Semiclassical Radiation and Dipole Selection Rules

Derivation

Semiclassical Radiation and Dipole Selection Rules

D-351 Home PU-401 Threads light · matter · symmetry Depends on Fermi's Golden Rule, hydrogen-atom-spectrum

Statement

Treating a hydrogenic atom coupled to a classical monochromatic radiation field in the electric-dipole approximation, the transition rate between bound states \(|n\ell m\rangle\) and \(|n'\ell'm'\rangle\) is governed by the matrix element of the position operator \(\langle n'\ell'm'|\,\vec{r}\,|n\ell m\rangle\). This matrix element vanishes unless the parity of the two states differs and the angular-momentum quantum numbers satisfy \(\Delta\ell=\pm 1\) and \(\Delta m=0,\pm 1\); these are the electric-dipole (E1) selection rules.

Why it matters

Selection rules are the bridge between the abstract spectrum of the hydrogen atom and what a spectrometer actually records. Of the enormous number of energetically allowed pairs of levels, only a small subset produce bright spectral lines; the rest are "forbidden" and appear only weakly through higher multipoles or not at all. Knowing which transitions are E1-allowed lets us assign every line in the Lyman, Balmer and Paschen series and read off angular-momentum labels directly from a spectrum.

The same algebra of parity and \(\Delta\ell,\Delta m\) recurs throughout physics: laser design, astrophysical line diagnostics, the polarization of emitted light in a magnetic field (the Zeeman pattern), and the metastability of states such as the \(2s\) level of hydrogen all follow from whether the dipole matrix element is forced to zero by symmetry.

Assumptions

The radiation wavelength greatly exceeds the atomic size, \(\lambda \gg a_0\).The dipole approximation \(e^{i\vec{k}\cdot\vec{r}}\approx 1\) fails; one must keep higher terms in \(\vec{k}\cdot\vec{r}\), producing magnetic-dipole (M1) and electric-quadrupole (E2) couplings with entirely different selection rules.

The perturbation is weak enough that first-order time-dependent perturbation theory (Fermi's golden rule) applies.At high intensity the linear rate is replaced by Rabi oscillations and multiphoton processes, and the single-photon selection rules no longer cap the allowed transitions.

The atomic states are exact parity eigenstates, i.e. the Hamiltonian is invariant under \(\vec{r}\to-\vec{r}\).Any static field or admixture that breaks parity (e.g. a strong external electric field, or the tiny parity-violating weak interaction) mixes opposite-parity states and opens nominally forbidden E1 lines.

Spin and the electron's magnetic coupling to the field are neglected, so the dipole operator acts only on the spatial coordinate.Including spin means \(\vec{r}\) cannot change the spin state, giving the auxiliary rule \(\Delta s=0\); when spin-orbit coupling is strong one must instead phrase the rules in \(j,m_j\), and the naive \(\Delta\ell\) statement is only approximate.

The field is treated classically (semiclassical approximation); the atom is quantized but the light is a c-number field.Spontaneous emission is not captured by the classical field and must be inserted by hand (or via the quantized-field mode density); the selection rules themselves are unchanged, but their microscopic origin in vacuum fluctuations is hidden.

Derivation

\[ \hat{H}(t)=\frac{1}{2m}\left(\hat{\vec{p}}-\frac{q}{c}\vec{A}(\vec{r},t)\right)^2+V(r) \]

Minimal coupling of a charge \(q=-e\) to the classical vector potential \(\vec{A}\); \(V(r)\) is the central Coulomb potential. A

\[ \hat{H}=\hat{H}_0-\frac{q}{mc}\,\vec{A}\cdot\hat{\vec{p}}+\frac{q^2}{2mc^2}\vec{A}^2,\qquad \hat{H}'(t)\approx-\frac{q}{mc}\,\vec{A}\cdot\hat{\vec{p}} \]

Working in Coulomb gauge \(\nabla\cdot\vec{A}=0\) so \([\vec{A},\hat{\vec{p}}]=0\); the diamagnetic \(\vec{A}^2\) term is second order in the field and dropped for weak fields. B

\[ \vec{A}(\vec{r},t)=A_0\,\hat{\boldsymbol\epsilon}\,\cos(\vec{k}\cdot\vec{r}-\omega t) \]

A single plane-wave mode with polarization \(\hat{\boldsymbol\epsilon}\perp\vec{k}\) and frequency \(\omega\); the golden rule will pick out the resonant Fourier component. A

\[ e^{i\vec{k}\cdot\vec{r}}=1+i\,\vec{k}\cdot\vec{r}+\cdots\;\approx\;1 \]

Dipole approximation: over the atomic volume \(|\vec{k}\cdot\vec{r}|\sim 2\pi a_0/\lambda\ll 1\), so the field is spatially uniform across the atom. Keeping only the leading term defines the E1 coupling. C

\[ W_{fi}=\frac{2\pi}{\hbar}\,\big|\langle f|\hat{H}'|i\rangle\big|^2\,\rho(E_f)\;\propto\;\big|\hat{\boldsymbol\epsilon}\cdot\langle f|\hat{\vec{p}}|i\rangle\big|^2 \]

Fermi's golden rule for the first-order rate; the field amplitude and density of states are common factors, so the state dependence lives entirely in the momentum matrix element. B

\[ \hat{\vec{p}}=\frac{m}{i\hbar}\,[\hat{\vec{r}},\hat{H}_0]\;\Longrightarrow\;\langle f|\hat{\vec{p}}|i\rangle=\frac{m}{i\hbar}(E_i-E_f)\,\langle f|\hat{\vec{r}}|i\rangle=i m\,\omega_{fi}\,\langle f|\hat{\vec{r}}|i\rangle \]

Heisenberg relation \(\hat{\vec{p}}=(m/i\hbar)[\hat{\vec{r}},\hat H_0]\) rewrites momentum as position; \(\hbar\omega_{fi}=E_f-E_i\). The rate is thus set by the electric-dipole matrix element \(\vec{d}_{fi}=q\langle f|\hat{\vec{r}}|i\rangle\). C

\[ \hat{\Pi}\,\hat{\vec{r}}\,\hat{\Pi}^{-1}=-\hat{\vec{r}},\qquad \hat{\Pi}\,|n\ell m\rangle=(-1)^{\ell}\,|n\ell m\rangle \]

The position operator is parity-odd, while a central-field eigenstate has definite parity \((-1)^\ell\) because \(Y_\ell^m(-\hat{r})=(-1)^\ell Y_\ell^m(\hat{r})\). B

\[ \langle n'\ell'm'|\hat{\vec{r}}|n\ell m\rangle=(-1)^{\ell+\ell'}(-1)\,\langle n'\ell'm'|\hat{\vec{r}}|n\ell m\rangle \]

Insert \(\hat\Pi^{-1}\hat\Pi=1\) on both sides of \(\hat{\vec r}\) and use step 7. The matrix element equals \((-1)^{\ell+\ell'+1}\) times itself, so it vanishes unless \((-1)^{\ell+\ell'+1}=+1\), i.e. \(\ell+\ell'\) is odd: the two states must have opposite parity. C

\[ z=r\cos\theta=r\sqrt{\tfrac{4\pi}{3}}\,Y_1^{0},\qquad x\pm iy=\mp r\sqrt{\tfrac{8\pi}{3}}\,Y_1^{\pm1} \]

Express the Cartesian components of \(\vec{r}\) as the spherical components \(r_{q}\;(q=0,\pm1)\), each proportional to a rank-1 spherical harmonic \(Y_1^{q}\). This is what makes \(\vec r\) a rank-1 (vector) tensor operator. B

\[ \langle \ell'm'|Y_1^{q}|\ell m\rangle\propto\int_0^{2\pi}\!\!e^{i(m-m'+q)\phi}\,d\phi \;=\;2\pi\,\delta_{m',\,m+q} \]

The \(\phi\)-integral of the three spherical harmonics is nonzero only when the azimuthal phases cancel, forcing \(m'=m+q\) with \(q\in\{0,\pm1\}\). Hence \(\Delta m=m'-m=0,\pm1\). C

Wigner–Eckart theorem: a rank-1 tensor between angular-momentum eigenstates factorizes into a Clebsch–Gordan coefficient times a reduced matrix element. The coefficient couples \(\ell\otimes 1\), which contains \(\ell'=\ell-1,\ell,\ell+1\). C

\[ \langle \ell\,0;1\,0|\ell'\,0\rangle=0 \ \text{ for } \ell'=\ell\ \Longrightarrow\ \Delta\ell=\pm1 \]

The parity rule of step 8 already forbids \(\Delta\ell=0\); the vanishing Clebsch–Gordan coefficient for \(\ell'=\ell\) (and the triangle rule excluding \(|\Delta\ell|>1\)) confirms it. Combined with step 10 this yields the full E1 rule set. C

\[ \boxed{\ \Delta\ell=\pm1,\qquad \Delta m=0,\pm1,\qquad \pi_i\pi_f=-1\ (\text{parity change}),\qquad \Delta s=0\ } \]

Reading. An electric-dipole photon carries one unit of angular momentum and odd parity. The atom must therefore change its orbital angular momentum by exactly one unit (\(\Delta\ell=\pm1\)), change parity, and change its magnetic quantum number by at most one unit (\(\Delta m=0\) for light linearly polarized along \(z\), \(\Delta m=\pm1\) for the two circular polarizations in the \(xy\)-plane). The spin is a spectator, \(\Delta s=0\). Any pair of hydrogen levels not meeting all of these is E1-forbidden and produces no first-order line.

Units check. The dipole matrix element \(\vec d_{fi}=q\langle f|\vec r|i\rangle\) carries units of charge \(\times\) length \(=\mathrm{C\,m}\), the SI unit of electric dipole moment. The quantum numbers \(\ell,m,s\) are dimensionless integers/half-integers and their selection rules are pure-number constraints, as required.

Limiting cases

Linearly polarized light along \(z\) (\(\hat{\boldsymbol\epsilon}=\hat z\)) drives only the \(q=0\) component, so only \(\Delta m=0\) transitions occur (the \(\pi\) line of a Zeeman triplet).
Circularly polarized light propagating along \(z\) drives \(q=+1\) or \(q=-1\) alone, giving pure \(\Delta m=+1\) (\(\sigma^+\)) or \(\Delta m=-1\) (\(\sigma^-\)) — the two Zeeman side lines with opposite handedness.
For \(\ell=0\to\ell'=0\) (e.g. \(1s\to2s\)) the rule \(\Delta\ell=\pm1\) forbids E1 entirely; the \(2s\) state is metastable and decays by two-photon emission.
When fine structure dominates, replace \((\ell,m)\) by \((j,m_j)\): the rules become \(\Delta j=0,\pm1\) (with \(j=0\to j'=0\) forbidden) and \(\Delta m_j=0,\pm1\), while parity must still flip.
In the classical (large-\(n\)) correspondence limit the discrete \(\Delta\ell=\pm1\) rule reproduces the single harmonic of a Keplerian orbit that radiates, matching the Larmor dipole picture.

Breaks when

Short wavelength / hard photons. When \(\lambda\) is no longer huge compared with \(a_0\) (X-ray and \(\gamma\)-ray transitions, inner shells of heavy atoms), \(e^{i\vec k\cdot\vec r}\neq1\) and the retained \(\vec k\cdot\vec r\) term drives M1 and E2 transitions obeying \(\Delta\ell=0,\pm2\) and \(\Delta j=0,\pm1,\pm2\) — the E1 rules simply do not describe the observed lines.
Strong external fields. A large static electric field (strong Stark regime) mixes states of opposite parity, so \(|i\rangle\) and \(|f\rangle\) are no longer parity eigenstates; the parity argument of step 8 collapses and "forbidden" lines gain E1 intensity.
Intense laser fields. When the coupling is not weak, first-order perturbation theory (and hence the single-photon rate) breaks down; Rabi flopping and multiphoton absorption allow net \(\Delta\ell=\pm2,\dots\) and \(\Delta m=\pm2,\dots\) because several dipole photons act coherently.
Strong configuration or spin-orbit mixing. If a nominal \(|n\ell m\rangle\) is actually a superposition of different \(\ell\) (relativistic or correlation mixing), the sharp \(\Delta\ell=\pm1\) statement is only approximate and intercombination lines (\(\Delta s\neq0\)) appear.

Failure modes

Confusing \(\Delta\ell\) with \(\Delta n\). There is no restriction on the principal quantum number; \(n\) can jump by any amount. Only \(\ell\) is constrained. Students often wrongly forbid, e.g., \(3p\to1s\).
Reading \(\Delta m=0,\pm1\) as three independent transitions for any polarization. A single fixed polarization selects one value of \(q\); the full set appears only when all polarizations are present or after summing over emission directions.
Forgetting the parity rule and allowing \(\Delta\ell=0\). The Wigner–Eckart triangle rule permits \(\ell'=\ell\), but parity kills it. Both arguments must be applied together.
Applying \(\Delta\ell=\pm1\) to a multi-electron atom without care. The single-electron orbital rule constrains the jumping electron; the atomic term rules are \(\Delta L=0,\pm1\) (not \(0\to0\)) and \(\Delta S=0\), which is a different statement.
Claiming \(2s\to1s\) is simply "allowed because energy is available." Energetics say nothing about rate; the symmetry veto makes it E1-forbidden regardless of the large energy gap.
Using the electric-dipole rules for magnetic transitions. M1 lines (e.g. the 21 cm line) obey \(\Delta\ell=0\); imposing \(\Delta\ell=\pm1\) on them predicts they cannot occur, which is false.

Discussion

The selection rules are, at heart, a conservation law dressed in the language of matrix elements. A single E1 photon carries angular momentum \(1\hbar\) and odd intrinsic parity. Demanding that the total angular momentum and parity of atom-plus-photon be conserved forces the atomic angular momentum to change by exactly one unit and its parity to flip. The Wigner–Eckart theorem makes this precise: because \(\vec r\) is a spherical tensor of rank 1, the angular part of every matrix element is a fixed Clebsch–Gordan coefficient, and the geometry of angular-momentum coupling — not the details of the radial wavefunction — decides whether a line exists.

This separation of "geometry decides existence, dynamics decides strength" is one of the most powerful ideas in atomic physics. The reduced matrix element \(\langle\ell'\|\hat r\|\ell\rangle\) contains all the messy radial integrals and sets the oscillator strength and lifetime, but it can only scale a line that the Clebsch–Gordan coefficient has already permitted. Two transitions with the same angular labels have intensities in a fixed ratio set purely by \(3j\) symbols — the basis of line-strength tables and the theory of the Zeeman and Stark patterns.

Physically, the rules explain why the hydrogen spectrum is orderly. Every bright line connects an \(s\) level to a \(p\) level, a \(p\) to \(s\) or \(d\), and so on, alternating parity as one climbs the ladder. Metastable states such as \(2s\), stranded with no E1-allowed downward partner, live many orders of magnitude longer than ordinary excited states and become reservoirs for two-photon decay and precision spectroscopy of the Lamb shift.

At the deepest level the rules are the low-energy face of gauge invariance and rotational symmetry. In the fully quantized theory the interaction \(-\tfrac{q}{mc}\vec A\cdot\vec p\) becomes emission or absorption of a photon whose polarization vector is itself a rank-1 object; the classical field of the semiclassical treatment is replaced by the mode of the quantized field, and spontaneous emission emerges from the vacuum term. Remarkably, the selection rules are identical in both pictures, because they follow from the transformation properties of \(\vec r\) under rotations and parity, not from how the field is quantized. The tiny parity-violating weak neutral current does open forbidden E1 amplitudes at the \(10^{-11}\) level, and measuring exactly these symmetry-forbidden lines is how atomic physics tests the electroweak theory. Common misconceptions: the rules are not about energy availability, they are not softened by making the light more intense within first order, and they constrain \(\ell\) and \(m\) but never the principal quantum number \(n\).

Worked examples

Example 1 — Which Balmer-region transitions from \(3d\) are E1-allowed?

\[ \text{Initial: } |n=3,\ell=2\rangle\ (3d),\qquad \text{candidates } n'=1,2 \]

List lower levels; apply \(\Delta\ell=\pm1\Rightarrow\ell'=1\ \text{or}\ 3\). Only \(\ell'=1\) (a \(p\) state) is available below \(n=3\). A

\[ 3d\to 2p:\quad \Delta\ell=1\ \checkmark,\ \ \pi:\,(-1)^2\to(-1)^1\ \text{flips}\ \checkmark \qquad 3d\to1s:\ \Delta\ell=2\ \times \]

Check parity flip and \(\Delta\ell=\pm1\) for each. \(3d\to1s\) needs \(\Delta\ell=-2\), forbidden; \(3d\to2s\) needs \(\Delta\ell=-2\), forbidden. B

\[ \lambda_{3\to2}=\frac{hc}{E_2-E_3}=\frac{hc}{13.6\,\mathrm{eV}\,(1/4-1/9)}=\frac{1240\ \mathrm{eV\,nm}}{1.89\ \mathrm{eV}} \]

Insert numbers using the hydrogen spectrum \(E_n=-13.6/n^2\,\mathrm{eV}\); the allowed line is the red Balmer-\(\alpha\). B

\[ 3d\to2p\ \text{allowed at } \lambda\approx 656\ \mathrm{nm};\quad 3d\to2s,\ 3d\to1s\ \text{E1-forbidden} \]

Reading. The only E1 decay of \(3d\) is to \(2p\); the \(d\to s\) and \(d\to s\) channels violate \(\Delta\ell=\pm1\). Units check. \(\mathrm{eV\,nm}/\mathrm{eV}=\mathrm{nm}\), a wavelength.

Example 2 — Polarization of the normal Zeeman \(\sigma^+\) line.

\[ B\hat z\ \Rightarrow\ E_{n\ell m}=E_n^{(0)}+m\,\mu_B B,\qquad \mu_B=\frac{e\hbar}{2m_e} \]

A weak field along \(z\) shifts each \(m\)-sublevel by \(m\mu_B B\) (normal Zeeman, spin ignored). Transitions split by \(\Delta m\). A

\[ \Delta m=+1\ \text{driven by } \hat r_{+1}\propto(x+iy)\ \Rightarrow\ \sigma^+\ \text{circular polarization} \]

The \(q=+1\) spherical component couples to left-circular light propagating along \(+z\); this fixes the emitted polarization. C

\[ \Delta E=E(\Delta m{=}{+}1)-E(\Delta m{=}0)=\mu_B B=(5.79\times10^{-5}\,\mathrm{eV/T})(1\ \mathrm{T}) \]

Numerical shift of the \(\sigma^+\) line relative to the unshifted \(\pi\) line at \(B=1\,\mathrm{T}\). B

\[ \Delta E\approx 5.8\times10^{-5}\ \mathrm{eV}\ \ (\Delta\nu\approx 14\ \mathrm{GHz}),\ \ \text{left-circularly polarized } \sigma^+ \]

Reading. The \(\Delta m=+1\) transition emerges displaced by one Bohr-magneton energy and carries \(\sigma^+\) circular polarization along the field. Units check. \((\mathrm{eV/T})(\mathrm{T})=\mathrm{eV}\); dividing by \(h\) gives \(\mathrm{GHz}\).

Problems

State, with justification, whether the transition \(4f\to2p\) in hydrogen is electric-dipole allowed.
Solution
\(4f\) has \(\ell=3\), \(2p\) has \(\ell=1\), so \(\Delta\ell=-2\). This violates \(\Delta\ell=\pm1\); moreover parity \((-1)^3=-1\to(-1)^1=-1\) does not flip. The transition is E1-forbidden. (It could proceed weakly as an electric quadrupole, \(\Delta\ell=\pm2\).)
For linearly polarized light along \(\hat z\), which \(\Delta m\) transitions are driven, and why is \(\Delta m=\pm1\) absent?
Solution
\(\hat z\) selects only the spherical component \(\hat r_0=z\propto Y_1^0\), which carries \(q=0\). The \(\phi\)-integral gives \(\delta_{m',m+q}=\delta_{m',m}\), so only \(\Delta m=0\) survives. The \(q=\pm1\) components \((x\pm iy)\) are orthogonal to \(\hat z\) and are simply not present in \(\hat{\boldsymbol\epsilon}\cdot\vec r\), so \(\Delta m=\pm1\) is not driven.
Explain quantitatively why the \(2s\) state of hydrogen is metastable, and estimate the order-of-magnitude enhancement of its lifetime relative to \(2p\) (\(\tau_{2p}\approx1.6\ \mathrm{ns}\)).
Solution
\(2s\) has \(\ell=0\); the only lower level is \(1s\) with \(\ell=0\), giving \(\Delta\ell=0\), which is E1-forbidden (no parity flip). E1 decay is impossible, so \(2s\) decays by two-photon E1E1 emission with rate \(\approx 8.2\ \mathrm{s^{-1}}\), i.e. \(\tau_{2s}\approx0.12\ \mathrm{s}\). The ratio \(\tau_{2s}/\tau_{2p}\approx0.12/(1.6\times10^{-9})\approx8\times10^{7}\), an enhancement of order \(10^{8}\).
A sodium-like atom shows a line at \(3p\to3s\). Using \(\Delta n=0\) here, confirm the transition is allowed and comment on whether \(\Delta n=0\) violates any selection rule.
Solution
\(3p\ (\ell=1)\to3s\ (\ell=0)\): \(\Delta\ell=-1\ \checkmark\), parity flips \((-1)^1\to(-1)^0\ \checkmark\), and \(\Delta m=0,\pm1\) is satisfiable. The transition is E1-allowed. There is no selection rule on the principal quantum number, so \(\Delta n=0\) is perfectly permitted (indeed this is the origin of the sodium D lines). Only \(\ell,m\) (and parity, spin) are constrained.
Light of a single circular polarization \(\sigma^-\) propagating along \(+z\) illuminates atoms initially in \(|\ell=1,m=+1\rangle\). To which final \(m\) states in an \(\ell'=2\) manifold can absorption occur, and what is the resulting \(m\)?
Solution
\(\sigma^-\) carries \(q=-1\), so \(\Delta m=-1\), giving final \(m'=m+q=+1-1=0\). The final state is \(|\ell'=2,m'=0\rangle\), which also satisfies \(\Delta\ell=+1\) and the parity flip \((-1)^1\to(-1)^2\). Thus absorption goes to exactly one substate, \(|2,0\rangle\); the transitions to \(m'=+2\) or \(m'=+1\) are not driven by this polarization.