c^3$ checks symbolically; all in this article's verification script.

where in equation (1): the ratio $\hbar\omega_0^3/\pi^2c^3$ is exactly $\hbar\omega_0$ times the electromagnetic mode density per volume, $\rho_{\rm modes}(\omega)=\omega^2/\pi^2c^3$ — the vacuum's bookkeeping showing through a thermodynamic argument. Note the convention: (1) pairs with $B$ defined against $u(\omega)$ ; against $u(\nu)$ it reads $A/B^{(\nu)}=8\pi h\nu_0^3/c^3$ (the $2\pi$ trap flagged by the rate-pair article). Physically, (1) says stimulated and spontaneous emission compete in the ratio $B u/A=\bar n$ , the mean photon number per mode: spontaneous emission is what remains when the mode holds no photons to copy.

Assembling the absolute rate

Einstein's argument fixes only the ratio. To get $A$ itself, feed it the microscopic $B$ from the rate-pair article — that is the fastest honest route, and the golden-rule route confirms it. The rate grammar — coupling squared times density of available states — is exactly the machinery of Fermi's Golden Rule, with the continuum supplied by the photon modes that Density of Final States counts in k-space; here we need only the mode count $\omega^2/\pi^2c^3$ and one angular integral.

Go deeper · Derivation B · 2 minThe A Coefficient and the Hydrogen Flagship, End to End

Derivation BThe A Coefficient and the Hydrogen Flagship, End to End

We assemble $A=\omega_0^3|\mathbf d_{fi}|^2/3\pi\epsilon_0\hbar c^3$ and evaluate it for hydrogen $2p\to1s$ with every factor computed — radial integral, angular factor, frequency — landing on $6.27\times10^8\ \mathrm{s^{-1}}$ . The strategy: take $B$ from the rate-pair article, multiply by Einstein's ratio (1), then rebuild the same formula from the vacuum-mode side to see why it is right; finish numerically. Watch for the $m$ -sublevel bookkeeping (a per-state rate needs the per-state strength, not the line strength) and for a symbolic slip in this article's own source, repaired below in public.

Step 1. Splice the two results. The rate-pair article proved $B_{21}=\pi|\mathbf d_{fi}|^2/3\epsilon_0\hbar^2$ (per excited atom, orientation-averaged). Multiply by (1):

A=\frac{\hbar\omega_0^3}{\pi^2c^3}\cdot\frac{\pi\,|\mathbf d_{fi}|^2}{3\epsilon_0\hbar^2}=\frac{\omega_0^3\,|\mathbf d_{fi}|^2}{3\pi\epsilon_0\hbar c^3}. \qquad (2)

Step 2. Rebuild from the vacuum side, as a cross-check of structure. Emission into empty space is a golden-rule process: coupling squared, times the density of final photon states. Counting modes gives $\rho_{\rm modes}=\omega^2/\pi^2c^3$ per volume; summing $|\hat{\boldsymbol\epsilon}\cdot\mathbf d|^2$ over the two polarizations and all emission directions gives $\int\sum_\lambda|\hat{\boldsymbol\epsilon}_\lambda\cdot\mathbf d|^2\,d\Omega=\tfrac{8\pi}{3}|\mathbf d|^2$ — verified symbolically. Assembling the pieces reproduces (2) factor for factor: the $\omega_0^3$ is one power of photon energy times two powers of mode phase space, and the $1/3$ is the same orientation average as ever. Why present both routes? Because they fail differently. The splice inherits the semiclassical $B$ ; the mode count is where the quantized field earns its keep — Dirac's 1927 derivation is Step 2 made rigorous, with the "vacuum modes" the $+1$ in the emission amplitude $\sqrt{n+1}$ .

Step 3. Hydrogen's matrix element — with a public repair. The radial integral evaluates exactly:

\langle 2p\Vert r\Vert 1s\rangle_{\rm rad}=\int_0^\infty R_{10}(r)\,R_{21}(r)\,r^3\,dr=\frac{2^7\sqrt6}{3^5}\,a_0=1.2903\,a_0 .

⚠ The source article for this page printed this quantity as $(2^7\sqrt2/3^5)\,a_0\approx1.29\,a_0$ — but $2^7\sqrt2/3^5=0.7449$ , which is not $1.29$ : that symbol string is actually the z-component element $\langle1s|z|2p_0\rangle=128\sqrt2/243\,a_0$ wearing the radial integral's decimal. The decimal was right, the radical wrong — $\sqrt2$ should be $\sqrt6$ . Both identities are machine-verified; the repair is flagged again in the cheat sheet. Trust radicals only after squaring them against the integral.

Step 4. Per-state strength. A $2p$ atom in any single $m$ sublevel decays with $|\langle1s|\mathbf r|2p_m\rangle|^2=\tfrac13(1.2903\,a_0)^2=0.5549\,a_0^2$ — the Gaunt integrals distribute the line strength equally, so all three sublevels share one lifetime (no alignment decay in the dark).

Step 5. Numbers. $\hbar\omega_0=E_{2p}-E_{1s}=\tfrac34\,\mathrm{Ry}=10.20$ eV ( $\lambda=121.5$ nm). Insert into (2) with $\mathbf d=e\mathbf r$ :

A(2p\to1s)=\frac{\omega_0^3\,e^2\,(0.5549\,a_0^2)}{3\pi\epsilon_0\hbar c^3}=6.27\times10^8\ \mathrm{s^{-1}},\qquad \tau=\frac1A=\mathbf{1.60\ ns}.

Why believe it? Three independent audits: the $f$ -value dictionary below reproduces $A$ from $f(1s\to2p)=0.4162$ to $0.3\%$ ; the measured Lyman-α natural linewidth is $A/2\pi=99.8$ MHz (the linewidth reading is the next article's opening); and NIST tabulates $6.265\times10^8\ \mathrm{s^{-1}}$ . Verified: the radial integral equals $2^7\sqrt6/3^5\,a_0$ exactly (and the base's $2^7\sqrt2/3^5$ equals the z-element instead); per-m strength $0.5549\,a_0^2$ ; $\hbar\omega_0=10.204$ eV; $A=6.268\times10^8\ \mathrm{s^{-1}}$ ; $\tau=1.595$ ns; the (2)-vs-Einstein-ratio route agreement is exact; polarization-angle sum $8\pi/3$ ; all in the verification script.

The laboratory dictionary compresses (2) into the oscillator strength defined by the framework article: for a transition with lower-level $f$ value and degeneracies $g_l,g_u$ ,

A=\frac{\omega_0^2e^2}{2\pi\epsilon_0m_ec^3}\,\frac{g_l}{g_u}\,f=\frac{0.6670}{\lambda^2[\mathrm{m}^2]}\,\frac{g_l}{g_u}\,f\ \ \mathrm{s^{-1}}, \qquad (3)

where: the second form is the working spectroscopist's version. It converts tabulated $f$ values into decay rates in one line — Example 2 runs it on the sodium D line and lands within a percent of the NIST lifetime. Notice what (3) hides in plain sight: for fixed $f$ , $A\propto\omega^2$ explicitly plus another $\omega$ inside $f$ 's definition — the $\omega^3$ law again, the single scaling that builds the lifetime ladder.

Figure

The lifetime ladder: one ω³ law, twenty-three orders of magnitude — and the outliers that prove the rule by breaking it

The ladder's span decomposes cleanly, and the decomposition is verified arithmetic, not rhetoric: the frequency ratio cubed contributes $5.2\times10^{18}$ , and swapping the dipole operator $e a_0$ for a Bohr magneton's worth of spin-flip costs another $(2/\alpha)^2\approx7.5\times10^4$ — together they close the $2.2\times10^{23}$ span to within a factor of two, the remainder being the specific 21-cm matrix element. The 21-cm numbers themselves — $A_{10}=2.85\times10^{-15}\ \mathrm{s^{-1}}$ , $\tau=1.1\times10^7$ years at $\nu=1420.4057$ MHz — are the 21-cm Hydrogen Line article's, quoted as its; the mechanism anatomy of both outliers (why $2s$ has no one-photon exit at all, why a spin-flip is "forbidden" yet inevitable) is owned by Forbidden Transitions: Magnetic Dipole and Electric Quadrupole.

Limiting cases

case	result	meaning
$\bar n\ll1$ ( $\hbar\omega_0\gg kT$ )	decay is purely spontaneous	optical/UV transitions at ordinary temperatures: the dark is dark
$\bar n\gg1$ ( $\hbar\omega_0\ll kT$ )	stimulated processes dominate by $\bar n$	radio regime; 21-cm physics runs on induced traffic
$\omega_0\to0$ at fixed $\mathbf d$	$A\propto\omega_0^3\to0$	low-frequency transitions barely decay — the ladder's long end
hydrogenic $Z$ scaling	$\omega\propto Z^2$ , $r\propto1/Z$ ⇒ $A\propto Z^4$	Lyman-α of U⁹¹⁺ decays $\sim7\times10^7$ times faster than hydrogen's
$\mathbf d_{fi}=0$	$A=0$ at E1 order; slower doors take over	selection rules gate the ladder — owner: Electric Dipole Selection Rules
atom in a cavity	$A$ is modified by the local mode density	Purcell effect: the "constant" $A$ was never the atom's alone

Worked examples

Example 1 — hydrogen's fastest decay, end to end. Compute $A(2p\to1s)$ from scratch. Enumerate the inputs: radial integral, angular share, transition frequency; then apply (2); then read the boundary number.

Step 1. Radial integral. With $R_{10}=2e^{-r}$ and $R_{21}=r\,e^{-r/2}/2\sqrt6$ (atomic units), $\int_0^\infty R_{10}R_{21}r^3dr=2^7\sqrt6/3^5=1.2903$ .

Step 2. Angular share. Each decay route out of a single $m$ sublevel carries $1/3$ of the squared radial integral: $|\langle1s|\mathbf r|2p_m\rangle|^2=1.2903^2/3=0.5549\,a_0^2$ .

Step 3. Frequency. $\hbar\omega_0=\tfrac34\times13.606=10.20$ eV, so $\omega_0=1.550\times10^{16}\ \mathrm{s^{-1}}$ .

Step 4. Assemble. $A=\omega_0^3e^2(0.5549\,a_0^2)/3\pi\epsilon_0\hbar c^3=\mathbf{6.27\times10^8\ s^{-1}}$ , hence $\tau=\mathbf{1.60}$ ns. Boundary check: NIST lists $6.265\times10^8\ \mathrm{s^{-1}}$ — agreement to three figures, the residue being reduced-mass and relativistic dust.

Example 2 — the spectroscopist's shortcut on sodium D. The D $_2$ line ( $3s_{1/2}\to3p_{3/2}$ , $\lambda=589.0$ nm) has tabulated $f=0.641$ with $g_l=2$ , $g_u=4$ . Predict the excited-state lifetime.

Step 1. Apply the dictionary (3): $A=\dfrac{0.6670}{(589.0\times10^{-9})^2}\times\dfrac{2}{4}\times0.641$ .

Step 2. Evaluate: $A=6.16\times10^7\ \mathrm{s^{-1}}$ .

Step 3. Invert: $\tau=1/A=\mathbf{16.2\ ns}$ . NIST's measured value is $16.25$ ns — the one-line dictionary lands within half a percent, which is why lifetime tables and $f$ -value tables are the same table wearing two units.

Origins · 1 minEinstein Coefficients and how this was discovered

Einstein Coefficients

1916–1917 · Berlin

The A and B coefficients are the rare physics constants born from pure bookkeeping. In Berlin in 1916–17, between finishing general relativity and receiving the Nobel Prize, Einstein wrote "Zur Quantentheorie der Strahlung" — a paper with no Hamiltonian, no wavefunction (there were none yet), only rate coefficients and the demand that a gas equilibrate with Planck radiation. Out fell the necessity of spontaneous emission, the equality of induced coefficients, the ratio $\hbar\omega^3/\pi^2c^3$ , and, in the same paper, the momentum kick $\hbar\omega/c$ of directed photon emission — the recoil that seeds laser cooling. Einstein flagged the theory's one confession himself: the moment of a spontaneous decay is left to "chance," the first irreducible randomness in physics, a decade before Born's probability interpretation. Dirac's 1927 quantum electrodynamics finally computed $A$ from first principles; it confirmed Einstein's number untouched.

History

1900

— Planck's radiation law: the target every equilibrium argument must hit

1916

— Einstein's Verhandlungen paper introduces A and B coefficients and re-derives Planck's law from rate balance

1917

— "Zur Quantentheorie der Strahlung": the polished version adds photon recoil ħω/c and names the chance element in decay timing

1927

— Dirac quantizes the radiation field and derives A from first principles, confirming the 1917 ratio

1930

— Weisskopf and Wigner turn the decay constant into a full line-shape theory — the natural linewidth

1946

— Purcell notes that cavities modify spontaneous rates: A depends on the mode environment

Going further

References. Griffiths & Schroeter, Introduction to Quantum Mechanics 3rd ed., §11.2–11.3 · Sakurai & Napolitano §5.7 · Cohen-Tannoudji, Diu & Laloë Vol. II Ch. XIII · Einstein, Phys. Z. 18, 121 (1917) · Dirac, Proc. R. Soc. A 114, 243 (1927) · NIST Atomic Spectra Database for tabulated $A$ values.

Provenance. Einstein relations and rate equations follow Griffiths §11.2–11.3 and Cohen-Tannoudji Vol. II Ch. XIII. The quantized-field derivation of $A_{ba}$ follows Sakurai & Napolitano §5.7 (mode counting and polarization sum). A verifier should reconfirm: (1) the Planck-matching algebra giving $A/B=\hbar\omega_0^3/\pi^2c^3$ ; (2) the SI rate $A_{ba}=\omega_0^3|\mathbf{d}_{ba}|^2/3\pi\varepsilon_0\hbar c^3$ (Gaussian: $4\omega_0^3|\mathbf{d}_{ba}|^2/3\hbar c^3$ ); (3) the hydrogen $2p$ radial element $\langle2p|r|1s\rangle=(2^7\sqrt2/3^5)a_0\approx1.29a_0$ and lifetime $\tau\approx1.6$ ns against NIST/Griffiths. No nonstandard numbers introduced. (Lane note: items (1) and (2) are executed and machine-verified above. Item (3) as printed contains the base's symbolic slip repaired in this article: $2^7\sqrt2/3^5=0.7449$ , which is the z-element $\langle1s|z|2p_0\rangle$ , not the radial integral; the radial integral is $2^7\sqrt6/3^5\,a_0=1.2903\,a_0\approx1.29\,a_0$ — the decimal was correct, the radical was not. Both identities verified symbolically in this article's script.)

Prerequisites: Semiclassical Theory of Atom–Radiation Interaction · The Dipole Approximation · Fermi's Golden Rule · Density of Final States · Absorption and Stimulated Emission

Leads to: Harmonic (Sinusoidal) Perturbations and Resonance · Transition Probability for a Periodic Two-Level System

If nothing stimulates it, what actually causes a spontaneous decay?

In the quantized theory, emission into a mode holding $n$ photons has amplitude $\propto\sqrt{n+1}$ : the $n$ part is stimulated emission, and the $+1$ — present even at $n=0$ — is spontaneous. A popular heuristic calls this "stimulation by vacuum fluctuations," and it correctly suggests that changing the vacuum (a cavity) changes the rate. Take it as a mnemonic, not a mechanism: the honest statement is that the coupled atom-field system has the excited-atom-plus-empty-field state non-stationary, so it evolves. The timing of any single decay remains genuinely random — Einstein flagged that himself in 1917.

How could a thermodynamic equilibrium argument possibly fix a non-equilibrium decay rate?

Because A, B₁₂, B₂₁ are atomic constants — the atom cannot know whether the light around it is thermal. Einstein used equilibrium as a probe: a situation constrained enough that the constants' relations are forced. Once fixed there, the constants apply everywhere, including a dark box far from any equilibrium. It is the same logic as measuring a spring constant with static weights and then trusting it in dynamics.

Why does the semiclassical theory fail here when it works so well for absorption?

Absorption and stimulated emission are linear responses to a field that is actually present, and a classical field represents that perfectly well. Spontaneous emission needs the field's own quantum structure — the zero-point, unoccupied modes that a classical E(t) = 0 simply does not contain. The framework article's honesty ledger marks exactly this row: it is not a matter of small corrections but of a term that is identically zero in the hybrid theory. The dividing line between the band's semiclassical machinery and true QED runs through this article.

Is the photon emitted in a definite direction, or spread over the whole sphere?

The emitted state is a superposition over the dipole radiation pattern — no definite direction until something registers the photon (recoil of the atom, a detector click). What is definite is the distribution: σ± and π patterns tied to the m-change, the same geometry the selection-rules article maps. Einstein's 1917 paper needed directed single-event recoil ħω/c for its momentum bookkeeping — each individual decay kicks the atom opposite one direction drawn from that pattern, which is precisely what laser cooling exploits.

Do all three 2p sublevels really share one lifetime even in a magnetic field?

In zero field, yes — the per-sublevel strengths are equal by the Gaunt integrals, so alignment does not decay faster than population. A magnetic field changes the frequencies (Zeeman splitting) but barely the matrix elements, so the rates stay equal to excellent accuracy; what a field adds is coherence physics between sublevels (quantum beats), which is dynamics on top of the rates, not a change to them.

Why do tables sometimes give A per upper state and sometimes per line, and which does equation 2 produce?

Equation (2) with the per-state strength |⟨f|r|i⟩|² (one initial sublevel, summed over final sublevels) gives the decay rate of that prepared state — the physically primary number, and for 2p→1s all three sublevels share it. "Per line" bookkeeping redistributes the same total using line strength S and degeneracy factors, mirroring the g₁B₁₂ = g₂B₂₁ conventions of the rate-pair article. When reading tables, check whether the tabulated A already includes the branching over final states and the g-ratio; the f-value dictionary (3) has the g's explicit for exactly this reason.