Transition Rates in Light–Matter Interaction

Light–matter transition rates connect three pieces of quantum mechanics:

1. electromagnetic fields provide a time-dependent interaction;
2. matter eigenstates provide transition frequencies and matrix elements;
3. perturbation theory converts those ingredients into probabilities or rates.

In the electric-dipole approximation, the central interaction is

$$H_{\mathrm{int}}(t) = -\mathbf d\cdot\mathbf E(t).$$

For a prescribed classical field, this Hamiltonian describes absorption and stimulated emission. When the electromagnetic field is quantized, the same dipole matrix element produces factors of $N$ for absorption, $N+1$ for emission, and a nonzero spontaneous-emission rate even when the relevant modes begin in vacuum.

Required background. Harmonic Perturbations fixes field-amplitude factors; Fermi’s Golden Rule and Density of States supply the rate limit and normalization. Dipole Approximation and Dipole Transitions explain the interaction and selection rules.

Electric-Dipole Coupling

For a nonrelativistic emitter whose internal size $a$ obeys $ka\ll1$, the leading length-gauge internal interaction is $-\mathbf d\cdot\mathbf E(\mathbf R,t)$, with

$$\mathbf d=\sum_a q_a(\mathbf r_a-\mathbf R).$$

For a neutral system the dipole is independent of a rigid origin shift. Center-of-mass motion, field gradients, and a spatially extended transition density need separate checks; Dipole Approximation develops them. Weak-field perturbation theory is an additional assumption, independent of the long-wavelength reduction.

Length and velocity gauges give equivalent results with consistently transformed states and complete bases. A finite matter-basis truncation can spoil that equivalence. If E1 coupling vanishes or is subleading, retain the relevant M1, E2, or higher operator using Multipole Expansion.

Transition Dipole Matrix Elements

Let

$$H_0\lvert i\rangle = E_i\lvert i\rangle, \qquad H_0\lvert f\rangle = E_f\lvert f\rangle.$$

Define the transition dipole

$$\mathbf d_{fi} = \langle f\rvert\mathbf d\lvert i\rangle$$

and the Bohr angular frequency

$$\omega_{fi} = \frac{E_f-E_i}{\hbar}.$$

The field does not couple to $\lvert\mathbf d_{fi}\rvert^2$ indiscriminately. It couples to a polarization projection,

$$\boldsymbol\epsilon\cdot\mathbf d_{fi},$$

where $\boldsymbol\epsilon$ is the polarization vector of the relevant field component. The transition requires both

$$\omega\approx\lvert\omega_{fi}\rvert$$

and a nonzero projected matrix element.

The two requirements answer different questions:

Ingredient	Physical role
$E_f-E_i$	line position or resonant frequency
$\mathbf d_{fi}$	intrinsic electric-dipole strength
$\boldsymbol\epsilon$	which tensor component is addressed
initial population	how many systems can undergo the process
final-state density	how many channels are available
pulse spectrum or linewidth	how energy matching is resolved

Hermiticity gives

$$\mathbf d_{if} = \mathbf d_{fi}^*.$$

This identity underlies the equality of upward and downward stimulated matrix-element strengths for a fixed pair of states. It does not make the observed upward and downward rates equal when populations, degeneracies, radiation occupation, or available channels differ.

A Classical Monochromatic Field

Write a real electric field at the system as

$$\begin{aligned} \mathbf E(t) ={}& \operatorname{Re} \left[ \mathcal E_0 \boldsymbol\epsilon e^{-i\omega t} \right] \\ ={}& \frac12 \left( \mathcal E_0\boldsymbol\epsilon e^{-i\omega t} + \mathcal E_0^*\boldsymbol\epsilon^*e^{i\omega t} \right), \end{aligned}$$

with

$$\boldsymbol\epsilon^*\cdot \boldsymbol\epsilon = 1.$$

The interaction has two harmonic components,

$$H_{\mathrm{int}}(t) = W e^{-i\omega t} + W^\dagger e^{i\omega t},$$

where

$$W = -\frac{\mathcal E_0}{2} \mathbf d\cdot\boldsymbol\epsilon.$$

The factor $1/2$ is easy to lose. It comes from expressing a real cosine-like field as the sum of positive- and negative-frequency components.

Absorption rate

For $E_f\gt E_i$, the $e^{-i\omega t}$ term is resonant with absorption. Fermi’s golden rule gives

$$\Gamma_{i\to f}^{\mathrm{abs}} = \frac{2\pi}{\hbar} \left| \langle f\rvert W\lvert i\rangle \right|^2 \delta(E_f-E_i-\hbar\omega).$$

Substituting $W$ gives

$$\Gamma_{i\to f}^{\mathrm{abs}} ={} \frac{\pi\lvert\mathcal E_0\rvert^2} {2\hbar} \times \left| \boldsymbol\epsilon\cdot\mathbf d_{fi} \right|^2 \delta(E_f-E_i-\hbar\omega).$$

Equivalently, using a delta function in angular frequency,

$$\Gamma_{i\to f}^{\mathrm{abs}} ={} \frac{\pi\lvert\mathcal E_0\rvert^2} {2\hbar^2} \times \left| \boldsymbol\epsilon\cdot\mathbf d_{fi} \right|^2 \delta(\omega-\omega_{fi}).$$

For a plane wave in vacuum with the normalized complex polarization above, the cycle-averaged intensity is

$$I = \frac12 \epsilon_0c \lvert\mathcal E_0\rvert^2.$$

Therefore

$$\Gamma_{i\to f}^{\mathrm{abs}} ={} \frac{\pi I} {\epsilon_0c\hbar^2} \times \left| \boldsymbol\epsilon\cdot\mathbf d_{fi} \right|^2 \delta(\omega-\omega_{fi}).$$

Here $|\mathcal E_0|$ is the norm of the complex field phasor. It equals the peak real field for linear polarization, but a circularly polarized field has constant real magnitude $|\mathcal E_0|/\sqrt2$. The intensity formula holds in either case.

The delta expression is a spectral contribution to be integrated over a continuum or a smooth incident spectrum. It is not a constant excitation rate for one isolated pair driven by a coherent monochromatic wave; a finite pulse produces the pulse’s spectral line shape instead.

Stimulated emission rate

For an initially excited state $\lvert f\rangle$, the conjugate field component can drive a transition to $\lvert i\rangle$. The stimulated-emission rate is

$$\Gamma_{f\to i}^{\mathrm{stim}} ={} \frac{\pi\lvert\mathcal E_0\rvert^2} {2\hbar^2} \times \left| \boldsymbol\epsilon^*\cdot\mathbf d_{if} \right|^2 \delta(\omega-\omega_{fi}).$$

Since $\mathbf d_{if}=\mathbf d_{fi}^*$,

$$\left| \boldsymbol\epsilon^*\cdot\mathbf d_{if} \right|^2 = \left| \boldsymbol\epsilon\cdot\mathbf d_{fi} \right|^2.$$

Thus a fixed classical mode gives equal stimulated rate coefficients for the two directions. Net absorption still depends on the population difference. This is the same absorption-minus-stimulated-emission structure developed in Linear Response Preview.

What a classical field cannot do

If $\mathcal E_0=0$, the classical interaction above vanishes. It therefore predicts neither spontaneous emission nor vacuum fluctuations. One may insert a decay rate phenomenologically, but deriving that rate requires quantized radiation modes or an equivalent quantum environment.

Finite Pulses and the Rate Limit

For a square pulse of duration $T$, the resonant integral is

$$\int_0^Tdt\, e^{i(\omega_{fi}-\omega)t} ={} T e^{i(\omega_{fi}-\omega)T/2} \times \operatorname{sinc} \left[ \frac{(\omega_{fi}-\omega)T}{2} \right].$$

The probability is proportional to

$$T^2 \operatorname{sinc}^2 \left[ \frac{(\omega_{fi}-\omega)T}{2} \right].$$

A constant golden-rule rate emerges only when:

• many final states lie inside the width $\Delta\omega\sim1/T$;
• the weighted final-state density varies slowly over that width;
• the total transition probability remains small;
• recurrences and coherent back-action are negligible.

For one isolated resonant transition, first-order probability initially grows as $T^2$ rather than linearly. Long-time coherent dynamics is described by Rabi oscillations, not by an irreversible constant rate. See Resonant Driving for the pulse-duration and depletion conditions.

Quantizing the Radiation Field

In a periodic normalization volume $\mathcal V$, a common transverse free-field mode expansion is

$$\mathbf E^{(+)}(\mathbf r) ={} i \sum_{\mathbf k,\lambda} \sqrt{ \frac{\hbar\omega_{\mathbf k}} {2\epsilon_0\mathcal V} } \times \boldsymbol\epsilon_{\mathbf k\lambda} a_{\mathbf k\lambda} e^{i\mathbf k\cdot\mathbf r},$$

with

$$\mathbf E^{(-)} = [\mathbf E^{(+)}]^\dagger, \qquad \mathbf E = \mathbf E^{(+)}+\mathbf E^{(-)}.$$

Here $\lambda$ labels the two transverse polarizations and

$$\omega_{\mathbf k}=c\lvert\mathbf k\rvert.$$

The overall phases in the mode expansion are conventional. Rates depend on absolute squares and are unchanged by a consistent phase redefinition.

For one mode with occupation number $N$, the ladder operators give

$$a\lvert N\rangle = \sqrt N\lvert N-1\rangle,$$

and

$$a^\dagger\lvert N\rangle = \sqrt{N+1}\lvert N+1\rangle.$$

These two square roots are the entire algebraic origin of the $N$ and $N+1$ rate factors.

Mode-Resolved Quantum Rates

Consider a lower matter state $\lvert g\rangle$, an upper state $\lvert e\rangle$, and

$$E_e-E_g = \hbar\omega_0.$$

For a mode $(\mathbf k,\lambda)$, absorption changes

$$\lvert g;N_{\mathbf k\lambda}\rangle \longrightarrow \lvert e;N_{\mathbf k\lambda}-1\rangle.$$

Its squared interaction matrix element is

$$\lvert M_{\mathrm{abs}}\rvert^2 ={} \frac{\hbar\omega_{\mathbf k}} {2\epsilon_0\mathcal V} N_{\mathbf k\lambda} \times \left| \mathbf d_{eg}\cdot \boldsymbol\epsilon_{\mathbf k\lambda} \right|^2.$$

Emission changes

$$\lvert e;N_{\mathbf k\lambda}\rangle \longrightarrow \lvert g;N_{\mathbf k\lambda}+1\rangle,$$

with

$$\lvert M_{\mathrm{em}}\rvert^2 ={} \frac{\hbar\omega_{\mathbf k}} {2\epsilon_0\mathcal V} (N_{\mathbf k\lambda}+1) \times \left| \mathbf d_{ge}\cdot \boldsymbol\epsilon_{\mathbf k\lambda}^* \right|^2.$$

The corresponding golden-rule expressions are

$$\Gamma_{g\to e}^{(\mathbf k\lambda)} ={} \frac{2\pi}{\hbar} \lvert M_{\mathrm{abs}}\rvert^2 \times \delta(E_e-E_g-\hbar\omega_{\mathbf k}),$$

and

$$\Gamma_{e\to g}^{(\mathbf k\lambda)} ={} \frac{2\pi}{\hbar} \lvert M_{\mathrm{em}}\rvert^2 \times \delta(E_e-E_g-\hbar\omega_{\mathbf k}).$$

Splitting

$$N+1 = \underbrace{N}_{\text{stimulated}} + \underbrace{1}_{\text{spontaneous}}$$

is physically meaningful here:

• the term proportional to $N$ is stimulated emission into an occupied mode;
• the remaining term is spontaneous emission into that mode;
• absorption has no vacuum term because $a\lvert0\rangle=0$.

The vacuum contribution should not be pictured as an ordinary photon already present in the mode. It follows from the quantum field’s operator algebra and the availability of a final one-photon state.

Single mode versus continuum

The golden-rule delta function is not reliable for a perfectly isolated atom coupled to one exactly resonant lossless mode. That problem can show coherent excitation exchange and belongs to the Jaynes–Cummings Model. Irreversible exponential decay emerges when many modes form an effectively continuous reservoir and the return amplitude can be neglected over the time window of interest.

Free-Space Spontaneous Emission

Set all photon occupations to zero and sum the mode-resolved emission rate over free-space wavevectors and polarizations. The continuum replacement is

$$\sum_{\mathbf k} \longrightarrow \frac{\mathcal V}{(2\pi)^3} \int_0^\infty k^2dk \int d\Omega.$$

The factor $1/\mathcal V$ in the squared mode field cancels the $\mathcal V$ in the mode density. For transverse polarizations,

$$\sum_{\lambda=1}^{2} \left| \mathbf d_{ge}\cdot \boldsymbol\epsilon_{\mathbf k\lambda} \right|^2 = \lvert\mathbf d_{ge}\rvert^2 - \left| \mathbf d_{ge}\cdot\widehat{\mathbf k} \right|^2.$$

The differential free-space rate is

$$\frac{d\Gamma_0}{d\Omega} ={} \frac{\omega_0^3} {8\pi^2\epsilon_0\hbar c^3} \times \left( \lvert\mathbf d_{ge}\rvert^2 - \left| \mathbf d_{ge}\cdot\widehat{\mathbf k} \right|^2 \right).$$

For a real dipole along $z$, the angular factor is $|d|^2\sin^2\theta$. A circular transition dipole $\mathbf d=d(\widehat{\mathbf x}+i\widehat{\mathbf y})/\sqrt2$ instead gives $|d|^2(1+\cos^2\theta)/2$. Both integrate to $8\pi|d|^2/3$.

Polar sections of the emission probability per solid angle, in units of the common prefactor times $|d|^2$. The angle $\theta$ is measured from $z$. Linear and circular transition dipoles have different directional patterns but the same total rate when $|\mathbf d|$ and $\omega_0$ agree.

Integrating over direction gives the standard electric-dipole rate

$$\Gamma_0 = \frac{ \omega_0^3 \lvert\mathbf d_{ge}\rvert^2 }{ 3\pi\epsilon_0\hbar c^3 }.$$

This formula assumes:

• free space in three spatial dimensions;
• electric-dipole coupling;
• weak atom–field coupling;
• a continuum of photon modes;
• Markov or Wigner–Weisskopf behavior over the decay window;
• no boundaries, cavities, waveguides, or material dispersion;
• a fixed transition dipole rather than an unresolved degenerate multiplet.

The $\omega_0^3$ scaling combines the electromagnetic mode density with the frequency dependence of the vacuum electric field per photon. The same normalization cancellation is a concrete application of Density of States in Transition Rates.

Lifetime, branching, and natural width

If an upper state can decay to several lower states,

$$\Gamma_e = \sum_g\Gamma_{e\to g}.$$

Its radiative lifetime is

$$\tau_e = \frac{1}{\Gamma_e},$$

and the branching fraction into channel $g$ is

$$b_{e\to g} = \frac{\Gamma_{e\to g}}{\Gamma_e}.$$

In the ideal Markov model, the excited amplitude and population behave as

$$c_e(t) \propto e^{-\Gamma_e t/2}, \qquad P_e(t) = e^{-\Gamma_e t}.$$

For a stable lower state and no additional dephasing, the lifetime-limited Lorentzian has angular-frequency full width at half maximum $\Gamma_e$. If both levels decay independently at rates $\Gamma_e$ and $\Gamma_g$, the optical coherence decays at $(\Gamma_e+\Gamma_g)/2$; an additional pure-dephasing rate $\gamma_\phi$ gives FWHM $\Gamma_e+\Gamma_g+2\gamma_\phi$ in the corresponding Markov model. Real spectra can also contain Doppler, collisional, power, transit-time, and inhomogeneous broadening. Those mechanisms should not be folded into $\Gamma_e$ without a model.

Structured electromagnetic environments

Cavities, waveguides, interfaces, photonic crystals, and absorptive media change the available mode density and field profiles. The replacement

$$\Gamma_0 \longrightarrow \Gamma(\mathbf r,\omega_0)$$

can change the total rate, radiation pattern, polarization, branching ratios, and Markov validity. The free-space formula is therefore a benchmark, not a universal law for every photonic environment.

The Quantum Optical Master Equation derives reduced dynamics; Thermal and Vacuum Noise expresses rates through bath correlations.

Thermal and Occupied Radiation Modes

For an equilibrium bosonic radiation bath,

$$\overline n(\omega,T) = \frac{1}{e^{\beta\hbar\omega}-1}.$$

Under the same weak-coupling and Markov assumptions used for $\Gamma_0$, the two-level rates are

$$\Gamma_\downarrow = \Gamma_0 [\overline n(\omega_0,T)+1],$$

and

$$\Gamma_\uparrow = \Gamma_0 \overline n(\omega_0,T).$$

Their ratio is

$$\frac{\Gamma_\uparrow} {\Gamma_\downarrow} = e^{-\beta\hbar\omega_0}.$$

At zero temperature, $\overline n=0$: upward thermal excitation vanishes, while downward spontaneous emission remains. At high thermal occupation, the $+1$ becomes negligible compared with $\overline n$, and upward and downward stimulated rates become nearly equal.

Einstein Coefficient Bridge

Spectroscopy and laser physics often package the same physics into Einstein coefficients. For lower and upper levels $g$ and $e$, write

$$R_{g\to e} = B_{ge}\,u(\omega_0),$$

and

$$R_{e\to g} = A_{eg} + B_{eg}\,u(\omega_0).$$

Here $u(\omega)$ is the total isotropic radiation energy density per unit angular frequency. With this convention, thermal equilibrium implies

$$g_gB_{ge} = g_eB_{eg},$$

where $g_g$ and $g_e$ are level degeneracies, and

$$\frac{A_{eg}}{B_{eg}} = \frac{\hbar\omega_0^3} {\pi^2c^3}.$$

Factors of $2\pi$, degeneracies, polarization averages, and line-profile normalizations change when one uses energy density per ordinary frequency, per angular frequency, per solid angle, or per polarization. An Einstein coefficient should never be quoted without its spectral-density convention.

The microscopic identifications are:

Einstein language	Microscopic content
$A_{eg}$	vacuum $+1$ term summed over final photon modes
$B_{eg}u$	stimulated emission from occupied modes
$B_{ge}u$	absorption from occupied modes
degeneracy relation	state counting plus microscopic reversibility

Einstein Coefficients derives these detailed-balance relations and converts spectral-density conventions.

Selection Rules, Polarization, and Degeneracy

For electric-dipole transitions, $\mathbf d$ is a rank-$1$ tensor and is odd under parity. In the usual angular-momentum basis, the leading rules are

$$\Delta J=0,\pm1, \qquad J=0\not\leftrightarrow J'=0,$$ $$\Delta M=q, \qquad q=0,\pm1,$$

and opposite initial and final parity.

Write $\mathbf d\cdot\mathbf E=\sum_q(-1)^q d_qE_{-q}$. The field component $E_{-q}$ multiplies the operator $d_q$, which changes $M$ by $q$. Relative to a chosen quantization axis:

Polarization component	Addressed dipole component	Magnetic rule
linear along the axis	$q=0$	$\Delta M=0$
one circular component	$q=+1$	$\Delta M=+1$
opposite circular component	$q=-1$	$\Delta M=-1$

The labels $\sigma^+$ and $\sigma^-$ depend on propagation and handedness conventions. The invariant statement is the spherical component $q$ and the matrix element it selects.

For an equally populated incoherent mixture of $g_i$ initial magnetic sublevels, with final sublevels unresolved, the polarization-resolved line strength is

$$S_{i\to f}^{(\boldsymbol\epsilon)} ={} \frac{1}{g_i} \sum_{m_i,m_f} \times \left| \langle f,m_f\rvert \mathbf d\cdot\boldsymbol\epsilon \lvert i,m_i\rangle \right|^2.$$

Use the actual populations for a nonuniform mixture, and a density-matrix expression retaining interference for coherent preparation. Unresolved sublevels alone do not justify the factor $1/g_i$. The Wigner–Eckart theorem separates reduced matrix elements from angular coefficients, but the derivation is given in Dipole Transitions.

An electric-dipole-forbidden line is not absolutely impossible. Magnetic-dipole, electric-quadrupole, two-photon, relativistic, hyperfine-mixing, or symmetry-breaking mechanisms can contribute at higher or different order.

Worked Scale Estimate

Take an optical transition with

$$\lambda_0=500\,\mathrm{nm}$$

and a transition dipole of one atomic unit,

$$\lvert\mathbf d_{ge}\rvert = ea_0 \approx 8.48\times10^{-30}\,\mathrm{C\,m}.$$

The angular frequency is

$$\omega_0 = \frac{2\pi c}{\lambda_0} \approx 3.77\times10^{15}\,\mathrm{s^{-1}}.$$

The free-space electric-dipole estimate is

$$\Gamma_0 \approx 1.62\times10^7\,\mathrm{s^{-1}}.$$

Thus

$$\tau \approx 6.17\times10^{-8}\,\mathrm{s} = 61.7\,\mathrm{ns},$$

and the lifetime-limited width in ordinary frequency is

$$\frac{\Gamma_0}{2\pi} \approx 2.58\,\mathrm{MHz}.$$

This is an order-of-magnitude example, not a prediction for a named line. Real rates depend on the actual many-electron or molecular matrix element, angular factors, branching channels, environment, and nonradiative decay.

What a Measured Spectral Line Contains

A measured line should not be identified with a bare golden-rule matrix element. Its interpretation typically combines:

Observable feature	Main theoretical ingredients
center frequency	energy difference plus shifts
integrated strength	matrix element, populations, degeneracy, geometry
line shape	pulse envelope, lifetime, motion, collisions, environment
polarization	tensor component and experimental axis conventions
emitted intensity	upper-state population, rate, branching, collection efficiency
absorbed intensity	lower-state population, optical depth, propagation, saturation

The historical and instrumental meaning of these quantities is discussed in Spectroscopy. Saturation and steady-state driven dynamics are described by the Optical Bloch Equations.

Exercises

Integrate the dipole radiation pattern

Take a real transition dipole along $z$. Use the differential rate to recover the total free-space spontaneous-emission rate.

Solution

For $\mathbf d=d\widehat{\mathbf z}$,

$$\lvert\mathbf d\rvert^2 - \lvert\mathbf d\cdot\widehat{\mathbf k}\rvert^2 = d^2\sin^2\theta.$$

The angular integral is

$$\begin{aligned} \int d\Omega\,\sin^2\theta &= 2\pi \int_0^\pi d\theta\,\sin^3\theta \\ &= \frac{8\pi}{3}. \end{aligned}$$

Therefore

$$\begin{aligned} \Gamma_0 &= \frac{\omega_0^3d^2} {8\pi^2\epsilon_0\hbar c^3} \frac{8\pi}{3} \\ &= \frac{\omega_0^3d^2} {3\pi\epsilon_0\hbar c^3}. \end{aligned}$$

Check thermal detailed balance

Starting from $\Gamma_\downarrow=\Gamma_0(\overline n+1)$ and $\Gamma_\uparrow=\Gamma_0\overline n$, show that their ratio is the Boltzmann factor.

Solution

For a Bose–Einstein occupation,

$$\overline n = \frac{1}{e^{\beta\hbar\omega_0}-1}.$$

Then

$$\overline n+1 = \frac{e^{\beta\hbar\omega_0}} {e^{\beta\hbar\omega_0}-1}.$$

Hence

$$\frac{\Gamma_\uparrow} {\Gamma_\downarrow} = \frac{\overline n}{\overline n+1} = e^{-\beta\hbar\omega_0}.$$

Compute a lifetime and branching fractions

An excited level has three independent radiative decay channels with rates

$$\Gamma_1=2.0\times10^6\,\mathrm{s^{-1}},$$ $$\Gamma_2=3.0\times10^6\,\mathrm{s^{-1}},$$

and

$$\Gamma_3=5.0\times10^6\,\mathrm{s^{-1}}.$$

Find the lifetime and branching fractions. State one reason the measured lifetime could be shorter.

Solution

The total radiative rate is

$$\Gamma_e = \Gamma_1+\Gamma_2+\Gamma_3 = 1.0\times10^7\,\mathrm{s^{-1}}.$$

Thus

$$\tau_e = \frac{1}{\Gamma_e} = 1.0\times10^{-7}\,\mathrm s = 100\,\mathrm{ns}.$$

The branching fractions are

$$\begin{gathered} b_1=0.20, \qquad b_2=0.30, \\ b_3=0.50. \end{gathered}$$

A nonradiative decay channel, collisional quenching, or an environment-enhanced radiative channel would add to the total decay rate and shorten the measured lifetime.

References

• C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Atom–Photon Interactions: Basic Processes and Applications, Wiley, 1992.
• R. D. Cowan, The Theory of Atomic Structure and Spectra, University of California Press, 1981.
• P. A. M. Dirac, “The quantum theory of the emission and absorption of radiation,” Proceedings of the Royal Society A 114, 243–265 (1927), doi:10.1098/rspa.1927.0039.
• C. J. Foot, Atomic Physics, Oxford University Press, 2005.
• National Institute of Standards and Technology, Atomic Spectra Database: Spectral Lines Help, transition probabilities, oscillator strengths, and line-strength conventions, accessed 13 October 2026.
• M. O. Scully and M. S. Zubairy, Quantum Optics, Cambridge University Press, 1997.
• V. Weisskopf and E. Wigner, “Berechnung der natürlichen Linienbreite auf Grund der Diracschen Lichttheorie,” Zeitschrift für Physik 63, 54–73 (1930), doi:10.1007/BF01336768.