Selection Rules and Forbidden Transitions

The dipole matrix element $d_{f i} = - e ⟨ f ∣ r ∣ i ⟩$ controls whether a transition radiates. For most pairs of atomic states it is exactly zero, and the transition is forbidden in the electric-dipole approximation. The pattern of nonzero matrix elements is the set of selection rules, and each rule traces to a symmetry: parity, rotational invariance, and the independence of the dipole operator from spin. This lesson derives each rule from the integral it enforces, then examines what drives a transition when the dipole element vanishes: the higher multipoles, weaker by powers of $α$ , and the long-lived metastable states they leave behind.

Parity and the change in orbital angular momentum

The position operator $r$ is odd under the parity transformation $r \to - r$ . A hydrogenic state $∣ n ℓ m ⟩$ has definite parity $(- 1)^{ℓ}$ , because the spherical harmonic $Y_{ℓ}^{m}$ satisfies $Y_{ℓ}^{m} (- \overset{r}{^}) = (- 1)^{ℓ} Y_{ℓ}^{m} (\overset{r}{^})$ . The matrix element $⟨ f ∣ r ∣ i ⟩$ integrates the product of three factors: the parity of $∣ i ⟩$ , the odd operator $r$ , and the parity of $∣ f ⟩$ . Under $r \to - r$ the integrand picks up $(- 1)^{ℓ_{f}} \cdot (- 1) \cdot (- 1)^{ℓ_{i}}$ . If this sign is $- 1$ the integral equals its own negative and must vanish. A nonzero dipole element therefore requires

(- 1)^{ℓ_{f} + ℓ_{i} + 1} = + 1 ⟺ ℓ_{f} + ℓ_{i} odd .

The initial and final orbital angular momenta must have opposite parity. This is the parity selection rule, and it holds for any system with a definite-parity Hamiltonian, not just hydrogen.

Parity gates the dipole integral. The operator r is odd; if the two states have the same parity the integrand is odd overall and integrates to zero, so only states of opposite parity radiate by the dipole.

Parity alone allows $ℓ_{f} - ℓ_{i} = \pm 1, \pm 3, \dots$ . The angular integral sharpens this to $\pm 1$ .

The angular integral: the change in the magnetic quantum number

The Cartesian components of $r$ are proportional to the $q = 0, \pm 1$ spherical harmonics of rank one:

z = r \frac{4 π}{3} Y_{1}^{0}, x \pm i y = \mp r \frac{8 π}{3} Y_{1}^{\pm 1} .

The dipole matrix element then contains an integral of three spherical harmonics, the Gaunt integral,

\int Y_{ℓ_{f}}^{m_{f} *} Y_{1}^{q} Y_{ℓ_{i}}^{m_{i}} d Ω,

which is nonzero only when two conditions hold. The azimuthal integral over the $e^{i (m_{i} + q - m_{f}) ϕ}$ phase forces $m_{f} = m_{i} + q$ , and the triangle condition on the three ranks forces $∣ ℓ_{f} - ℓ_{i} ∣ \leq 1 \leq ℓ_{f} + ℓ_{i}$ . Combined with parity, which already excluded $ℓ_{f} = ℓ_{i}$ , the surviving cases are

Δ ℓ = ℓ_{f} - ℓ_{i} = \pm 1, Δ m = m_{f} - m_{i} = 0, \pm 1.

The three values of $Δ m$ correspond to the three polarizations of the emitted or absorbed photon. The $q = 0$ component ( $z$ , hence $Δ m = 0$ ) radiates linearly polarized light along the quantization axis, the $π$ component. The $q = \pm 1$ components ( $x \pm i y$ , hence $Δ m = \pm 1$ ) radiate circularly polarized light, the $σ^{\pm}$ components. This polarization structure is exactly what the Zeeman effect resolves when a magnetic field splits the $m$ sublevels.

Total angular momentum and the photon's spin

An emitted photon carries one unit of angular momentum. Conservation of total angular momentum between the atom-plus-photon initial and final states restricts how the atom's total angular momentum $j$ can change. Adding the photon's spin-1 to the initial $j_{i}$ by the triangle rule gives

Δ j = 0, \pm 1, j_{i} = 0 \neq \to j_{f} = 0,

the $0 \to 0$ case excluded because a single spin-1 photon cannot connect two scalar states (there is no way to carry off one unit of angular momentum). The same argument applied to the projection gives $Δ m_{j} = 0, \pm 1$ . In LS-coupled many-electron atoms the rules extend to the term symbols: because the dipole operator is a sum of one-electron position operators, a single electron changes its orbital by one and the spin is a spectator, giving

Δ L = 0, \pm 1 (L = 0 \neq \to 0), Δ S = 0, Δ J = 0, \pm 1 (J = 0 \neq \to 0),

with a parity change and $Δ ℓ = \pm 1$ for the jumping electron. The $Δ S = 0$ rule forbids intercombination lines between different spin multiplicities, which is why singlet-triplet transitions in helium are so weak, and it fails only when spin-orbit coupling mixes the multiplicities (progressively for heavier atoms).

Rule	Quantity	Allowed change	Origin
Parity	$π = (- 1)^{\sum ℓ}$	must change	$r$ is odd
Orbital	$Δ ℓ$	$\pm 1$	Gaunt integral, parity
Magnetic	$Δ m, Δ m_{j}$	$0, \pm 1$	rank-1 azimuthal integral
Total $J$	$Δ J$	$0, \pm 1$ ( $0 \neq \to 0$ )	photon spin 1
Total $L$	$Δ L$	$0, \pm 1$ ( $0 \neq \to 0$ )	one electron jumps
Spin	$Δ S$	$0$	$r$ spin-independent

A Grotrian diagram makes the rules visible: allowed lines connect adjacent orbital columns, and no allowed line runs vertically within a single column.

A Grotrian diagram. Solid arrows are allowed E1 transitions between adjacent orbital columns (Delta-ell = one); the dashed vertical transition within the s column violates the parity and orbital rules and is dipole-forbidden.

The Wigner-Eckart theorem and hyperfine rules

The angular selection rules are one instance of the Wigner-Eckart theorem: a rank- $k$ spherical tensor operator has matrix elements between angular-momentum eigenstates that factor into a geometric Clebsch-Gordan coefficient and a state-independent reduced matrix element. The dipole operator is a rank-1 tensor, so its Clebsch-Gordan factor vanishes unless the triangle rule $∣ j_{f} - j_{i} ∣ \leq 1 \leq j_{f} + j_{i}$ and $Δ m_{j} = 0, \pm 1$ hold — exactly the $Δ j$ and $Δ m_{j}$ rules. The same theorem applied to the total angular momentum $F = I + J$ of a hyperfine level gives the further rule

Δ F = 0, \pm 1, F = 0 \neq \to F = 0,

for an E1 transition, because the photon still carries one unit of angular momentum regardless of how the nuclear spin is coupled in. The reduced matrix element carries all the radial physics; the geometry is universal.

Higher multipoles

The selection rules are rules for the leading E1 term. When the dipole element vanishes, the neglected term $i k \cdot r$ in the expansion of $e^{i k \cdot r}$ can still drive the transition. That term splits into two pieces with distinct symmetry:

Magnetic dipole (M1). The antisymmetric part couples to the magnetic moment operator $- μ_{B} (L + 2 S) /ℏ$ . It has even parity, so it connects states of the same parity ( $Δ ℓ = 0$ ), with $Δ j = 0, \pm 1$ and $Δ m = 0, \pm 1$ . It drives transitions within a fine-structure or hyperfine multiplet, including the 21 cm line.
Electric quadrupole (E2). The symmetric part couples to the quadrupole moment $\sim r^{2} Y_{2}^{q}$ . It also has even parity ( $Δ ℓ = 0, \pm 2$ , but $0 \neq \to 0$ and $ℓ = 0 \neq \to ℓ = 0$ ), with $Δ j = 0, \pm 1, \pm 2$ (and $0 \neq \to 0, 1$ ).

Both are suppressed relative to E1 by the small factor that measures the field's variation across the atom. In rate the suppression is of order

\frac{A _{M1}}{A _{E1}} \sim α^{2} \sim 5 \times 1 0^{- 5}, \frac{A _{E2}}{A _{E1}} \sim (k a_{0})^{2} \sim (α Z)^{2},

because each higher multipole carries an extra factor of $k a_{0} \sim α Z$ in amplitude, hence its square in rate. A forbidden line is not strictly forbidden; it is weaker by four or five orders of magnitude, and it appears when the geometry suppresses competing allowed decays (in dilute astrophysical plasmas, or in trapped single ions where an E2 or M1 clock transition is a feature rather than a defect).

Radiative-rate hierarchy. Each successive multipole (E1, M1, E2) is suppressed by roughly the square of the fine-structure constant, spanning about ten orders of magnitude in transition rate.

Metastable states

A state with no allowed E1 decay to any lower level cannot radiate at the strong $1 0^{8} s^{- 1}$ rate. Its only decay channels are the suppressed multipoles or multi-photon processes, so it lives orders of magnitude longer than an ordinary excited state. Such a state is metastable.

The canonical example is the hydrogen $2 s_{1/2}$ level. The only lower state is $1 s_{1/2}$ , and the $2 s \to 1 s$ transition has $Δ ℓ = 0$ : dipole-forbidden by parity. Nor can it proceed by M1 or E2 to any useful rate (the M1 element between $2 s$ and $1 s$ nearly vanishes because the radial wavefunctions are nearly orthogonal to the operator). The dominant decay is two-photon emission, $2 s \to 1 s + 2 γ$ , with a rate of about $8 s^{- 1}$ , giving a lifetime of $0.12 s$ — roughly a hundred million times the few-nanosecond lifetime of the neighboring $2 p$ state.

A metastable level. The 2s state cannot decay to 1s by an electric dipole (Delta-ell = 0, parity forbidden); the 2p state decays promptly in nanoseconds, while 2s survives about 0.1 s via two-photon emission.

Metastable states are the raw material of much of atomic physics. Population accumulates in them because it cannot leak away quickly, and that is the condition for a population inversion in a laser. The metastable $2^{3} S$ level of helium stores energy in discharge lamps and is the reservoir for the helium-neon laser; metastable levels of trapped ions, connected to the ground state by an ultra-narrow E2 transition, are the reference oscillators of optical clocks. The long lifetime that makes a transition spectroscopically faint also makes its natural line width extraordinarily narrow, the theme of the final lesson.

The selection rules are the grammar of atomic spectra. A term diagram with its levels drawn is not yet a spectrum; the selection rules decide which of the many conceivable transitions actually appear, how strong each is, and how it is polarized. What they leave over — the forbidden lines, the metastable reservoirs — is not an afterthought but the working substance of lasers, of the interstellar medium's emission, and of the most precise clocks yet built. The last lesson turns from which lines appear to the shape each line has once it does.