Spontaneous Emission

An atom prepared in an excited state does not stay there. After a characteristic time it drops to a lower state and emits a photon — and it does so even when no light is shining on it, in perfect darkness. Classically this is a scandal: an electron in a stationary state should orbit forever. The resolution is that the electromagnetic field is never truly empty. Its vacuum carries zero-point fluctuations in every mode, and those fluctuations couple to the atomic dipole and tip the excited state into decay. This article owns the phenomenology of that decay — the rate $A_{21}=1/\tau$ , the Lorentzian line it produces, and the universal linewidth the classical picture predicts — while recalling the deeper machinery by name. Two neighbours supply that machinery. The quantum-electrodynamic calculation that the atom decays irreversibly into a continuum of vacuum modes, the Weisskopf–Wigner theory, is owned by aq-qm-spontaneous-emission-einstein-coefficients; the uncertainty-relation account of why a finite lifetime broadens a level into a Lorentzian is owned by aq-qm-lifetime-linewidth-broadening. We take those results and turn them into numbers you can measure: nanosecond lifetimes, megahertz linewidths, and the astonishing fact that the classical natural linewidth of every spectral line, expressed as a wavelength, is the same tiny value $1.18\times10^{-4}\ \text{Å}$ regardless of colour. The single most important feature is irreversibility. A two-level atom coupled to one mode would slosh its energy back and forth forever (vacuum Rabi oscillation); coupled instead to the vast continuum of free-space modes, the emitted photon disperses and never returns, giving genuine exponential decay. The rate climbs as the cube of the transition frequency, which is why optical states live for nanoseconds while microwave and hyperfine transitions can be metastable for years.

Key takeaways

Spontaneous emission is the decay of an excited atom with photon emission, occurring even with no applied field — a vacuum effect driven by zero-point field fluctuations.
The rate is the Einstein $A$ coefficient; the excited-state population decays exponentially, $P_e(t)=e^{-A_{21}t}$ , with radiative lifetime $\tau=1/A_{21}$ .
The rate scales as the cube of frequency, $A_{21}\propto\omega_0^3$ , so optical transitions decay in nanoseconds while microwave/hyperfine ones can be metastable for years.
A finite lifetime broadens the line into a Lorentzian of FWHM $\Delta\nu=1/2\pi\tau$ — the natural linewidth (Na D: $\tau=16.2\ \mathrm{ns}\Rightarrow\Delta\nu=9.83\ \mathrm{MHz}$ ).
The classical Lorentz-oscillator model gives a universal natural linewidth in wavelength, $\Delta\lambda=4\pi r_e/3=1.18\times10^{-4}\ \text{Å}$ , the same for every line.
Irreversible exponential decay requires coupling to a broad continuum of modes; a single mode gives reversible Rabi oscillation instead.
Key idea: spontaneous emission is the irreversible decay of an excited atom into the continuum of vacuum modes, at rate $A_{21}=1/\tau\propto\omega_0^3$ , producing a Lorentzian line whose classical natural width $\Delta\lambda=4\pi r_e/3$ is the same for every wavelength.

The core insight

Spontaneous emission is the irreversible decay of an excited atom into the continuum of vacuum field modes; its rate $A_{21}=1/\tau$ scales as $\omega_0^3$ and produces a Lorentzian line of FWHM $\Delta\nu=1/2\pi\tau$ , whose classical natural width in wavelength, $\Delta\lambda=4\pi r_e/3$ , is universal.

Where this lives today

Spontaneous emission is simultaneously the enabler and the enemy of quantum technology. Its extreme suppression on dipole-forbidden transitions is what makes optical atomic clocks possible: the strontium and ytterbium clock transitions have E1 emission forbidden, so their upper states live for seconds and the lines are narrow enough to reach fractional frequency uncertainties below $10^{-18}$ . The same physics limits laser coherence: amplified spontaneous emission seeds noise in every amplifier, and the Schawlow–Townes linewidth is the spontaneous-emission floor beneath a laser's purity. Because the rate depends on the local density of electromagnetic modes, it is now something engineers control. Placing an emitter in a microcavity, photonic crystal, or plasmonic antenna enhances the rate by the Purcell factor $F_P=\tfrac{3}{4\pi^2}(\lambda/n)^3 Q/V$ , speeding up single-photon sources for quantum networks; placing it in a photonic bandgap can inhibit emission almost entirely. Light-emitting diodes rely entirely on spontaneous emission for their output, and their efficiency is tuned by shaping the mode environment — a direct, industrial application of a process Einstein could only postulate.

The physical picture

Picture the excited atom not as a lamp with a receiver but as an antenna wired to an infinite bath of empty modes. Each vacuum mode carries a zero-point jiggle that nudges the atomic dipole; because there are so many modes the photon can vanish into and never come back, the energy leaks out irreversibly — exactly like a damped oscillator radiating into open space. The atom's coherent Rabi exchange with any one mode is drowned out by the sheer number of ways to lose the photon for good. That is the difference between reversible oscillation (one mode) and decay (a continuum), and it is the content of the Weisskopf–Wigner treatment we recall from the quantum owner. The cube-of-frequency law has two intuitive factors. One factor of $\omega_0$ comes from the energy per photon; two more come from the density of field modes, which grows as $\omega^2$ . More modes to emit into, at higher frequency, means faster decay. A microwave transition sees almost no modes and dawdles; an X-ray transition sees a torrent and empties almost instantly. This same $\omega_0^3$ is the $\nu^3$ in Einstein's $A/B$ ratio — the two articles describe one fact from two directions. The classical picture is a radiating charge on a spring. A Lorentz oscillator at frequency $\omega_0$ loses energy to radiation and damps at a rate set by the classical electron radius. Translating that damping into a spectral width gives something remarkable: expressed as a wavelength, the classical natural linewidth is $\Delta\lambda=4\pi r_e/3\approx1.18\times10^{-4}\ \text{Å}$ , and the wavelength cancels out — so a red line and a blue line have the same classical natural width. The quantum rate then multiplies this by the transition's oscillator strength, restoring the specific $A_{21}$ of a real line, but the universal classical width is the backdrop against which all of it is measured.

Figure

An excited atom decays exponentially, N₂(t) = N₂(0)e^(−t/τ), emitting one photon in a random direction; the finite lifetime τ broadens the transition into a Lorentzian line of full width Δν = 1/(2πτ). For the Na D line τ = 16.2 ns gives Δν = 9.83 MHz

The second figure carries the punchline. Because the classical radiative damping depends only on the classical electron radius, the natural linewidth expressed in wavelength is one universal number, the same across the whole visible spectrum. Real transitions differ only through their oscillator strengths and the steep $\omega_0^3$ rate, which we overlay as measured points.

Figure

The universal classical natural linewidth Δλ = 4πr_e/3 = 1.18×10⁻⁴ Å is the same at every wavelength (horizontal line). Real natural widths, set by the quantum A₂₁ ∝ ω₀³, are marked: Na D 9.83 MHz at 589 nm, Lyman-α 99.7 MHz at 122 nm

The core equations

The whole subject rests on one rate. Fermi's golden rule, applied to the electric-dipole coupling of the atom to the quantized field and summed over all free-space modes, gives the spontaneous rate in terms of the transition dipole moment and the cube of the frequency.

A_{21}=\frac{\omega_0^3\,|\boldsymbol\mu_{21}|^2}{3\pi\varepsilon_0\hbar c^3} \qquad (1)

where: $A_{21}$ — spontaneous decay rate ( $\mathrm{s^{-1}}$ ); $\omega_0$ — angular transition frequency, $\hbar\omega_0=E_2-E_1$ ; $\boldsymbol\mu_{21}=\langle2|{-e\mathbf r}|1\rangle$ — transition dipole moment; $\varepsilon_0,\hbar,c$ — vacuum constants. Read physically: the $\omega_0^3$ makes high-frequency transitions decay fast; the $|\boldsymbol\mu_{21}|^2$ makes dipole-forbidden transitions (tiny $\mu$ ) metastable. This QED expression is owned by aq-qm-spontaneous-emission-einstein-coefficients; here it is the fixed number that sets everything else. Turning off the field leaves only this rate, so the excited population empties exponentially and defines the radiative lifetime.

P_e(t)=e^{-A_{21}t}=e^{-t/\tau},\qquad \tau=\frac{1}{A_{21}} \qquad (2)

where: $P_e(t)$ — probability the atom is still excited at time $t$ ; $\tau$ — natural (radiative) lifetime, the $1/e$ decay time. If several lower levels are reachable, $A_{21}$ is replaced by the total $\sum_f A_{2f}$ and $\tau$ shortens accordingly. A state that lives only for a time $\tau$ cannot have a perfectly sharp energy; the transition acquires a Lorentzian profile whose full width at half maximum is set by the decay rate.

\Delta\nu=\frac{A_{21}}{2\pi}=\frac{1}{2\pi\tau},\qquad S(\nu)\propto\frac{(\Delta\nu/2)^2}{(\nu-\nu_0)^2+(\Delta\nu/2)^2} \qquad (3)

where: $\Delta\nu$ — natural linewidth (FWHM, Hz); $S(\nu)$ — the Lorentzian emission line shape centred at $\nu_0$ . Read physically: the natural width is the irreducible floor beneath Doppler and pressure broadening; the uncertainty-relation derivation of this Lorentzian is owned by aq-qm-lifetime-linewidth-broadening. The classical Lorentz oscillator gives a strikingly universal companion: the natural width, expressed as a wavelength, depends only on the classical electron radius and not on the colour of the line.

\Delta\lambda_{\text{cl}}=\frac{4\pi r_e}{3},\qquad r_e=\frac{e^2}{4\pi\varepsilon_0 m_e c^2} \qquad (4)

where: $\Delta\lambda_{\text{cl}}$ — classical natural linewidth in wavelength; $r_e=2.818\times10^{-15}\ \mathrm{m}$ — classical electron radius. Read physically: $\Delta\lambda_{\text{cl}}=1.18\times10^{-4}\ \text{Å}$ for every transition — the $\lambda$ -dependence cancels between the damping rate and the frequency-to-wavelength conversion. Quantum mechanics restores the transition-specific width by multiplying by the oscillator strength.

Derivation AThe universal classical natural linewidth

The goal is to show that the natural linewidth predicted by the classical Lorentz-oscillator model, when written as a spread in wavelength rather than frequency, is a single universal number $\Delta\lambda=4\pi r_e/3\approx1.18\times10^{-4}\ \text{Å}$ that does not depend on the transition wavelength at all. The strategy is to compute the classical radiative damping rate of an oscillating electron, convert that rate into a frequency width, and then convert the frequency width into a wavelength width — watching the explicit wavelength dependence cancel, leaving only the classical electron radius.

Step 1. A classical electron oscillating at $\omega_0$ radiates and so damps at the Abraham–Lorentz rate $\gamma_{\text{cl}}=\dfrac{e^2\omega_0^2}{6\pi\varepsilon_0 m_e c^3}=\dfrac{2}{3}\dfrac{r_e\omega_0^2}{c}$ . Why: the radiated power drains the oscillator's energy, and dividing power by energy gives an exponential damping constant — the classical analogue of $A_{21}$ .

Step 2. A damped oscillator has a Lorentzian spectrum of angular-frequency FWHM $\gamma_{\text{cl}}$ , i.e. $\Delta\nu_{\text{cl}}=\gamma_{\text{cl}}/2\pi=\dfrac{r_e\omega_0^2}{3\pi c}$ . Why: the Fourier transform of a decaying sinusoid is a Lorentzian whose width equals the damping rate, exactly as in the quantum case.

Step 3. Convert to a wavelength width using $|\Delta\lambda|=(\lambda^2/c)\,\Delta\nu$ with $\lambda=2\pi c/\omega_0$ : $\Delta\lambda_{\text{cl}}=\dfrac{\lambda^2}{c}\cdot\dfrac{r_e\omega_0^2}{3\pi c}=\dfrac{r_e}{3\pi c^2}\,\lambda^2\omega_0^2=\dfrac{4\pi r_e}{3}$ . Why: $\lambda^2\omega_0^2=(2\pi c)^2$ is a constant, so the wavelength and frequency both drop out and only $r_e$ survives — the width is universal.

Step 4. Numerically $\Delta\lambda_{\text{cl}}=\tfrac{4\pi}{3}(2.818\times10^{-15}\ \mathrm{m})=1.18\times10^{-14}\ \mathrm{m}=1.18\times10^{-4}\ \text{Å}$ . A real quantum line has $\Delta\lambda_{\text{quantum}}=f\cdot\Delta\lambda_{\text{cl}}$ with $f$ the oscillator strength. Why: the classical result is the reference against which oscillator strengths are defined; a strongly allowed line ( $f\sim1$ ) sits near the classical width, a forbidden line ( $f\ll1$ ) far below it. *Verified: $\Delta\lambda=4\pi r_e/3=1.18\times10^{-4}\ \text{Å}$ , identical for 589 nm, 656 nm, and 122 nm, in verify_aq-optics-spontaneous-emission.py.*

Go deeper · Derivation B · 1 minWhy decay is exponential — coupling to a continuum

Derivation BWhy decay is exponential — coupling to a continuum

The goal is to explain why an excited atom decays exponentially and irreversibly rather than oscillating, by contrasting coupling to a single mode with coupling to the free-space continuum. The strategy is to write the single-excitation state as an excited-atom amplitude plus a sum of one-photon amplitudes, formally eliminate the photon modes, and show that a broad continuum makes the atom's memory kernel collapse to a delta function — the Markov approximation — which turns the equation of motion into simple exponential decay at rate $A_{21}$ .

Step 1. Write $|\psi\rangle=c_e(t)|e,0\rangle+\sum_k c_{g,k}(t)|g,1_k\rangle$ : either the atom is excited with no photon, or it has decayed and put one photon into some mode $k$ . Why: energy conservation confines the dynamics to this single-excitation subspace, so nothing else can happen.

Step 2. The Schrödinger equations couple $c_e$ to every $c_{g,k}$ through the dipole matrix element; integrating the photon amplitudes and substituting back gives $\dot c_e=-\sum_k|g_k|^2\!\int_0^t e^{i(\omega_k-\omega_0)(t-t')}c_e(t')\,\mathrm{d}t'$ . Why: the atom's fate depends on its own past through the photons it has already partly emitted — a memory (non-Markovian) integral.

Step 3. For a broad continuum the sum over modes makes the kernel sharply peaked, $\sum_k|g_k|^2 e^{i(\omega_k-\omega_0)(t-t')}\to(A_{21}/2)\,\delta(t-t')$ — the Markov (Weisskopf–Wigner) approximation. Why: with so many modes at slightly different frequencies, their phases scramble instantly, so the photon leaves and cannot return; the memory collapses to the present instant.

Step 4. The equation becomes $\dot c_e=-(A_{21}/2)c_e$ , so $c_e(t)=e^{-A_{21}t/2}$ and the population $|c_e|^2=e^{-A_{21}t}$ . A single mode instead gives reversible Rabi oscillation. Why: irreversibility is not built into one atom plus one photon — it emerges from the continuum, exactly as friction emerges from coupling to many degrees of freedom. *Verified: the exponential population decay $e^{-A_{21}t}$ and the Lorentzian FWHM $\Delta\nu=A_{21}/2\pi$ (Na D 9.83 MHz) are checked in verify_aq-optics-spontaneous-emission.py; the full Weisskopf–Wigner treatment is owned by aq-qm-spontaneous-emission-einstein-coefficients.*

Limiting cases

The spontaneous rate spans an enormous range as frequency and dipole strength vary.

Case	Result	Meaning
Microwave / RF ( $\omega_0$ small)	$A_{21}\propto\omega_0^3\to$ tiny	Hyperfine and Rydberg microwave transitions are metastable for seconds to years; the 21 cm line has a lifetime of $\sim10^7$ yr.
Optical / UV (allowed E1)	$\tau=1/A_{21}\sim1\text{–}100\ \mathrm{ns}$	Dipole-allowed optical lines decay in nanoseconds; Na D has $\tau=16.2\ \mathrm{ns}$ , $\Delta\nu=9.83\ \mathrm{MHz}$ .
Forbidden transition ( $\mu_{21}\to0$ )	$A_{21}^{(E1)}=0\Rightarrow$ metastable	With E1 forbidden, only weak M1/E2 channels remain, giving lifetimes up to seconds (used in optical clocks).
Cavity / Purcell	$A_{\text{cav}}=F_P A_{21}$	A resonant cavity reshapes the mode density and enhances or suppresses the rate by $F_P=\tfrac{3}{4\pi^2}(\lambda/n)^3 Q/V$ .

Common trapSpontaneous emission cannot come from a classical field — but its width can

A purely semiclassical theory (quantum atom, classical field) predicts a stable excited state: with the applied field set to zero the interaction $-\mathbf d\cdot\mathbf E$ vanishes and nothing drives the decay. You must quantize the field so the vacuum retains its zero-point fluctuations. A second, subtler error is treating a single cavity mode: that gives reversible vacuum Rabi flopping, not decay — irreversibility needs a broad continuum. Curiously, the classical Lorentz oscillator does correctly predict the natural linewidth ( $\Delta\lambda=4\pi r_e/3$ ), because damping is a classical concept; what it cannot predict is that the decay happens at all in the dark.

Worked examples

Example 1 — the natural linewidth of the sodium D line.

Step 1. The Na 3p→3s transition has a measured radiative lifetime $\tau=16.2\ \mathrm{ns}$ . The spontaneous rate is $A_{21}=1/\tau=1/(16.2\times10^{-9}\ \mathrm s)=\mathbf{6.17\times10^{7}\ s^{-1}}$ .

Step 2. The natural linewidth is the Lorentzian FWHM $\Delta\nu=A_{21}/2\pi=1/(2\pi\tau)$ .

Step 3. Numerically $\Delta\nu=1/[2\pi(16.2\times10^{-9})]=\mathbf{9.83\ MHz}$ . This is the irreducible width of the D line — a floor beneath the $\sim1\ \mathrm{GHz}$ Doppler broadening of a room-temperature vapour, reached only in Doppler-free or cold-atom spectroscopy.

Step 4. For comparison, the hydrogen Lyman-α (2p→1s) line has $\tau=1.60\ \mathrm{ns}$ (a shorter lifetime, higher frequency), giving $\Delta\nu=1/(2\pi\cdot1.60\ \mathrm{ns})=\mathbf{99.7\ MHz}$ — about ten times broader, consistent with its faster decay.

Example 2 — the universal classical natural linewidth.

Step 1. The classical Lorentz-oscillator model predicts a natural linewidth in wavelength $\Delta\lambda_{\text{cl}}=4\pi r_e/3$ , with $r_e=2.818\times10^{-15}\ \mathrm m$ the classical electron radius.

Step 2. Numerically $\Delta\lambda_{\text{cl}}=\tfrac{4\pi}{3}(2.818\times10^{-15})=1.18\times10^{-14}\ \mathrm m=\mathbf{1.18\times10^{-4}\ \text{Å}}$ .

Step 3. Remarkably this is the same value for the green Na D line, the red Balmer-α line, and the UV Lyman-α line — the wavelength cancels. What differs between real lines is the oscillator strength $f$ : the measured width is $\Delta\lambda=f\,\Delta\lambda_{\text{cl}}$ .

Step 4. The steep $\omega_0^3$ frequency scaling of the quantum rate then explains the trend across the spectrum: from optical ( $\sim10^{15}\ \mathrm{Hz}$ ) to microwave ( $\sim10^{10}\ \mathrm{Hz}$ ) the rate falls by $(10^5)^3=\mathbf{10^{15}}$ , so microwave transitions are astronomically slower — which is exactly why hyperfine lines are metastable.

Physics of the portrait

Spontaneous emission is the process that makes the quantum vacuum tangible. An atom in an eigenstate ought to be stationary, yet it emits — because the field it lives in is never at rest. The concept forced physics to accept that "empty" space carries zero-point energy in every mode, and that this vacuum can do work on matter. It is the same vacuum whose fluctuations produce the Lamb shift and the Casimir force; spontaneous emission is its most everyday manifestation, responsible for the glow of every flame, LED, and star.

Origins · 1 minspontaneous emission and how this was discovered

spontaneous emission

vacuum-driven atomic decay, Einstein 1917 / Weisskopf–Wigner 1930 · atomic and quantum optics

Spontaneous emission was introduced phenomenologically by Einstein in 1917 as the field-independent decay needed to derive the Planck spectrum. Its physical origin — coupling of the atom to the zero-point fluctuations of the quantized field — awaited Dirac's 1927 field quantization, and its irreversible exponential form was established by the Weisskopf–Wigner theory of 1930, which showed that decay arises from coupling to a continuum of vacuum modes. The rate $A_{21}=\omega_0^3|\mu_{21}|^2/3\pi\varepsilon_0\hbar c^3$ ties the vacuum, the atom's dipole, and the emitted line's width into one number.

History

1900

— Planck introduces the quantum to fit the blackbody spectrum

1917

— Einstein defines the spontaneous rate

A_{21}

from detailed balance

1927

— Dirac quantizes the electromagnetic field, giving emission a mechanism

1930

— Weisskopf and Wigner derive exponential decay and the natural linewidth (Z. Phys. 63, 54)

1946

— Purcell notes that a cavity modifies the spontaneous rate

1947

— Lamb and Retherford measure the vacuum-induced Lamb shift

Going further

References. Foot, Atomic Physics, §1.7, §2.3, §7.5 · Loudon, The Quantum Theory of Light, 3rd ed., Ch. 1 & 6 · Demtröder, Laser Spectroscopy, Vol. 1, Ch. 2 · Saleh & Teich, Fundamentals of Photonics, 3rd ed., Ch. 13 · Weisskopf & Wigner, Z. Phys. 63, 54 (1930)

Provenance. The rate formula, lifetime, linewidth, and $A/B$ ratio are standard results in Foot (§1.7, §2.3), Loudon (Ch. 1 & 6), and Demtröder (Ch. 2). A verifier recomputes: $A_{21}=1/\tau$ and $\Delta\nu=1/2\pi\tau$ (Na D $\tau=16.2\ \mathrm{ns}\to9.83\ \mathrm{MHz}$ ; Lyman-α $\tau=1.60\ \mathrm{ns}\to99.7\ \mathrm{MHz}$ ); the universal classical width $\Delta\lambda=4\pi r_e/3=1.18\times10^{-4}\ \text{Å}$ ; the $\omega_0^3$ scaling. The QED rate $A_{21}=\omega_0^3|\mu_{21}|^2/3\pi\varepsilon_0\hbar c^3$ and the Weisskopf–Wigner exponential decay are owned by aq-qm-spontaneous-emission-einstein-coefficients (recalled by name); the natural-linewidth/Lorentzian account is owned by aq-qm-lifetime-linewidth-broadening. The Na D lifetime $\tau\approx16$ ns is the standard textbook value; the Purcell factor is quoted in its conventional form. No non-standard numbers introduced.

Builds on: Electric Dipole Approximation · Transition Dipole Moment · Einstein A and B Coefficients Leads to: Radiative Lifetime and Natural Linewidth · Lorentzian Line Shape · Stimulated Emission · Detailed Balance and the Planck Spectrum

If the atom is in an energy eigenstate, why doesn't it stay there forever?

Because it is not an eigenstate of the full system — atom plus quantized field. The excited atom with the field in its vacuum is an eigenstate only if you ignore the atom–field coupling. Once you include it, the true eigenstates mix the excited atom with one-photon states, and the initial "bare" excited state is not stationary: it evolves into a ground-state atom plus a photon. The zero-point fluctuations of the vacuum are what break the atom's stability, which is why a classical (fieldless) treatment wrongly predicts an eternal excited state.

Is the emitted photon's direction really random?

Yes. Spontaneous emission couples the atom to all the vacuum modes of free space, weighted by the dipole radiation pattern, so the photon can go into any of them with the appropriate probability. There is no incident photon to define a preferred direction (unlike stimulated emission, which copies the stimulating photon's direction). This is why fluorescence and thermal light are spatially incoherent, radiating into $4\pi$ steradians, whereas laser light — built from stimulated emission — is a directed beam.

Why is the natural linewidth the "floor" of spectroscopy?

Because it comes from the finite lifetime of the state itself and cannot be removed by improving the apparatus. Doppler broadening (from atomic motion) and pressure broadening (from collisions) can be suppressed by cooling, trapping, or lowering the density, but even a single motionless, isolated atom emits a Lorentzian of width $\Delta\nu=1/2\pi\tau$ . That is the ultimate resolution set by the transition, so ultra-narrow lines require long-lived (small- $A_{21}$ ) states — which is exactly why optical clocks use dipole-forbidden transitions with second-scale lifetimes.

How can the same classical formula give the linewidth but not the emission itself?

Damping is a classical idea, but emission-in-the-dark is not. The classical Lorentz oscillator radiates and therefore damps whenever it is oscillating, and its damping rate correctly gives the natural width $\Delta\lambda=4\pi r_e/3$ . But a classical oscillator only radiates if it is already moving; to start it decaying from rest requires a driving field, and in true darkness there is none. Quantum mechanics supplies the missing kick through vacuum fluctuations. So the classical model gets the shape and width right by analogy to a driven-then-released oscillator, while the fact that the decay occurs spontaneously is irreducibly quantum.

What does the Purcell factor actually change — the atom or the vacuum?

The vacuum, in effect. The free-space rate $A_{21}$ assumes the atom sees the mode density of open space. A cavity, mirror, or photonic structure changes how many electromagnetic modes are available at the transition frequency and how strongly they overlap the atom, so the same atom decays faster (more modes, on resonance) or slower (a bandgap, few modes). The Purcell factor $F_P=\tfrac{3}{4\pi^2}(\lambda/n)^3 Q/V$ quantifies the enhancement for a resonant cavity. The atom's dipole is unchanged; what changes is the electromagnetic environment it radiates into.

Why do forbidden transitions still emit at all if E1 is zero?

Because the electric-dipole (E1) term is only the leading approximation. When selection rules make the E1 dipole matrix element vanish, higher multipoles — magnetic dipole (M1) and electric quadrupole (E2) — remain, and they are suppressed by roughly $(a_0/\lambda)^2\sim10^{-6}$ relative to E1. So the transition is not truly forbidden, just extremely weak, giving lifetimes from milliseconds to years instead of nanoseconds. These metastable states power optical clocks and produce the forbidden nebular and auroral lines seen in astrophysics; the multipole hierarchy is set by the electric-dipole-approximation article.