Lorentzian Line Shape

No spectral line is a mathematical spike. The instant you accept that an excited atom lives only a finite time — the result aq-optics-radiative-lifetime-natural-linewidth hands over, that spontaneous emission at rate $A_{21}=1/\tau$ empties the upper state exponentially — you have already committed to a line of finite width and to one particular shape for it. That shape is the Lorentzian: not one profile among many, but the profile any first-order rate process must produce, because a quantity decaying as $e^{-\gamma t}$ has a Fourier spectrum that is Lorentzian and nothing else. The mechanism is one sentence of physics in two languages. Classically, a bound electron is a damped harmonic oscillator: struck once, it rings at $\omega_0$ while its amplitude bleeds away as $e^{-\gamma t/2}$ , and the radiated power spectrum is the squared Fourier transform of that ring-down — a Lorentzian of full width $\gamma$ centred on $\omega_0$ . Quantum-mechanically, Weisskopf–Wigner theory follows the excited-state amplitude leaking irreversibly into the vacuum modes and returns the identical profile with $\gamma=A_{21}$ . The classical oscillator is not a cartoon of the quantum result — it is exactly right about the line shape. What sets the Lorentzian apart is its wings: far from centre it falls only as $(\omega-\omega_0)^{-2}$ , a power law, where a Gaussian has already collapsed super-exponentially. Those heavy tails carry far-off-resonant scattering, set pressure-broadened absorption in planetary atmospheres, and refuse to vanish even when Doppler broadening buries the core. Because natural and collisional broadening act identically on every atom, the Lorentzian is the archetype of homogeneous broadening — the width every atom in the ensemble shares.

Key takeaways

The Lorentzian $g(\omega)=\dfrac{\gamma/2\pi}{(\omega-\omega_0)^2+(\gamma/2)^2}$ is the profile produced whenever the radiating dipole loses coherence exponentially in time.
It is the Fourier transform of a damped oscillation: a decay rate $\gamma$ gives FWHM $\Delta\omega=\gamma$ centred on $\omega_0$ , so a shorter lifetime is a broader line.
Natural broadening fixes the irreducible $\gamma_{\mathrm{nat}}=A_{21}=1/\tau$ ; phase-interrupting collisions add $\gamma_{\mathrm{col}}=2n\sigma\bar v$ , the rates simply adding while the profile stays Lorentzian.
The signature is the wing: $g\propto(\omega-\omega_0)^{-2}$ , still 2.7 percent of peak three widths out — heavy tails dominating far-detuned absorption.
In a warm gas the homogeneous Lorentzian convolves with the inhomogeneous Doppler Gaussian into the Voigt profile — Gaussian core, Lorentzian wings.
Key idea: exponential decay in time Fourier-transforms into a Lorentzian of FWHM equal to the decay rate; a line is Lorentzian if and only if its coherence dies by a single first-order rate process.

The core insight

Exponential decay in time ( $e^{-\gamma t/2}$ ) Fourier-transforms into a Lorentzian in frequency of FWHM $\gamma$ ; a line is Lorentzian if and only if the coherence decays as a simple first-order rate process.

Where this lives today

The Lorentzian is the working profile of precision spectroscopy and quantum technology. Every laser-cooling and atomic-clock scheme negotiates with a natural linewidth: the sodium and rubidium cooling transitions have Lorentzian widths of a few megahertz that set the Doppler cooling limit, while optical-lattice-clock transitions are chosen for their millihertz natural widths — ultra-long lifetimes giving razor-sharp Lorentzians that anchor frequency standards good to eighteen digits. In the other direction, atmospheric remote sensing lives on the wings: HITRAN radiative-transfer codes model every $\mathrm{CO_2}$ and $\mathrm{H_2O}$ line as a Voigt profile whose far $(\nu-\nu_0)^{-2}$ Lorentzian tail carries a disproportionate share of the absorbed sunlight, essential to climate modelling. The same Fourier relationship governs hardware with nothing to do with atoms. A Fabry–Pérot or laser cavity has resonances whose transmission peaks are Lorentzian, with linewidth $\Delta\nu=1/2\pi\tau_c$ set by the photon lifetime $\tau_c$ — the finesse is the ratio of free spectral range to that width. Gravitational-wave detectors, frequency combs, and cavity-QED experiments all stabilise against these Lorentzian modes, and a stabilised laser's own linewidth is quoted as the FWHM of its (near-)Lorentzian spectrum. Whenever a physicist writes "the linewidth", a Lorentzian is almost always the shape assumed.

The physical picture

Strike a bell once. It sounds its resonant pitch, but the tone dies away — and a tone lasting only a finite time is not a single pure frequency but a band whose width is set by how fast the ringing fades. A bell that rang forever would emit one infinitely sharp pitch; damping smears it into a spread inversely proportional to the ring-down time. An atom is exactly this bell: the lifetime $\tau$ is how long the dipole rings, and the line has width $\Delta\omega\sim1/\tau$ . The first figure draws both halves of that reciprocal relation — the ring-down in time and the Lorentzian it becomes in frequency.

Figure

A struck dipole rings down as E₀ e^(−γt/2) cos ω₀t and its power spectrum is the squared Fourier transform — a Lorentzian on ω₀ whose full width at half maximum equals the decay rate γ, so a shorter lifetime makes a broader line

The slow $(\omega-\omega_0)^{-2}$ wings are the fingerprint of the sharp switch-on at $t=0$ : emission begins abruptly, and that near-discontinuity injects high-frequency content a smooth pulse would never carry. A Gaussian envelope, smooth at every order, gives wings that vanish super-exponentially; the abruptly-switched exponential gives heavy power-law wings instead. The second figure lays a Lorentzian and a Gaussian of identical width side by side, scaled to a real atomic linewidth.

Homogeneity is the conceptual payoff. Because natural decay and collisions act on every atom the same way, the entire ensemble radiates one and the same Lorentzian. This is why a saturating laser can burn a spectral "hole" narrower than the Doppler width: it bleaches one velocity class, and the hole it leaves has the homogeneous Lorentzian width beneath the broad inhomogeneous Gaussian.

The core equations

Everything begins with the profile itself. The normalised Lorentzian carries unit area and is written most cleanly in angular frequency, with a single width parameter $\gamma$ that is simultaneously the FWHM and the total coherence-decay rate.

g(\omega)=\frac{\gamma/2\pi}{(\omega-\omega_0)^2+(\gamma/2)^2} \qquad (1)

where: $\omega_0$ — line-centre (resonance) angular frequency; $\gamma$ — FWHM in angular frequency, equal to the total decay rate of the dipole coherence; $g(\omega)$ — line-shape function, normalised so $\int_{-\infty}^{\infty}g\,\mathrm{d}\omega=1$ . Physical reading: the line is fixed by two numbers — where it sits and how wide it is — and because $\gamma/2$ enters squared, the half-power points fall exactly $\gamma/2$ from centre. A spectrometer reads hertz, not radians per second. Rewriting (1) in ordinary frequency introduces a $2\pi$ that is the most common source of factor errors in the subject, so it is worth stating explicitly.

g(\nu)=\frac{\Delta\nu/2\pi}{(\nu-\nu_0)^2+(\Delta\nu/2)^2},\qquad \Delta\nu=\frac{\gamma}{2\pi}=\frac{1}{2\pi\tau} \qquad (2)

where: $\nu$ — ordinary frequency in Hz, $\omega=2\pi\nu$ ; $\Delta\nu$ — FWHM in Hz; $\tau$ — coherence lifetime (for radiative decay, the excited-state lifetime). Physical reading: the width in hertz is $1/2\pi\tau$ , not $1/\tau$ — dropping the $2\pi$ inflates every linewidth by six. The profile is not postulated but forced by the time behaviour of the field. A dipole switched on at $t=0$ and damped at the amplitude rate $\gamma/2$ radiates a one-sided, exponentially decaying sinusoid — the field decaying at half the intensity rate.

E(t)=E_0\,e^{-\gamma t/2}\,e^{-i\omega_0 t}\,\Theta(t) \qquad (3)

where: $\gamma/2$ — field (amplitude) decay rate, half the intensity rate $\gamma$ ; $\Theta(t)$ — Heaviside step, emission switched on at $t=0$ ; $E_0$ — initial field amplitude. Physical reading: $\gamma$ and $\gamma/2$ are not interchangeable, and the abrupt switch-on $\Theta(t)$ is what seeds the heavy wings. The detected spectrum is the squared modulus of the Fourier transform of that field. The transform of a single exponential is a single pole, and its modulus squared is a Lorentzian.

|I(\omega)|\propto|\tilde E(\omega)|^2=\frac{|E_0|^2}{(\omega-\omega_0)^2+(\gamma/2)^2} \qquad (4)

where: $\tilde E(\omega)$ — Fourier transform of $E(t)$ ; $I(\omega)$ — spectral intensity, proportional to $g(\omega)$ . Physical reading: this is the origin of the shape — the unavoidable transform of exponential decay; normalising to unit area returns equation (1). With the shape fixed, the physics enters through $\gamma$ . Its irreducible floor is the natural width, set by spontaneous emission alone: the state decays at the Einstein rate $A_{21}$ whatever else happens.

\gamma_{\mathrm{nat}}=A_{21}=\frac{1}{\tau},\qquad \Delta\nu_{\mathrm{nat}}=\frac{A_{21}}{2\pi} \qquad (5)

where: $A_{21}$ — Einstein spontaneous-emission rate (upper $\to$ lower); $\gamma_{\mathrm{nat}}$ — natural linewidth (angular); $\tau$ — radiative lifetime. Physical reading: the natural width belongs to the transition alone, fixed by $A_{21}$ , and cannot be reduced without changing the atom. In a real gas a second channel opens: collisions that scramble the dipole phase. Each truncates the coherent wave-train, adding a decay rate that piles onto the radiative one and, scaling with perturber density, scales with pressure.

\gamma_{\mathrm{col}}=\frac{2}{\tau_{\mathrm{col}}}=2\,n\sigma\bar v,\qquad \gamma=\gamma_{\mathrm{nat}}+\gamma_{\mathrm{col}} \qquad (6)

where: $\tau_{\mathrm{col}}$ — mean time between phase-interrupting collisions; $n$ — perturber number density; $\sigma$ — broadening cross-section; $\bar v$ — mean relative speed. Physical reading: independent decay channels add their rates, so natural and collisional broadening sum linearly and leave the profile Lorentzian, growing linearly in pressure. Two relations close the toolkit: the peak height, which follows from unit area, and the Voigt profile, the convolution of the homogeneous Lorentzian with the inhomogeneous Doppler Gaussian.

g_{\max}=g(\nu_0)=\frac{2}{\pi\,\Delta\nu},\qquad V(\nu)=\int_{-\infty}^{\infty}g_{L}(\nu')\,g_{G}(\nu-\nu')\,\mathrm{d}\nu' \qquad (7)

where: $g_{\max}$ — peak of the normalised profile, in Hz $^{-1}$ ; $g_L$ — homogeneous Lorentzian (natural plus collisional); $g_G$ — inhomogeneous Doppler Gaussian; $V$ — observed Voigt line shape. Physical reading: unit area forces a narrow line to be tall, and combining homogeneous with inhomogeneous broadening is a convolution, never a sum of widths.

Derivation AThe Lorentzian from a damped classical oscillator

The goal is to show that the Lorentzian is not a modelling choice but a mathematical inevitability, extracted from nothing more than the equation of motion of a radiating dipole with weak radiative damping. We strike the bound electron impulsively at $t=0$ , follow the field it radiates, and Fourier-transform that field into a spectrum — and the classical result turns out identical to the quantum Weisskopf–Wigner one, which is why this picture is the right one to carry in your head.

Step 1. Write the electron as a damped harmonic oscillator, $\ddot x+\gamma\dot x+\omega_0^2 x=0$ , with $\omega_0$ the natural frequency and $\gamma$ the small radiative damping rate. Why: a bound charge that is restored toward equilibrium while radiating energy is exactly a damped oscillator, and the whole line shape is latent in this one linear equation.

Step 2. Solve in the underdamped regime $\gamma\ll\omega_0$ : $x(t)=x_0\,e^{-\gamma t/2}\cos(\omega_0 t)$ for $t\ge0$ . Why: the amplitude decays at $\gamma/2$ while the energy decays at $\gamma$ — the factor of two between field and intensity rates enters here.

Step 3. The radiated far field mirrors the dipole acceleration, so $E(t)\propto e^{-\gamma t/2}e^{-i\omega_0 t}\Theta(t)$ . Why: the detector integrates the field, which inherits the ring-down envelope, so the light's spectrum is the spectrum of this switched-on decaying sinusoid.

Step 4. Fourier-transform: $\tilde E(\omega)=\int_0^\infty E_0\,e^{-\gamma t/2}e^{-i(\omega_0-\omega)t}\,\mathrm{d}t=\dfrac{E_0}{\gamma/2-i(\omega-\omega_0)}$ . Why: the one-sided exponential integrates to a single simple pole, and that single pole — not two, not a Gaussian — dictates everything downstream.

Step 5. Take the squared modulus, $S(\omega)\propto\dfrac{|E_0|^2}{(\omega-\omega_0)^2+(\gamma/2)^2}$ , and normalise to unit area to recover $g(\omega)$ with FWHM $\gamma$ . Why: the modulus of a single pole squared is the Lorentzian denominator; $\int\mathrm{d}x/(x^2+a^2)=\pi/a$ with $a=\gamma/2$ fixes the prefactor to $\gamma/2\pi$ . *Verified: half-maximum at $|\omega-\omega_0|=\gamma/2$ , so FWHM $=\gamma$ ; the area integrates to 1 by $\int\mathrm{d}x/(x^2+a^2)=\pi/a$ ; and Weisskopf–Wigner reproduces the same profile with $\gamma=A_{21}$ .*

Go deeper · Derivation B · 1 minCollisions add a rate yet keep the line Lorentzian

Derivation BCollisions add a rate yet keep the line Lorentzian

The aim is to explain why phase-interrupting collisions broaden a line without ever changing its shape — the width grows linearly with pressure while the profile stays stubbornly Lorentzian. We model the wave as a sinusoid whose phase is randomised at each collision, compute the field autocorrelation, and invoke Wiener–Khinchin. The punchline: two independent decay channels multiply in the correlation and therefore add in rate — the algebraic reason natural and collisional widths sum.

Step 1. Model collisions as Poisson-distributed phase-scrambling events, so the average phase factor decays as $\langle e^{i\phi(t)}\rangle=e^{-t/\tau_{\mathrm{col}}}$ at rate $1/\tau_{\mathrm{col}}$ . Why: a collision that randomises the dipole phase destroys coherence just as decay does, and randomly-timed events give exponential survival — the same form as radiative decay.

Step 2. Combine with radiative decay: $\langle E^*(0)E(t)\rangle\propto e^{-i\omega_0 t}\,e^{-(\gamma_{\mathrm{nat}}/2+1/\tau_{\mathrm{col}})t}$ . Why: independent losses multiply their survival probabilities, so the two exponentials combine into one whose rate is the sum — this is where "rates add" is born.

Step 3. Apply Wiener–Khinchin, $S(\omega)=\int\langle E^*(0)E(t)\rangle e^{i\omega t}\,\mathrm{d}t$ , transforming the combined exponential back into a Lorentzian. Why: the transform of an exponential correlation is a Lorentzian regardless of what set its rate, so the shape is preserved.

Step 4. Read off the total width $\gamma=\gamma_{\mathrm{nat}}+2/\tau_{\mathrm{col}}=A_{21}+2n\sigma\bar v$ , linear in density $n$ and hence pressure. Why: with $\tau_{\mathrm{col}}=1/(n\sigma\bar v)$ the collisional term is $2n\sigma\bar v$ , so the FWHM rises in direct proportion to pressure — the empirical broadening law drops out. *Verified: as $\gamma_{\mathrm{col}}\to0$ the natural width is recovered; the width scales as $\propto P$ through $n\propto P$ , reproducing the pressure-broadening law; and the profile stays a single Lorentzian.*

Limiting cases

How the Lorentzian $g(\omega)=\dfrac{\gamma/2\pi}{(\omega-\omega_0)^2+(\gamma/2)^2}$ behaves in the regimes that matter:

Case	Result	Meaning
On resonance	$g(\omega_0)=\dfrac{2}{\pi\gamma}$	Peak height scales as $1/\gamma$ : narrower lines are taller, so unit area is preserved.
Half maximum	$\vert\omega-\omega_0\vert=\gamma/2$	The two half-power points straddle centre by $\gamma/2$ each, giving FWHM $\Delta\omega=\gamma$ .
Far wing	$g\xrightarrow{\vert\omega-\omega_0\vert\gg\gamma}\dfrac{\gamma}{2\pi(\omega-\omega_0)^2}$	Power-law $(\omega-\omega_0)^{-2}$ tails, far heavier than a Gaussian's — still 2.7 percent of peak three widths out.
Vanishing damping	$\lim_{\gamma\to0}g(\omega)=\delta(\omega-\omega_0)$	An infinitely long-lived state gives a monochromatic line — the Dirac delta.

Common trapWidths that add versus profiles that must be convolved

The commonest error is to add all broadening as widths. Two homogeneous Lorentzians on the same centre — natural plus collisional — do add their rates, $\gamma=\gamma_{\mathrm{nat}}+\gamma_{\mathrm{col}}$ , because the convolution of two Lorentzians is a Lorentzian whose width is the sum. But a homogeneous Lorentzian and an inhomogeneous Doppler Gaussian do not add — they must be convolved into a Voigt profile, whose width is neither the sum nor the quadrature sum. A second trap is in the factors: the field decays at $\gamma/2$ while intensity decays at $\gamma$ , and the width in hertz is $\Delta\nu=\gamma/2\pi=1/2\pi\tau$ , not $1/\tau$ — drop the $2\pi$ and every linewidth is wrong by a factor of six.

Worked examples

1 · The natural linewidth of the sodium D line (confidence builder)

The sodium $3p\to3s$ D transition has wavelength $\lambda=589\ \mathrm{nm}$ and upper-state radiative lifetime $\tau=16.2\ \mathrm{ns}$ . Find its natural linewidth in hertz, the peak of the normalised profile, and the on-resonance absorption cross-section.

Step 1. The emission rate is the reciprocal lifetime: $A_{21}=1/\tau=1/(16.2\times10^{-9}\ \mathrm{s})=6.17\times10^{7}\ \mathrm{s^{-1}}$ , which is $\gamma_{\mathrm{nat}}$ in angular units. Fixed by the transition dipole; it cannot be lowered without changing the atom.

Step 2. Convert to hertz through the all-important $2\pi$ : $\Delta\nu_{\mathrm{nat}}=A_{21}/2\pi=(6.17\times10^{7})/6.283=9.8\times10^{6}\ \mathrm{Hz}$ . So the sodium D line has natural width Δν ≈ 9.8 MHz, the floor beneath every Doppler-free measurement on it.

Step 3. The peak of the normalised profile is $g_{\max}=2/(\pi\Delta\nu)=$ 6.5×10⁻⁸ Hz⁻¹ — a line ten times narrower would be ten times taller, conserving unit area.

Step 4. The on-resonance cross-section for an ideal closed transition — fixed by the oscillator-strength sum rule (owned by aq-optics-oscillator-strength) and evaluated at the Lorentzian peak — is $\sigma_0=3\lambda^2/2\pi=3(589\times10^{-9})^2/(2\pi)=\mathbf{1.7\times10^{-13}\ m^2}$ , about a million times the atom's geometric area, which is why a thin sodium vapour is jet-black on resonance and transparent a few linewidths away. Δν ≈ 9.8 MHz, g(ν₀) ≈ 6.5×10⁻⁸ Hz⁻¹, σ₀ ≈ 1.7×10⁻¹³ m².

2 · When do collisions take over from natural broadening?

A sodium atom in a buffer gas suffers phase-interrupting collisions with cross-section $\sigma\approx10^{-18}\ \mathrm{m^2}$ at mean relative speed $\bar v\approx600\ \mathrm{m/s}$ . Find the collisional FWHM at one atmosphere (300 K), and the pressure at which it equals the 9.8 MHz natural width.

Step 1. Perturber density at 1 atm, 300 K from the ideal-gas law: $n=P/k_BT=(1.013\times10^{5})/[(1.381\times10^{-23})(300)]=2.45\times10^{25}\ \mathrm{m^{-3}}$ . Colliders per cubic metre the atom runs into.

Step 2. Collisional angular width: $\gamma_{\mathrm{col}}=2n\sigma\bar v=2(2.45\times10^{25})(10^{-18})(600)=2.9\times10^{10}\ \mathrm{s^{-1}}$ . Each collision truncates the wave-train, so the rate tracks how often they occur.

Step 3. In hertz: $\Delta\nu_{\mathrm{col}}=\gamma_{\mathrm{col}}/2\pi=\mathbf{4.7\times10^{9}\ Hz}$ , about 4.7 GHz at one atmosphere — swamping the 9.8 MHz natural width by nearly three orders of magnitude.

Step 4. Since $\Delta\nu_{\mathrm{col}}\propto P$ , collisions match the natural width at $P_{\times}=(1\ \mathrm{atm})(9.8\ \mathrm{MHz})/(4.7\ \mathrm{GHz})\approx\mathbf{1.6\ Torr}$ . Below a torr the line is natural-width-limited; above it, pressure-broadened — Lorentzian either way. Δν_col ≈ 4.7 GHz at 1 atm; the widths cross near 1.6 Torr.

Physics of the portrait

The line shape that bears his name was, for Hendrik Lorentz, a by-product of a deeper program: reducing all of optics to the motion of charged particles inside matter. The damped oscillator that gives the Lorentzian is the same "elastically bound electron" he used to explain dispersion, the Zeeman effect, and the refractive index.

Origins · 1 minHendrik Lorentz and how this was discovered

Hendrik Lorentz

1853–1928, Dutch physicist, Nobel Prize 1902 · classical electron theory and the optics of matter

Hendrik Antoon Lorentz built the bridge between Maxwell's fields and the atoms that carry them. From the late 1870s he modelled matter as charged particles — "ions", later electrons — elastically bound so each behaves as a harmonic oscillator driven by the passing light wave. This single idea, the elastically bound electron, let him derive the refractive index as a sum over resonances (the Lorentz–Lorenz relation, 1878–1880), and adding radiative damping gave emitted and absorbed lines exactly the profile we now call Lorentzian. His 1906 collision theory extended the picture to gases — the phase-interrupting encounters reconstructed in this article's second derivation — and in 1902 he shared the Nobel Prize with his student Pieter Zeeman for the magnetic splitting of spectral lines, again through the driven bound electron. The remarkable fact, clear only after Weisskopf and Wigner's 1930 quantum treatment, is that his classical oscillator gets the line shape exactly right: the quantum theory reinterprets $\gamma$ but not the Lorentzian it produces. A century on, "Lorentzian" remains the default profile of every resonance in physics, from atomic transitions to cavity modes to nuclear decays.

History

1878

— Lorentz introduces the elastically bound electron and the Lorentz–Lorenz relation for the refractive index

1896

— Lorentz's electron theory ties spectral emission and absorption to damped charge motion

1902

— Lorentz and Zeeman share the Nobel Prize for the magnetic splitting of spectral lines

1906

— Lorentz formulates the collision theory of pressure broadening

1912

— Voigt convolves the Lorentzian with the Gaussian Doppler profile to model observed lines

1930

— Weisskopf and Wigner derive the natural Lorentzian line shape from quantum mechanics

Going further

Texts. Foot, Atomic Physics, §7.1–7.4 (natural and collisional broadening) · Loudon, The Quantum Theory of Light, 3rd ed., Ch. 2 (Weisskopf–Wigner, Lorentzian line shape) · Demtröder, Laser Spectroscopy, Vol. 1, Ch. 3 (homogeneous versus inhomogeneous profiles) · Saleh & Teich, Fundamentals of Photonics, Ch. 13 · Hecht, Optics, 5th ed., §3.5 (classical dipole damping).

Provenance. The natural width $\gamma_{\mathrm{nat}}=A_{21}=1/\tau$ is recalled by name from aq-optics-radiative-lifetime-natural-linewidth, aq-optics-spontaneous-emission, and aq-optics-einstein-a-b-coefficients; the oscillator-strength sum rule behind $\sigma_0$ is owned by aq-optics-oscillator-strength and aq-optics-transition-dipole-moment. A verifier should confirm: (1) the transform of $e^{-\gamma t/2}e^{-i\omega_0 t}\Theta(t)$ gives the single pole and its squared modulus the Lorentzian (Loudon Ch. 2); (2) normalisation via $\int\mathrm{d}x/(x^2+a^2)=\pi/a$ ; (3) FWHM $=\gamma$ with $g(\omega_0)=2/\pi\gamma$ ; (4) $\tau=16.2$ ns $\Rightarrow\Delta\nu_{\mathrm{nat}}\approx9.8$ MHz and $\sigma_0=3\lambda^2/2\pi\approx1.7\times10^{-13}\ \mathrm{m^2}$ (Foot §7.1); (5) $\gamma_{\mathrm{col}}=2n\sigma\bar v\approx4.7$ GHz at 1 atm for $\sigma=10^{-18}\ \mathrm{m^2},\ \bar v=600\ \mathrm{m/s}$ (Demtröder Ch. 3); (6) the Voigt profile as the Lorentzian–Gaussian convolution.

Prerequisites. Radiative Lifetime and Natural Linewidth · Spontaneous Emission · Einstein A and B Coefficients · Transition Dipole Moment. Leads to. Doppler Broadening and Gaussian Line Shape · Resonant Absorption · Oscillator Strength · Detailed Balance and the Planck Spectrum.

Ask the tutor: Want me to work a Voigt-profile fit that separates the Lorentzian and Doppler contributions of a real spectral line?

Why does a finite lifetime force a finite linewidth at all?

Because a wave lasting only a finite time cannot be a single pure frequency. A monochromatic wave must oscillate identically from the infinite past to the infinite future; truncate it — switch it on at $t=0$ and let it decay — and you have mixed in a spread of frequencies. This is Fourier's trade-off, a signal confined to a time $\sim\tau$ occupying a band $\sim1/\tau$ — the same relation as the Heisenberg bound $\Delta E\,\Delta t\gtrsim\hbar$ . The exponential character then fixes the shape to be Lorentzian, but a finite width follows from finiteness of the lifetime alone.

What is the difference between the rates γ and γ/2?

They decay at different speeds. The upper-state population, and hence the intensity, decays at the full rate $\gamma=A_{21}=1/\tau$ : after a time $\tau$ the population has fallen to $1/e$ . The field amplitude (and the dipole coherence that radiates it) decays at half that rate, $\gamma/2$ , because intensity is amplitude squared and $e^{-\gamma t}=(e^{-\gamma t/2})^2$ . That is why equation (3) carries $e^{-\gamma t/2}$ while the natural width is $\gamma$ — and confusing the two is the classic factor-of-two slip.

Why do natural and collisional widths add, but Lorentzian and Doppler do not?

It comes down to homogeneous versus inhomogeneous. Natural decay and collisions are both homogeneous: they act identically on every atom, so each atom already radiates a Lorentzian, and two independent decay channels multiply their survival probabilities in the autocorrelation — the rates add, and the convolution of two Lorentzians is one wider Lorentzian. Doppler broadening is inhomogeneous: different atoms have different velocities and hence different centre frequencies, spreading identical narrow lines across a Gaussian of shifts. Combining a per-atom Lorentzian with that distribution of centres is a genuine convolution — the Voigt profile.

Why are the Lorentzian wings so much heavier than a Gaussian's?

The wings encode the sharpest feature of the signal in time. The field switches on abruptly at $t=0$ — a near-discontinuity, rich in high-frequency content, which shows up as slowly-falling $(\nu-\nu_0)^{-2}$ wings. A Gaussian pulse is infinitely smooth, so it has no such content and its wings die super-exponentially. Concretely, three FWHM from centre a Lorentzian is still at 2.7 percent of peak while a same-width Gaussian has fallen to about one part in a hundred thousand — which is why far-off-resonant absorption and scattering are always Lorentzian-dominated, even when the core is Gaussian.

If the line has a width, is energy not conserved in emission?

Energy is conserved perfectly — the width describes the distribution of photon energies, not a violation. A single emitted photon carries a definite energy, but because the excited state is not a perfect energy eigenstate (it decays), each photon's energy is drawn from a Lorentzian centred on $\hbar\omega_0$ with FWHM $\hbar\gamma$ , averaging to exactly $\hbar\omega_0$ over many emissions. The finite width is the energy uncertainty of a finite-lifetime state, $\Delta E=\hbar\gamma=\hbar/\tau$ , fully consistent with conservation event by event.

Does the classical oscillator model really get the quantum line shape right?

Yes — remarkably, it gets the shape exactly right, which is why physicists still teach it. The classical damped oscillator predicts a Lorentzian of FWHM $\gamma$ , and the quantum Weisskopf–Wigner treatment returns the identical Lorentzian with $\gamma$ reinterpreted as $A_{21}$ . What the classical model cannot supply is the value of $\gamma$ (that needs the transition dipole and the mode density) or the discrete, one-photon nature of emission. But the line's functional form, its $(\nu-\nu_0)^{-2}$ wings, and the FWHM-equals-decay-rate relation are all classical-oscillator-exact.