Selection Rules for Atomic Transitions

An excited atom is like a ball perched on a ledge: it will roll down, but only along paths the physics allows. The overwhelmingly fastest path is the electric-dipole transition, in which the atom's oscillating charge cloud radiates like a tiny antenna. That antenna only turns on for certain level pairs. The controlling quantity is the transition dipole matrix element $\langle f|\mathbf{r}|i\rangle$: it is a specific integral over the initial and final wavefunctions, and it is exactly zero unless the quantum numbers change in the permitted ways. When it vanishes, the antenna is dark, and the atom must find a slower path down. The most robust of the rules is about parity. The dipole operator $\mathbf{r}$ is odd — it changes sign under inversion $\mathbf{r}\to-\mathbf{r}$ — so the integrand $\psi_f^*\,\mathbf{r}\,\psi_i$ is odd overall (and integrates to zero) unless $\psi_f$ and $\psi_i$ have opposite parity. Since an orbital's parity is $(-1)^\ell$, opposite parity means $\Delta\ell$ must be odd, and the angular integral then narrows this to exactly $\Delta\ell=\pm1$. The remaining rules ($\Delta m$, $\Delta S$, $\Delta L$, $\Delta J$) follow from $\mathbf{r}$ being a spin-independent rank-one (vector) operator — the domain of the Wigner–Eckart theorem, which aq-qm-electric-dipole-selection-rules develops in full.

When a level cannot decay by E1 to anything below it, it does not simply stay excited forever. Weaker interactions — the magnetic-dipole (M1) and electric-quadrupole (E2) terms that E1 outshines by a factor of roughly a million — eventually carry it down. The result is a metastable state with an anomalously long lifetime. The punchline figure lays these lifetimes on a single logarithmic axis, spanning twelve orders of magnitude from a normal allowed line to the second-scale forbidden lines that only appear when nothing collides with the atom first.

Whether a transition radiates by the electric-dipole mechanism is set by a single matrix element, and every selection rule is a statement about when that element is nonzero. The rate itself is proportional to its square.

where: $\mathbf{d}$ is the electric-dipole operator and $\langle f|\mathbf{d}|i\rangle$ the transition dipole moment. If this vanishes the E1 channel is closed and the transition is "forbidden." The full rate (the Einstein $A$ coefficient) carries a factor $\omega^3|\langle f|\mathbf{d}|i\rangle|^2$; the selection rules are precisely the conditions under which the matrix element survives. Working out when $\langle f|\mathbf{r}|i\rangle\ne0$ gives the rule set, stated here for a one-electron transition and, for total angular momenta, for the atom as a whole.

where: $\Delta\ell=\pm1$ enforces the parity flip on the active electron; the three values of $\Delta m$ correspond to the three polarizations ($\pi$ for $\Delta m=0$, $\sigma^\pm$ for $\Delta m=\pm1$); $\Delta S=0$ holds because the dipole operator does not act on spin; $\Delta L=0,\pm1$ and $\Delta J=0,\pm1$ come from $\mathbf{r}$ being a vector (rank-1) operator, with the strict exclusion of $J=0\to J=0$ because a rank-1 operator cannot connect two spinless-angular-momentum states. When E1 is forbidden, the next terms take over with their own weaker rules.

where: M1 (magnetic dipole) and E2 (electric quadrupole) are the leading corrections beyond E1. They connect same-parity states — exactly the pairs E1 cannot — which is why a level with no opposite-parity neighbour below it decays only through these slow channels and becomes metastable. Their rates are suppressed relative to E1 by roughly $(\alpha)^2$ or $(a_0/\lambda)^2\sim10^{-6}$–$10^{-8}$ (see the electric-dipole approximation).

Example 1 — Is the sodium D line allowed? The bright yellow doublet of a sodium lamp comes from $3p\to3s$. Check it against the selection rules and note its lifetime.

Step 1. Orbital: $3p\ (\ell=1)\to3s\ (\ell=0)$, so $\Delta\ell=-1$ — parity flips, rule satisfied.

Step 2. Angular momenta: the term is $^2P\to{}^2S$, so $\Delta S=0$ (both doublets), $\Delta L=-1$, and $\Delta J=0,\pm1$ ($3p_{3/2}\to3s_{1/2}$ has $\Delta J=-1$; $3p_{1/2}\to3s_{1/2}$ has $\Delta J=0$, neither being $0\to0$).

Step 3. All rules satisfied ⇒ the transition is E1-allowed. Consistent with a fast, bright line, the measured lifetime of the $3p$ state is about 16 ns ($A\approx6\times10^{7}\,\text{s}^{-1}$).

Result: the Na D doublet is a fully allowed electric-dipole transition — which is why sodium lamps are so efficient and bright.

Example 2 — Why hydrogen's 2s state is a trap. Hydrogen in the $2s$ state wants to fall to $1s$. Explain why it cannot do so by E1 and estimate how much longer it lives than the $2p$ state.

Step 1. Orbital: $2s\ (\ell=0)\to1s\ (\ell=0)$, so $\Delta\ell=0$ — same parity, the E1 matrix element vanishes. The transition is forbidden to electric dipole.

Step 2. No lower level of opposite parity exists (the only level below is $1s$, also even parity), so the atom cannot reach the ground state by any single-photon E1 step. It decays instead by simultaneous two-photon emission ($2s\to1s+2\gamma$).

Step 3. Compare lifetimes. The $2p$ state decays by allowed E1 in $\tau_{2p}=1/A_{2p}=1/(6.265\times10^{8}\,\text{s}^{-1})=\textbf{1.60 ns}$. The $2s$ state's two-photon lifetime is $\tau_{2s}=\textbf{0.12 s}$.

Result: the ratio is $\tau_{2s}/\tau_{2p}=0.12/(1.60\times10^{-9})\approx\textbf{7.5}\times\textbf{10}^{7}$ — the $2s$ state lives nearly $10^{8}$ times longer, a spectacular illustration of a selection rule turning a level into a trap.

A metastable state is the direct fingerprint of a selection rule at work: a level that has no allowed electric-dipole path down, so it must wait — sometimes for seconds — for a weak forbidden channel. Metastable states are not laboratory curiosities. They are the upper levels of the forbidden lines that let astronomers read the composition of nebulae, the reservoirs that make helium–neon and CO₂ lasers possible, and the clock levels of the world's most accurate optical atomic clocks, whose entire precision rests on a transition being just barely allowed.

Forbidden transitions are the beating heart of the most precise measurements ever made. Optical atomic clocks based on trapped Sr, Yb⁺, and Al⁺ ions run on ultra-narrow clock transitions that are E1-forbidden — their whole stability comes from a metastable upper level with a lifetime of seconds, giving a natural linewidth below a hertz. The same physics reads the cosmos: the forbidden [OIII] lines at 495.9 and 500.7 nm and the [NII] and [SII] lines are how astronomers measure the temperature, density, and composition of planetary nebulae and star-forming regions, because those dilute plasmas are the one place metastable levels survive long enough to radiate. Closer to home, the green (557.7 nm) and red (630.0 nm) auroral lines are forbidden transitions of atomic oxygen high in the thermosphere, where collisions are rare enough to let a metastable atom wait out its long radiative lifetime.

References. Foot, Atomic Physics (2005), §2.2 and §7.5 (selection rules, forbidden transitions) · Demtröder, Atoms, Molecules and Photons, Ch. 7 · Bransden & Joachain, Physics of Atoms and Molecules, Ch. 4 · Loudon, The Quantum Theory of Light, Ch. 2 · Corney, Atomic and Laser Spectroscopy.

Provenance. The E1 rules, parity argument, and Wigner–Eckart $\Delta J$ rule are standard (Foot §2.2; Bransden & Joachain Ch. 4). This upgrade recomputes: the parity factor $(-1)^{\ell_f+\ell_i+1}$ and the Gaunt triangle rule giving $\Delta\ell=\pm1$; the $J=0\to0$ prohibition from the Clebsch–Gordan triangle; the hydrogen $2p$ lifetime $\tau=1/A_{2p}=1.60$ ns from $A_{2p}=6.265\times10^{8}\,\text{s}^{-1}$, Lyman-$\alpha$ at 121.57 nm; and the $2s$/$2p$ lifetime ratio $0.12\,\text{s}/1.60\,\text{ns}\approx7.5\times10^{7}$. The matrix-element derivation and the M1/E2 machinery are recalled by name from aq-qm-electric-dipole-selection-rules and aq-qm-forbidden-transitions-multipole. The auroral (557.7/630.0 nm) and He $2^3S$ (~7900 s) values are standard (Corney; Foot).

Builds on: LS Coupling and Term Symbols · Fine Structure and Spin–Orbit Coupling · Hydrogenic Wavefunctions. Leads to: The Electric Dipole Approximation · Spontaneous Emission · Einstein A and B Coefficients.