The Dipole Approximation and Einstein Coefficients

The golden rule delivers a transition rate once the coupling operator $V$ between the atom and the perturbing field is known. That operator is the interaction of the atomic electron with an electromagnetic wave. This lesson derives it, reduces it to the electric-dipole form that dominates optical transitions, and extracts the two quantities every measured line reports: the oscillator strength and the Einstein coefficients. The oscillator strength packages the dipole matrix element into a dimensionless number obeying a sum rule; the Einstein coefficients package the same physics into rates and connect spontaneous emission to stimulated emission through thermodynamics.

The atom-field interaction

A charged particle in an electromagnetic field described by vector potential $A$ and scalar potential $ϕ$ has the Hamiltonian obtained by minimal coupling, $p \to p + e A$ for an electron of charge $- e$ :

H = \frac{1}{2 m} (p + e A)^{2} - e ϕ + V_{atom} (r) .

Work in the Coulomb gauge, $\nabla \cdot A = 0$ and $ϕ = 0$ for the radiation field, so $A$ commutes with $p$ . Expanding the square and dropping the term quadratic in $A$ (second order in the field, negligible for weak light and irrelevant to single-photon transitions) leaves

H = H_{0} + H^{'} (t), H^{'} (t) = \frac{e}{m} A (r, t) \cdot p,

with $H_{0}$ the atomic Hamiltonian. For a plane wave of angular frequency $ω$ , wavevector $k$ , and polarization $\overset{ε}{^}$ ,

A (r, t) = A_{0} \overset{ε}{^} cos (k \cdot r - ω t),

the perturbation oscillates harmonically, and the golden-rule matrix element is $⟨ f ∣ (e / m) A_{0} e^{i k \cdot r} \overset{ε}{^} \cdot p ∣ i ⟩$ .

The long-wavelength approximation

The matrix element carries the factor $e^{i k \cdot r}$ , the spatial phase of the wave across the atom. Its size is set by $k a$ , where $a \sim a_{0}$ is the atomic radius and $k = 2 π / λ$ . For an optical transition $λ \sim 500 nm$ and $a_{0} \sim 0.05 nm$ , so

k a_{0} = \frac{2 π a _{0}}{λ} \sim \frac{2 π ( 0.05 )}{500} \approx 6 \times 1 0^{- 4} .

The wave is essentially uniform across the atom. Expanding

e^{i k \cdot r} = 1 + i k \cdot r + \dots

and keeping the leading term is the electric-dipole (E1) approximation. The neglected term $i k \cdot r$ is smaller by $\sim k a_{0} \sim α Z$ (the next lesson shows it generates the magnetic-dipole and electric-quadrupole transitions, weaker by $\sim 1 0^{- 5}$ in rate).

An optical wavelength (about 500 nm) is roughly ten thousand times the Bohr radius, so the field is uniform across the atom and only its value at the nucleus matters. The dipole approximation replaces exp(i k . r) by 1.

In the dipole approximation the interaction can be rewritten in a more transparent form. The momentum matrix element relates to the position matrix element through the commutator $[r, H_{0}] = i ℏ p / m$ , so

⟨ f ∣ p ∣ i ⟩ = \frac{m}{i ℏ} ⟨ f ∣ [r, H_{0}] ∣ i ⟩ = im ω_{f i} ⟨ f ∣ r ∣ i ⟩ .

Substituting turns the $A \cdot p$ coupling into the equivalent length-gauge dipole interaction

H^{'} = - d \cdot E (t), d = - e r,

where $d$ is the electron's electric-dipole moment and $E$ is the electric field of the wave. The transition is governed by the dipole matrix element $d_{f i} = ⟨ f ∣ d ∣ i ⟩ = - e ⟨ f ∣ r ∣ i ⟩$ . When this vector vanishes, the E1 transition is forbidden, the subject of the selection-rule lesson.

Oscillator strength and the sum rule

The dipole matrix element carries dimensions and depends on the pair of states. It is conventional to fold it into a dimensionless oscillator strength, which compares the quantum transition with a classical charged oscillator of the same frequency. For a transition $i \to f$ ,

f_{f i} = \frac{2 m ω _{f i}}{3ℏ} ∣ ⟨ f ∣ r ∣ i ⟩ ∣^{2},

the factor $\frac{1}{3}$ averaging over the three Cartesian directions of an unpolarized or randomly oriented sample. Absorption gives $f_{f i} > 0$ ; emission ( $ω_{f i} < 0$ ) gives $f_{f i} < 0$ . The oscillator strength is what spectroscopic tables report, because a strong line has $f$ of order unity and a weak one has $f$ orders of magnitude smaller.

The oscillator strengths from any fixed level obey an exact constraint.

The sum rule is a conservation law for spectral weight: an atom has a fixed budget of oscillator strength, shared among all its transitions (including those into the continuum). A few strong resonance lines exhaust most of the budget; higher members of a series get progressively less.

The oscillator-strength budget of a level distributed over its transitions. The strong resonance line takes most of the total; higher series members and the continuum share the remainder, and the signed sum is fixed at one.

Einstein's three coefficients

Einstein derived the relations among absorption, stimulated emission, and spontaneous emission in 1917 from thermodynamics alone, before quantum mechanics could compute any of them.¹ Consider two levels, lower $∣ 1 ⟩$ and upper $∣ 2 ⟩$ with $E_{2} - E_{1} = ℏ ω_{0}$ , in equilibrium with blackbody radiation of spectral energy density $ρ (ω)$ . Three processes change the populations $N_{1}, N_{2}$ :

Absorption, rate per lower atom $B_{12} ρ (ω_{0})$ : the atom absorbs a photon and climbs.
Stimulated emission, rate per upper atom $B_{21} ρ (ω_{0})$ : the field drives the atom down, adding a photon coherent with the field.
Spontaneous emission, rate per upper atom $A_{21}$ : the atom decays with no field present, emitting into a random mode.

The absorption and stimulated rates are proportional to the radiation density and follow directly from the golden rule; spontaneous emission has no classical driving term and is the process that requires the quantized field.

The three radiative processes between two levels bathed in radiation of density rho. Absorption and stimulated emission scale with rho; spontaneous emission proceeds at the rate A even in the dark.

Detailed balance fixes the ratios

In equilibrium the upward and downward transition rates balance:

N_{1} B_{12} ρ (ω_{0}) = N_{2} (A_{21} + B_{21} ρ (ω_{0})) .

Solve for the radiation density,

ρ (ω_{0}) = \frac{A _{21} / B _{21}}{( N _{1} B _{12} ) / ( N _{2} B _{21} ) - 1} .

The populations in thermal equilibrium follow the Boltzmann distribution, $N_{2} / N_{1} = (g_{2} / g_{1}) e^{- ℏ ω_{0} / k_{B} T}$ for levels of degeneracy $g_{1}, g_{2}$ , so $N_{1} B_{12} / (N_{2} B_{21}) = (g_{1} B_{12} / g_{2} B_{21}) e^{ℏ ω_{0} / k_{B} T}$ . This must reproduce the Planck spectrum

ρ (ω) = \frac{ℏ ω ^{3}}{π ^{2} c ^{3}} \frac{1}{e ^{ℏ ω / k_{B} T} - 1}

at every temperature. Matching term by term forces two relations that hold independently of temperature.

Two consequences follow immediately. First, $B_{12} = B_{21}$ for non-degenerate levels: absorption and stimulated emission are the same process run in opposite directions, exactly as the two counter-rotating terms in the harmonic perturbation predicted. Second, the ratio $A / B$ grows as $ω^{3}$ . The mode density of the electromagnetic field rises steeply with frequency, so spontaneous emission dominates in the optical and ultraviolet while stimulated processes dominate in the microwave. This $ω^{3}$ scaling is why building a laser (which needs stimulated emission to win) grows harder toward short wavelengths, and why an X-ray laser is a far greater engineering feat than a microwave maser.

The ratio of spontaneous to stimulated emission grows as the cube of the frequency. Spontaneous decay dominates in the optical and ultraviolet; stimulated processes dominate at microwave frequencies where the mode density is low.

From matrix elements to the spontaneous rate

Combining the golden rule with the Einstein relations gives the spontaneous emission rate in terms of the dipole matrix element. The absorption coefficient computed from $H^{'} = - d \cdot E$ and averaged over polarizations is

B_{12} = \frac{π}{3 ε _{0} ℏ ^{2}} d_{21}^{2},

and multiplying by $A / B = ℏ ω_{0}^{3} / π^{2} c^{3}$ gives the spontaneous rate

A_{21} = \frac{ω _{0}^{3}}{3 π ε _{0} ℏ c ^{3}} d_{21}^{2} = \frac{ω _{0}^{3} e ^{2}}{3 π ε _{0} ℏ c ^{3}} ∣ ⟨ 2 ∣ r ∣ 1 ⟩ ∣^{2} .

This is the master formula of radiative decay. It has two ingredients: the $ω_{0}^{3}$ factor from the vacuum mode density, and the squared dipole matrix element that carries all the atomic structure. Written through the oscillator strength,

A_{21} = \frac{e ^{2} ω _{0}^{2}}{2 π ε _{0} m c ^{3}} ∣ f_{12} ∣,

so a transition with $f \sim 1$ and $ω_{0}$ in the visible has $A_{21} \sim 1 0^{8} s^{- 1}$ , a lifetime of a few nanoseconds — the order of magnitude of a strong allowed line. The reciprocal of $A_{21}$ is the natural lifetime that sets the natural line width.

Quantity	Symbol	Scaling	Typical value (strong optical line)
Dipole matrix element	$d_{21}$	$\sim e a_{0}$	$\sim 8 \times 1 0^{- 30} C m$
Oscillator strength	$f_{12}$	$≲ 1$	$\sim 0.1$ – $1$
Spontaneous rate	$A_{21}$	$\propto ω_{0}^{3} d_{21}^{2}$	$\sim 1 0^{8} s^{- 1}$
Natural lifetime	$τ = 1/ A_{21}$	$\propto ω_{0}^{- 3}$	$\sim 1$ – $100 ns$

Oscillator strength and the absorption cross section

The oscillator strength also fixes how strongly an atom absorbs. Integrating the absorption cross section $σ (ω)$ over the whole line gives a result that depends only on $f_{12}$ and universal constants,

\int σ (ω) d ω = \frac{π e ^{2}}{2 ε _{0} m c} f_{12} = 2 π^{2} c r_{e} f_{12},

with $r_{e} = e^{2} /4 π ε_{0} m c^{2}$ the classical electron radius. The integrated absorption is independent of the line shape, so it is unchanged by Doppler or pressure broadening: broadening redistributes the same total absorption over a wider frequency range, lowering the peak but conserving the area. This is why the equivalent width of an absorption line, not its peak depth, measures the column density of atoms, and it is the basis of quantitative astrophysical and laboratory absorption spectroscopy.

The three coefficients and the dipole matrix element are the working currency of atomic radiation. Absorption and stimulated emission share one coefficient $B$ and underlie spectroscopy and lasers; spontaneous emission, fixed by $A = (ℏ ω^{3} / π^{2} c^{3}) B$ , sets natural lifetimes and line widths. The next lesson asks the sharper question the matrix element $d_{21} = - e ⟨ 2 ∣ r ∣ 1 ⟩$ raises: for which pairs of states does it vanish, and what happens to a transition when it does.

Einstein, A. (1917), Zur Quantentheorie der Strahlung, Physikalische Zeitschrift 18, 121. The $A$ and $B$ coefficients and their detailed-balance relations predate the quantum-mechanical calculation of either. The derivation here follows Foot, Atomic Physics, §7.5, and the matrix-element formulae Bransden & Joachain, Physics of Atoms and Molecules, 2nd ed., §4.6. The Thomas-Reiche-Kuhn sum rule: Bransden & Joachain, §4.6; Foot, §7.4. ↩

The atom-field interaction

The long-wavelength approximation

Oscillator strength and the sum rule

Einstein's three coefficients

Detailed balance fixes the ratios

From matrix elements to the spontaneous rate

Oscillator strength and the absorption cross section

Footnotes