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Absorption, stimulated and spontaneous emission, the $\omega^3$ spontaneous rate, the dipole selection rules, lasers and forbidden lines.
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By the end of this lesson you will be able to decide which transitions are allowed, compute spontaneous emission rates and lifetimes, compare stimulated and spontaneous emission, and estimate radiation forces on atoms.
You know Fermi's golden rule, resonance, the hydrogen atom's states and the angular momentum algebra. You also know that light is an oscillating electric field. This lesson couples the two: light drives transitions in atoms, atoms emit light, and the rules of angular momentum decide which transitions can happen at all.
| Term | What it means |
|---|---|
| Electric-dipole approximation | Treating the light field as uniform across the atom, so the coupling is $-\vec{d} \cdot \vec{E}$ with $\vec{d} = -e\vec{r}$. |
| Absorption | A transition upward driven by incident light. |
| Stimulated emission | A transition downward driven by incident light, adding a photon identical to those present. |
| Spontaneous emission | A transition downward with no incident light, driven by the vacuum's field. |
| Einstein coefficients | $A$ for spontaneous emission and $B$ for absorption and stimulated emission, related by $A/B = \hbar\omega^3/\pi^2c^3$. |
| Selection rules | Conditions, such as $\Delta l = \pm 1$, under which the dipole matrix element can be nonzero. |
| Metastable state | An excited state with no allowed dipole decay, which therefore lives unusually long. |
The wavelength of visible light is thousands of times larger than an atom, so the field is nearly uniform across it and couples through the electric dipole: $\hat{H}' = -\hat{\vec{d}} \cdot \vec{E}(t)$ with $\hat{\vec{d}} = -e\hat{\vec{r}}$. Time-dependent perturbation theory then gives three processes, as Einstein reasoned in 1917. Absorption and stimulated emission occur at equal rates per atom, proportional to the light's intensity at the transition frequency. Spontaneous emission occurs even in the dark, at a rate
$$A = \frac{\omega^3|\vec{d}_{ba}|^2}{3\pi\varepsilon_0\hbar c^3},$$
which quantum electrodynamics derives by applying the golden rule to the continuum of photon modes, whose density grows as $\omega^2$. For hydrogen's $2p \to 1s$ transition this gives $A = 6.3 \times 10^8$ s⁻¹, a lifetime of $1.6$ ns.
The dipole matrix element $\langle n'l'm'|\hat{\vec{r}}|nlm\rangle$ is zero unless
$$\Delta l = \pm 1, \qquad \Delta m = 0, \pm 1.$$
These selection rules come from two facts: $\hat{\vec{r}}$ is odd under parity, so it connects states of opposite parity, $(-1)^l$; and it behaves like a vector, carrying one unit of angular momentum, so the angular momenta must combine like $l$ and $1$. Transitions that break them are forbidden — not impossible, but slowed by many orders of magnitude.
Another way: picture
Picture an atom as a tiny antenna. An electron in a superposition of $1s$ and $2p$ has a charge cloud that sloshes back and forth at the transition frequency, an oscillating dipole that radiates like a radio antenna. A superposition of $1s$ and $2s$, both spherical, only breathes in and out, with no dipole at all, so it cannot radiate this way. Selection rules are the rules for which pairs of states make a working antenna.
Another way: steps
Checks. Optical transitions in atoms have lifetimes of nanoseconds to tens of nanoseconds. Microwave and radio transitions, with frequencies a million times lower, have spontaneous rates $10^{18}$ times slower. Hydrogen-like ions decay $Z^4$ times faster. And forbidden lines, when observed, must come from low-density gas, where collisions do not quench the long-lived states first.
Spontaneous emission is the golden rule applied to emission into empty photon modes. The coupling to each mode involves the dipole times the vacuum field of that mode, whose strength grows as $\sqrt{\omega}$, giving a factor $\omega$ in $|V|^2$. The number of photon modes per unit frequency in a volume grows as $\omega^2$. Together they give $\omega^3$.
The consequences are dramatic. Hydrogen's $2p$ state lives $1.6$ ns, while the hyperfine transition of the 21 cm line, with a frequency a million times lower and a magnetic rather than electric dipole, lives about $11$ million years. X-ray transitions in heavy atoms, with frequencies a thousand times higher, decay in femtoseconds. That scaling is also why X-ray lasers are so hard to build: the upper state empties before a population inversion can be maintained.
Einstein found the relations between his coefficients without quantum mechanics, by requiring that atoms in thermal radiation reach the Boltzmann populations. In equilibrium, absorption up must balance stimulated and spontaneous emission down, and that works for every temperature only if stimulated emission exists, with $B_{\text{up}} = B_{\text{down}}$ and $A/B = \hbar\omega^3/(\pi^2c^3)$.
The ratio of stimulated to spontaneous emission in thermal light is the mean photon number per mode, $\bar{n} = 1/(e^{\hbar\omega/k_BT} - 1)$. For visible light at ordinary temperatures, $\bar{n} \ll 1$, and spontaneous emission dominates: flames and bulbs emit incoherently. For microwaves at room temperature, $\bar{n}$ is in the hundreds, and stimulated emission dominates — which is why the first maser, working at $24$ GHz, came before the first laser.
In a laser, stimulated emission is made to beat absorption. That requires a population inversion, more atoms in the upper state than the lower, which thermal equilibrium never provides. A three- or four-level scheme does it: atoms are pumped to a high level, fall quickly to a long-lived upper laser level, and lase to a lower level that empties quickly. Each stimulated photon is a copy of the one that triggered it — same frequency, direction, phase and polarization — because photons are bosons and prefer to enter occupied modes.
Mirrors at each end of the gain medium send the light back and forth, and the photon number in one mode grows until losses balance gain. The selection rules matter: the upper laser level should be long-lived, often because its fastest decays are forbidden, as in the ruby and neodymium lasers. Theodore Maiman built the first laser, a ruby laser, at Hughes Research Laboratories in Malibu in 1960.
Forbidden transitions still happen, through the magnetic-dipole or electric-quadrupole couplings neglected in the dipole approximation, but millions of times more slowly. In a laboratory gas, collisions knock atoms out of such long-lived states before they can radiate. In the near-vacuum of nebulae, with only a few thousand atoms per cubic centimeter, there are no collisions to interfere, and the forbidden lines shine.
The green glow of planetary nebulae, once attributed to an unknown element called "nebulium," was identified by Ira Bowen in 1927 as forbidden lines of doubly ionized oxygen. The green and red aurora come from forbidden transitions of atomic oxygen, whose upper states live $0.7$ s and $110$ s, possible only in the thin upper atmosphere. Astronomers use the ratios of forbidden lines to measure the temperatures and densities of gas throughout the universe.
The dipole operator's components can be written as $z$ and $x \pm iy$, which are proportional to $rY_1^0$ and $rY_1^{\pm 1}$. The angular part of the matrix element is then an integral of three spherical harmonics, $\int Y_{l'}^{m'*}Y_1^{q}Y_l^m\,d\Omega$, which vanishes unless $m' = m + q$, giving $\Delta m = 0, \pm 1$, and unless $l'$ is one of the values produced by adding angular momenta $l$ and $1$, giving $l' = l, l \pm 1$. Parity removes $l' = l$.
The polarization of the light follows too. $\Delta m = 0$ transitions are driven by light polarized along $z$; $\Delta m = \pm 1$ by circularly polarized light, which carries angular momentum $\pm\hbar$ along its direction. That link lets experimenters pump atoms into chosen $m$ states with polarized light, the optical pumping invented by Alfred Kastler, and read out the Zeeman components of spectral lines from their polarization.
Every photon an atom absorbs from a laser beam pushes it along the beam by $h/\lambda$; the photons it re-emits spontaneously go in random directions and average to zero. A saturated atom absorbs and emits at half its spontaneous rate, about $3 \times 10^7$ times a second for sodium, so the force is enormous for such a small particle: a deceleration of about $9 \times 10^5$ m/s², nearly a hundred thousand times Earth's gravity.
In a Zeeman slower, a magnetic field that changes along the beam keeps the Doppler-shifted atoms in resonance as they slow, bringing an oven's $800$ m/s atoms almost to rest in a meter. William Phillips developed the technique at the National Institute of Standards and Technology in the 1980s, sharing the 1997 Nobel Prize. Laser-slowed atoms are the starting point of atomic clocks, Bose–Einstein condensates and neutral-atom quantum computers — all resting on the spontaneous emission rate of a single allowed transition.
The green and red light of the aurora comes from oxygen atoms high in the atmosphere, excited by electrons streaming in from the solar wind. Both lines are forbidden. The green line at $557.7$ nm comes from a state that lives about $0.7$ s; the red line at $630.0$ nm from one that lives about $110$ s. At ground level an excited oxygen atom collides with a neighbor billions of times a second and loses its energy without radiating.
That is why the colors layer by height. Above about $200$ km, collisions are rare enough for even the $110$-second state to radiate, and the aurora glows red. Between $100$ and $200$ km the green line dominates. Lower still, where collisions quench both, allowed transitions of molecular nitrogen produce the purple and blue fringe at the bottom of intense displays. Photographers in Alaska and the northern states record the altitude structure of the atmosphere every time they capture the aurora's colors.
The Schrödinger equation for an isolated atom has stationary excited states, which suggests they should last forever. They do not, because the atom is never truly isolated: it is coupled to the electromagnetic field, whose vacuum state still fluctuates. That coupling makes every excited state with an allowed downward transition decay at the rate $A$. Spontaneous emission is not caused by the light around the atom; it happens in perfect darkness.
A second misconception is that forbidden transitions never occur. The selection rules apply to the electric-dipole coupling only. Weaker couplings — magnetic dipole, electric quadrupole, two-photon processes — let forbidden transitions proceed, just millions of times more slowly. In the right conditions, such as a nebula or the upper atmosphere, they produce some of the most striking light in the sky.
Find the angular frequency of Lyman-alpha.
$\omega = \dfrac{10.2\ \text{eV}}{6.582 \times 10^{-16}\ \text{eV s}} = 1.550 \times 10^{16}\ \text{rad/s}$
$E_2 - E_1 = \tfrac{3}{4} \times 13.6$ eV.
Write the squared dipole matrix element.
$|\langle 1s|\vec{r}|2p\rangle|^2 = \dfrac{2^{15}}{3^{10}}a^2 = 0.5549 \times (5.29 \times 10^{-11})^2 = 1.554 \times 10^{-21}\ \text{m}^2$
From the hydrogen wave functions.
Assemble the rate.
$A = \dfrac{\omega^3e^2|\vec{r}|^2}{3\pi\varepsilon_0\hbar c^3}$
The spontaneous emission formula.
Evaluate the expression.
$A = \dfrac{(1.550 \times 10^{16})^3(1.602 \times 10^{-19})^2(1.554 \times 10^{-21})}{3\pi(8.854 \times 10^{-12})(1.0546 \times 10^{-34})(3.00 \times 10^{8})^3} = 6.26 \times 10^{8}\ \text{s}^{-1}$
SI units throughout.
Invert for the lifetime.
$\tau = \dfrac{1}{A} = 1.60\ \text{ns}$
Matching the measured value.
List the states of hydrogen's $n = 3$ level.
$3s, \ 3p, \ 3d$
All at the same energy in the simple model.
Find where $3s$ can decay.
$3s \to 2p$
Only $p$ states are reachable from $s$.
Find where $3p$ can decay.
$3p \to 2s, \quad 3p \to 1s$
Both change $l$ by one.
Find where $3d$ can decay.
$3d \to 2p$
$3d \to 1s$ would need $\Delta l = 2$.
Find the Balmer-alpha contributors.
$3s \to 2p, \quad 3p \to 2s, \quad 3d \to 2p$
All produce the $656$ nm line.
Note the stuck state.
$2s: \text{ no allowed dipole decay}$
It decays by two-photon emission, living $0.12$ s.
Write the ratio in thermal light.
$\dfrac{R_{\text{stim}}}{A} = \bar{n} = \dfrac{1}{e^{\hbar\omega/k_BT} - 1}$
The mean photon number per mode.
Evaluate for red light at room temperature.
$\dfrac{\hbar\omega}{k_BT} = \dfrac{1.96}{0.0259} = 75.7, \qquad \bar{n} = e^{-75.7} \approx 10^{-33}$
Stimulated emission is utterly negligible.
Evaluate for red light in the Sun's photosphere.
$\dfrac{\hbar\omega}{k_BT} = \dfrac{1.96}{0.500} = 3.9, \qquad \bar{n} = 0.020$
Still small, at $5800$ K.
Evaluate for a $10$ GHz microwave at room temperature.
$\dfrac{\hbar\omega}{k_BT} = \dfrac{4.14 \times 10^{-5}}{0.0259} = 0.0016, \qquad \bar{n} = 625$
Stimulated emission dominates.
Find the crossover.
$\bar{n} = 1 \quad\Rightarrow\quad \hbar\omega = k_BT\ln 2$
Below this frequency, thermal light stimulates more than atoms emit on their own.
Evaluate the crossover at room temperature.
$f = \dfrac{0.0259 \times 0.693}{4.136 \times 10^{-15}} = 4.3\ \text{THz}$
In the far infrared.
Relate this to the laser's difficulty.
$\text{visible: } \bar{n} \ll 1 \text{ unless the mode is filled artificially}$
A laser cavity builds up enormous $\bar{n}$ in a single mode.
State the condition for net gain.
$N_{\text{upper}} > N_{\text{lower}}$
Population inversion: stimulated emission then outpaces absorption.
Recall the scaling.
$\tau \propto \dfrac{1}{\omega^3} \propto \lambda^3$
For a fixed dipole matrix element.
Multiply by the cube of the ratio.
$\tau' = 16 \times 2^3$
Twice the wavelength.
Evaluate the lifetime.
Which of these hydrogen transitions is allowed for electric-dipole radiation?
Complete the worked solution: an allowed transition has a lifetime of $17$ ns. Another transition with the same dipole matrix element emits at twice the wavelength, and a third at three times. Find their lifetimes and the difference between them, in ns.
Scale the lifetime for twice the wavelength.
$\tau_2 = 2^3 \times 17 =$ a
Half the frequency, an eighth of the rate.
Scale the lifetime for three times the wavelength.
$\tau_3 = 3^3 \times 17 =$ b
A third of the frequency.
Subtract the two lifetimes.
$\tau_3 - \tau_2 =$ c
Long-wavelength transitions are slow to decay spontaneously.
Match each rule of radiative transitions to its content.
| $\Delta l = \pm 1$ | $\Delta m = 0, \pm 1$ | $\propto \omega^3|d|^2$ | $1/(e^{\hbar\omega/k_BT} - 1)$ | |
|---|---|---|---|---|
| the selection rule on $l$ | ||||
| the selection rule on $m$ | ||||
| the spontaneous rate | ||||
| stimulated per spontaneous in thermal light |
For each hydrogen transition, enter $1$ if it is electric-dipole allowed and $0$ if it is forbidden.
| allowed (1) or not (0) | |
|---|---|
| $2\text{p} \to 1\text{s}$ | |
| $2\text{s} \to 1\text{s}$ | |
| $4\text{f} \to 3\text{d}$ | |
| $3\text{d} \to 1\text{s}$ |
For the sodium yellow line in the Sun's photosphere, with photon energy $2.104$ eV and radiation at $5800$ K, how many stimulated emissions occur for each spontaneous one? Use $k_B = 8.617 \times 10^{-5}$ eV/K.
Answer:
Estimate the lifetime of the $2\text{p}$ state of the beryllium ion Be³⁺ ($Z = 4$) against spontaneous decay to $1\text{s}$, in picoseconds. The photon energy is $10.2Z^2$ eV and the dipole matrix element obeys $|\langle 1\text{s}|\vec{r}|2\text{p}\rangle|^2 = \tfrac{2^{15}}{3^{10}}(a/Z)^2$ with $a = 0.0529$ nm.
Answer: ps
A strong laser beam tuned to the $670.8$ nm transition of lithium-7 (upper-state lifetime $27.1$ ns, atomic mass $7.016$ u) slows a beam of atoms. What is the maximum deceleration, in km/s²? Use $h = 6.626 \times 10^{-34}$ J s and $1$ u $= 1.6605 \times 10^{-27}$ kg.
Answer: km/s²
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
Complete the worked solution: an allowed transition has a lifetime of $24$ ns. Another transition with the same dipole matrix element emits at twice the wavelength, and a third at three times. Find their lifetimes and the difference between them, in ns.
Scale the lifetime for twice the wavelength.
$\tau_2 = 2^3 \times 24 =$ a
Half the frequency, an eighth of the rate.
Scale the lifetime for three times the wavelength.
$\tau_3 = 3^3 \times 24 =$ b
A third of the frequency.
Subtract the two lifetimes.
$\tau_3 - \tau_2 =$ c
Long-wavelength transitions are slow to decay spontaneously.
You can predict how atoms emit and absorb light. Explain to someone why hydrogen's $2s$ state lives a hundred million times longer than its $2p$ state.
17. Your turn: a transition has a lifetime of $16$ ns. What would the lifetime be for a transition with the same dipole at twice the wavelength?, step 3
$\tau' = 128\ \text{ns}$
Eight times longer.