Electric-Dipole Selection Rules

Derivation

Electric-Dipole Selection Rules

D-308 Home PU-306 Threads light · symmetry · waves Depends on The Wigner-Eckart Theorem, spherical-harmonics, Bound-State Spectrum of the Hydrogen Atom

Statement

For a one-electron atom the electric-dipole transition amplitude between stationary states \(|n\,l\,m\rangle\) and \(|n'\,l'\,m'\rangle\) is governed by the matrix element \(\langle n'l'm'|\,\hat{\vec r}\,|nlm\rangle\). Because \(\hat{\vec r}\) is an odd-parity rank-1 spherical tensor operator, the Wigner–Eckart theorem forces this amplitude to vanish unless \(\Delta l = l'-l = \pm 1\), \(\Delta m = m'-m = 0,\pm 1\), and the two states have opposite parity. These are the electric-dipole (E1) selection rules.

Why it matters

Almost every spectral line you can name — the sodium D doublet, the hydrogen Balmer series, the light emitted by a fluorescent tube — is an E1 transition, and its very existence is decided by these three rules. They tell you which lines appear, which are forbidden (and therefore weak and long-lived), and how a line splits and polarises in a magnetic field.

The rules also showcase the deepest labour-saving device in atomic physics: the Wigner–Eckart theorem separates the geometry of a transition (angular momentum, encoded in a Clebsch–Gordan coefficient you look up) from its dynamics (a single radial reduced matrix element). All of angular structure follows from symmetry alone, before any radial integral is attempted.

Assumptions

The radiation couples to the atom through the electric-dipole term \(\hat H' = -\hat{\vec d}\cdot\vec E = e\,\hat{\vec r}\cdot\vec E\). If dropped, one is no longer computing E1 rates; magnetic-dipole (M1) and electric-quadrupole (E2) terms obey different selection rules (\(\Delta l = 0,\pm2\) for E2) and the whole analysis changes. The states are exact eigenstates of a central, parity-symmetric Hamiltonian, so \(|nlm\rangle\) carries definite \(l\), \(m\) and parity \((-1)^l\). If the potential is not central (external field, Stark mixing), \(l\) is no longer good and states are superpositions of different \(l\); the "\(\Delta l\)" rule dissolves while the \(m\)-rule survives if axial symmetry remains. The long-wavelength (dipole) approximation \(e^{i\vec k\cdot\vec r}\approx 1\) holds, i.e. the photon wavelength greatly exceeds the atomic size, \(k a_0 \ll 1\). If dropped, retardation reinstates the higher multipoles kept in \(e^{i\vec k\cdot\vec r}=1+i\vec k\cdot\vec r+\cdots\); the neglected terms are exactly the M1/E2 amplitudes, suppressed by \(\sim (ka_0)\sim \alpha Z\). Spin is a spectator: we treat \(\hat{\vec r}\) acting only on the spatial part, so spin gives \(\Delta s=0,\ \Delta m_s=0\) trivially. If dropped (spin–orbit coupling strong), the good labels become \(j,m_j\) and the rules re-emerge as \(\Delta j=0,\pm1\) (no \(0\to0\)), \(\Delta m_j=0,\pm1\), with the same parity change.

Derivation

\[ w_{fi}\;\propto\;\big|\langle n'l'm'|\,\hat{\vec\epsilon}\cdot\hat{\vec r}\,|nlm\rangle\big|^2 \]

Fermi's golden rule for the perturbation \(\hat H'=e\,\vec E\cdot\hat{\vec r}\); \(\hat{\vec\epsilon}\) is the photon polarisation. The rate vanishes exactly when the matrix element does, so the selection rules are the zeros of \(\langle n'l'm'|\hat r_q|nlm\rangle\). A

\[ \hat r_{0}=z,\qquad \hat r_{\pm 1}=\mp\tfrac{1}{\sqrt2}\,(x\pm i y) \]

Assemble the three Cartesian components of \(\hat{\vec r}\) into the standard spherical basis. Any \(\hat{\vec\epsilon}\cdot\hat{\vec r}\) is a linear combination of these three, so it suffices to treat each \(\hat r_q\). A

\[ \hat r_{q}=r\sqrt{\frac{4\pi}{3}}\;Y_1^{q}(\theta,\phi)\;\equiv\; r\,C^{(1)}_q,\qquad q=0,\pm1 \]

Rewrite the spherical components using the \(l=1\) spherical harmonics: \(z=r\cos\theta=r\sqrt{4\pi/3}\,Y_1^0\), and \(x\pm iy=r\sin\theta\,e^{\pm i\phi}=\mp r\sqrt{8\pi/3}\,Y_1^{\pm1}\). This exhibits \(\hat r_q\) as a rank-1 spherical tensor operator \(\hat T^{(1)}_q\). B

\[ U(R)\,\hat T^{(1)}_q\,U^\dagger(R)=\sum_{q'} D^{(1)}_{q'q}(R)\,\hat T^{(1)}_{q'} \]

Under a rotation \(R\), the components \(\hat r_q\) mix among themselves exactly as the \(Y_1^q\) do — the defining property of a rank-1 tensor operator (\(D^{(1)}\) is the spin-1 Wigner matrix). This is the hypothesis the Wigner–Eckart theorem requires. C

\[ \langle n'l'm'|\,\hat T^{(1)}_q\,|nlm\rangle=\langle l\,m;1\,q\,|\,l'\,m'\rangle\;\frac{\langle n'l'\,\|\,\hat T^{(1)}\,\|\,nl\rangle}{\sqrt{2l'+1}} \]

Apply the Wigner–Eckart theorem (assumed prior result). The entire \(m,m',q\) dependence is carried by one Clebsch–Gordan coefficient; the reduced matrix element \(\langle n'l'\|\hat T^{(1)}\|nl\rangle\) is a single number independent of all three magnetic quantum numbers. Every selection rule is now a property of the CG coefficient. C

\[ \langle l\,m;1\,q\,|\,l'\,m'\rangle=0\quad\text{unless}\quad m'=m+q \]

A Clebsch–Gordan coefficient is nonzero only when the \(z\)-projections add: \(m'=m+q\). This is the statement \([\hat L_z,\hat r_q]=q\,\hbar\,\hat r_q\), i.e. conservation of \(m\) up to the unit carried by the operator. B

\[ q\in\{0,\pm1\}\;\Longrightarrow\;\boxed{\;\Delta m=m'-m=0,\pm1\;} \]

Since the operator has only three components \(q=0,\pm1\), the allowed projection changes are exactly these. \(q=0\) (the \(z\)/\(\pi\) component) gives \(\Delta m=0\); \(q=\pm1\) (the \(\sigma^\pm\) components) give \(\Delta m=\pm1\). A

\[ \langle l\,m;1\,q\,|\,l'\,m'\rangle=0\quad\text{unless}\quad |l-1|\le l'\le l+1 \]

The triangle condition on the CG coefficient: coupling \(l\otimes 1\) can only produce total \(l'\in\{l-1,\,l,\,l+1\}\). This narrows \(\Delta l\) to \(0,\pm1\) but does not yet exclude \(\Delta l=0\). B

\[ \hat P\,\hat{\vec r}\,\hat P^{-1}=-\hat{\vec r},\qquad \hat P\,|nlm\rangle=(-1)^{l}\,|nlm\rangle \]

Now bring in parity, independent of the CG algebra. The position operator is parity-odd; a central-potential eigenstate has parity \((-1)^l\) (assumed: spherical harmonics obey \(Y_l^m(-\hat n)=(-1)^l Y_l^m(\hat n)\)). B

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

Insert \(\hat P^{-1}\hat P=\mathbf 1\) on both sides and let \(\hat P\) act. The matrix element equals \((-1)^{\,l'+l+1}\) times itself, so it must vanish unless that factor is \(+1\), i.e. unless \(l'+l\) is odd. Equivalently, initial and final states must have opposite parity. C

\[ \big(|l-1|\le l'\le l+1\big)\;\wedge\;\big(l'+l\ \text{odd}\big)\;\Longrightarrow\;\boxed{\;\Delta l=\pm 1\;} \]

Parity kills \(l'=l\) (that would make \(l'+l=2l\), even), leaving only \(l'=l\pm1\) from the triangle. The two arguments intersect to give the sharp rule. Equivalently the Gaunt integral \(\int Y_{l'}^{m'*}Y_1^q Y_l^m\,d\Omega\) requires \(l+l'+1\) even. B

Result

\[ \langle n'l'm'|\,\hat{\vec r}\,|nlm\rangle\neq 0 \iff \Delta l=\pm1,\quad \Delta m=0,\pm1,\quad \pi'=-\pi \]

Reading. An E1 photon carries one unit of orbital angular momentum and negative intrinsic parity. Conservation forces the atom to change \(l\) by exactly one unit and flip parity, while the projection \(m\) can change by \(-1,0,+1\) depending on the photon's polarisation (\(\sigma^-,\pi,\sigma^+\)). The \(\Delta m\) rule is pure rotational geometry (holds even in a field that breaks the \(\Delta l\) rule); the \(\Delta l\) rule needs both the triangle inequality and parity.

Units check. The matrix element of \(\hat{\vec r}\) has dimensions of length; multiplied by \(-e\) it is a dipole moment \([\,\text{C·m}\,]\). The selection rules themselves are conditions on dimensionless integers \(l,m\), so they are units-consistent by construction — they gate a quantity of dimension length, not a rate. The reduced matrix element \(\langle n'l'\|\hat T^{(1)}\|nl\rangle\) likewise has dimension length (it is \(\int R_{n'l'} r R_{nl} r^2\,dr\) up to angular factors).

Limiting cases

\(q=0\) (linearly \(z\)-polarised light): only \(\Delta m=0\) transitions — the \(\pi\) lines, which vanish when viewed along the field axis.
\(q=\pm1\) (circularly polarised light): \(\Delta m=\pm1\) — the \(\sigma^\pm\) lines, giving the classic Zeeman triplet of one \(\pi\) plus two \(\sigma\) components.
\(l=0\to l'=1\) (e.g. \(1s\to 2p\)): allowed; the \(s\)-\(s\) transition \(l=0\to l'=0\) is strictly forbidden (no \(l'=l-1\) exists and parity forbids \(l'=l\)).
Large \(l\): the rule \(\Delta l=\pm1\) is exact for all \(l\); it never softens, unlike \(\Delta n\) which is unrestricted (any \(n'\) is allowed, weighted only by the radial overlap).

Breaks when

The dipole approximation fails. When \(ka_0\) is not negligible (hard X-rays, or forbidden lines where E1 is zero), the higher multipoles in \(e^{i\vec k\cdot\vec r}\) dominate: M1 obeys \(\Delta l=0\), E2 obeys \(\Delta l=0,\pm2\) with no parity change. The E1 rules simply do not apply to those photons.
\(l\) ceases to be a good quantum number. In an external electric field (Stark effect) or with strong configuration mixing, eigenstates are superpositions of different \(l\); "forbidden" lines borrow strength and appear. Only rules tied to still-good symmetries (e.g. \(\Delta m\) under residual axial symmetry) survive.
Strong spin–orbit or hyperfine coupling. With \(j\) (or \(F\)) the good label, the operational rules become \(\Delta j=0,\pm1\) (no \(0\to0\)) and \(\Delta m_j=0,\pm1\); the bare orbital rule \(\Delta l=\pm1\) applies only to the leading LS term and is relaxed by intermediate coupling.

Failure modes

"\(\Delta n=\pm1\)." There is no selection rule on \(n\); \(1s\to 3p\), \(2s\to 5p\), etc. are all allowed. Only the angular numbers are constrained.
Forgetting parity and keeping \(\Delta l=0\). The triangle condition alone permits \(l'=l\); students who skip the parity argument wrongly allow \(p\to p\). Parity is what deletes \(\Delta l=0\).
Applying \(j,m_j\) rules to a spinless orbital problem (or vice versa) — mixing the two label schemes and, e.g., forbidding \(\Delta j=0\) when only \(l\) is relevant.
Reading \(\Delta m=0,\pm1\) as three separate transitions for unpolarised light without noting that a given polarisation selects a single \(q\); the three appear together only when all polarisations are collected.
Confusing "matrix element zero" with "rate small." A selection-rule zero is exact (to E1 order); it is not a smallness estimate.

Discussion

The physical content is angular-momentum bookkeeping for the photon. A dipole photon is a spin-1, negative-parity object; when the atom absorbs or emits one, the atom must supply the compensating change. The three components \(q=0,\pm1\) are precisely the three spin states of the photon projected on the quantisation axis, which is why polarisation and \(\Delta m\) are locked together. Circularly polarised \(\sigma^+\) light carries \(+\hbar\) of projection and drives \(\Delta m=+1\); this is the working principle of optical pumping and of laser cooling.

The Wigner–Eckart theorem is doing the heavy lifting: it guarantees that every E1 line in a multiplet shares one reduced matrix element, so relative line strengths within a multiplet are fixed ratios of Clebsch–Gordan (or \(3j\)) coefficients — pure numbers, computable without ever touching a radial wavefunction. The radial integral only sets the overall scale. This factorisation of "geometry" from "dynamics" is a recurring theme across nuclear, particle and condensed-matter physics.

Parity deserves emphasis as an independent selection principle. The triangle inequality \(|l-1|\le l'\le l+1\) comes from \(SU(2)\) representation theory and would permit \(\Delta l=0\); it is the discrete symmetry \(\hat P\hat{\vec r}\hat P^{-1}=-\hat{\vec r}\) that removes it. This is why the general statement of the E1 rule is "opposite parity" rather than "\(\Delta l=\pm1\)": in atoms with more than one electron, or with configuration mixing, the good quantum number is total parity, and the Laporte rule (parity must change) is the primary law, with the single-electron \(\Delta l=\pm1\) as its most familiar special case.

Common misconceptions. (i) The rules constrain a change per photon, not per transition — a two-photon process can connect same-parity states (\(2s\to1s\)), which is exactly why the hydrogen \(2s\) state is metastable. (ii) "Forbidden" means E1-forbidden, not impossible: such lines proceed by M1/E2 or multi-photon channels, just far more slowly. (iii) The \(\Delta m\) rule is not weaker than the \(\Delta l\) rule; it is in fact more robust, surviving in fields where \(\Delta l\) breaks.

Worked examples

\[ \text{Which of }2p\to1s,\ 2s\to1s,\ 3d\to1s\ \text{are E1-allowed?} \]

Apply \(\Delta l=\pm1\) and the parity change to each candidate. Set up symbolically: final \(1s\) has \(l'=0\). A

\[ 2p\to1s:\ \Delta l=0-1=-1\ \checkmark;\quad 2s\to1s:\ \Delta l=0-0=0\ \times;\quad 3d\to1s:\ \Delta l=0-2=-2\ \times \]

Only \(\Delta l=\pm1\) survives. Parity confirms: \(p\to s\) flips parity (odd\(\to\)even); \(s\to s\) and \(d\to s\) do not change parity, so both are E1-forbidden. A

\[ 2p\to1s\ \text{allowed};\qquad 2s\to1s,\ 3d\to1s\ \text{forbidden (E1)} \]

Reading. The Lyman-\(\alpha\) line \(2p\to1s\) is strong; \(2s\to1s\) is E1-forbidden, so the \(2s\) level is metastable and decays only by the slow two-photon channel (lifetime \(\sim 0.12\) s versus \(\sim1.6\) ns for \(2p\)).

Units check. Selection is a comparison of integers; no dimensions involved. The surviving amplitude \(\langle 1s|\hat{\vec r}|2p\rangle\) has dimension length, as required.

\[ \text{A } 3d\ (l=2)\text{ state decays. To which lower }l'\text{ and with what }\Delta m\ \text{options?} \]

Impose \(\Delta l=\pm1\): from \(l=2\) the allowed final orbital angular momenta are \(l'=1\) or \(l'=3\). Only \(l'=1\) is energetically lower for the same-\(n\)-decreasing cascade to \(p\). A

\[ 3d\to 2p:\quad \Delta l=1-2=-1\ \checkmark,\qquad \Delta m\in\{-1,0,+1\} \]

Parity: \(d\) (even) \(\to p\) (odd) flips — allowed. For a given initial \(m\), the three polarisation channels \(q=0,\pm1\) give \(m'=m,\ m\pm1\), each weighted by the Clebsch–Gordan coefficient \(\langle 2\,m;1\,q|1\,m'\rangle\). B

\[ \text{Take initial } m=+2:\quad m'=m+q\in\{+1,+2,+3\}\ \Rightarrow\ \text{only } m'=+1\ (q=-1) \text{ reaches a valid } 2p\ (|m'|\le1) \]

The final \(2p\) manifold only has \(m'=-1,0,+1\), so \(m'=+2,+3\) are impossible: from stretched \(m=+2\) the sole channel is \(\sigma^-\) to \(m'=+1\). This is why the top Zeeman sublevel decays with pure circular polarisation. B

\[ 3d\to2p\ \text{allowed};\quad \text{from } m=+2\ \text{only } m'=+1\ (\sigma^-)\ \text{survives} \]

Reading. The stretched state \(|3d,m=+2\rangle\) can only emit a \(\sigma^-\) photon to \(|2p,m=+1\rangle\); the availability of only one channel makes stretched states a clean single-frequency, single-polarisation source, exploited in optical pumping.

Units check. All quantities are integer angular-momentum labels; the transition amplitude they gate, \(\langle 2p|\hat r_{-1}|3d\rangle\), carries dimension length.

Problems

State whether \(4f\to3d\) is E1-allowed and give the parity change.
Solution
\(4f\) has \(l=3\), \(3d\) has \(l'=2\). \(\Delta l=2-3=-1\) satisfies \(\Delta l=\pm1\). Parity: \((-1)^3=-1\) (odd) \(\to (-1)^2=+1\) (even) — parity flips. Both conditions met, so the transition is E1-allowed. \(\Delta n=-1\) is irrelevant to selection.
Explain, using the parity argument alone, why \(2s\to1s\) is E1-forbidden without invoking the triangle rule.
Solution
Both \(2s\) and \(1s\) have \(l=0\), hence identical parity \((-1)^0=+1\). The dipole operator is parity-odd, so \(\langle1s|\hat{\vec r}|2s\rangle=(-1)^{0+0+1}\langle1s|\hat{\vec r}|2s\rangle=-\langle1s|\hat{\vec r}|2s\rangle\), forcing the matrix element to zero. Same-parity states cannot connect by E1 regardless of the triangle rule.
For an initial state \(|l=1,m=0\rangle\), list the allowed final \((l',m')\) for E1 emission to \(l'=0\), and identify the required photon polarisations.
Solution
With \(l'=0\): \(\Delta l=-1\) allowed, parity flips (\(p\to s\)). Only \(m'=0\) exists in the \(l'=0\) manifold. Then \(\Delta m=m'-m=0\), so \(q=0\): the \(\pi\) (linearly \(z\)-polarised) component. Result: single line \((0,0)\), \(\pi\)-polarised.
An atom in \(|2p,m=+1\rangle\) decays to \(1s\). Determine \(m'\), \(\Delta m\) and the photon polarisation.
Solution
Final \(1s\) has \(l'=0,\ m'=0\). \(\Delta l=0-1=-1\) allowed; parity flips. \(\Delta m=0-(+1)=-1\Rightarrow q=-1\): a \(\sigma^-\) (left-circular relative to the axis) photon. The emitted photon carries away \(+1\) unit of \(z\)-projection to conserve total \(m\).
The reduced matrix element for \(2p\to1s\) in hydrogen is set by \(R=\int_0^\infty R_{10}(r)\,r\,R_{21}(r)\,r^2\,dr\). Using \(R_{10}=2a_0^{-3/2}e^{-r/a_0}\) and \(R_{21}=\tfrac{1}{\sqrt{24}}\,a_0^{-3/2}(r/a_0)e^{-r/2a_0}\), evaluate \(R\) in units of \(a_0\).
Solution
\(R=\displaystyle\int_0^\infty \big(2a_0^{-3/2}e^{-r/a_0}\big)\,r\,\big(\tfrac{1}{\sqrt{24}}a_0^{-3/2}\tfrac{r}{a_0}e^{-r/2a_0}\big)r^2\,dr = \frac{2}{\sqrt{24}\,a_0^{4}}\int_0^\infty r^4 e^{-3r/2a_0}\,dr.\) Using \(\int_0^\infty r^4 e^{-\beta r}dr=24/\beta^5\) with \(\beta=3/2a_0\): \(\int=24\,(2a_0/3)^5=24\cdot\frac{32}{243}a_0^5=\frac{768}{243}a_0^5\). Then \(R=\frac{2}{\sqrt{24}\,a_0^4}\cdot\frac{768}{243}a_0^5=\frac{1536}{243\sqrt{24}}\,a_0\). Numerically \(\sqrt{24}=4.899\), so \(R=\frac{1536}{243\cdot4.899}a_0=\frac{1536}{1190.5}a_0\approx1.29\,a_0\). (The often-quoted radial factor \(\langle r\rangle_{2p,1s}\approx1.29\,a_0\), giving a dipole length \(\sim\) atomic scale, consistent with the units check.)