Dipole Transitions

Electric-dipole transitions are transitions driven by the leading coupling between a charged system and an electric field. In the dipole approximation, the relevant operator is the electric dipole moment

$$\mathbf d = \sum_a q_a\mathbf r_a,$$

or, for a single particle of charge $q$,

$$\mathbf d=q\mathbf r.$$

This page derives the symmetry selection rules for electric-dipole matrix elements. Transition rates, linewidths, oscillator strengths, and detailed spectroscopy belong to transition theory and spectroscopy pages.

Dipole Matrix Element

In the electric-dipole approximation, the transition amplitude is controlled by a matrix element of the form

$$\langle f| \mathbf d\cdot\boldsymbol\epsilon |i\rangle,$$

where $\boldsymbol\epsilon$ is the polarization vector of the applied or emitted radiation. If this matrix element vanishes by symmetry, the electric-dipole transition is forbidden in that approximation.

The symmetry content comes from two facts:

• $\mathbf d$ is a vector under rotations;
• $\mathbf d$ is odd under parity.

These two transformation properties are enough to derive the standard angular and parity selection rules.

Spherical Components and Polarization

A vector operator can be written in spherical components:

$$d_0=d_z, \qquad d_{\pm1} = \mp\frac{1}{\sqrt2} \left( d_x\pm id_y \right).$$

The components $d_q$ form a rank-$1$ spherical tensor:

$$k=1, \qquad q=-1,0,1.$$

The component selected by the light depends on polarization relative to the chosen quantization axis:

Component	Common label	Magnetic rule
$q=0$	linear, or $\pi$ polarization	$\Delta m=0$
$q=+1$	one circular component	$\Delta m=+1$
$q=-1$	the opposite circular component	$\Delta m=-1$

The association of $\sigma^\pm$ labels with $q=\pm1$ depends on propagation direction and polarization convention, so the invariant statement is the $q$ label and the rule $\Delta m=q$.

Rotational Selection Rules

Because $d_q$ is a rank-$1$ tensor, the Wigner–Eckart theorem gives

$$m_f=m_i+q,$$

and

$$|j_i-1| \le j_f \le j_i+1.$$

Equivalently, the total angular momentum can change by

$$\Delta j=0,\pm1,$$

subject to the triangle condition. In particular, a $j_i=0\to j_f=0$ electric-dipole transition is forbidden by angular momentum because a rank-$1$ operator cannot connect two scalar angular-momentum multiplets.

For orbital angular momentum in a central potential, replace $j$ by $\ell$ when spin is not included:

$$|\ell_i-1| \le \ell_f \le \ell_i+1.$$

This rotation-only rule allows $\Delta\ell=0,\pm1$ where valid. Parity removes one of these options.

Parity Selection Rule

The electric dipole operator is odd under parity:

$$\Pi\mathbf d\Pi^{-1} = -\mathbf d.$$

If the initial and final states have definite parities $\pi_i$ and $\pi_f$, the parity rule for an odd operator requires

$$\pi_f=-\pi_i.$$

Thus electric-dipole transitions connect states of opposite parity.

For central-potential orbital states,

$$\pi_\ell=(-1)^\ell.$$

Opposite parity requires $\ell_f$ and $\ell_i$ to differ by an odd integer. Combining this with the vector triangle rule gives

$$\Delta\ell=\pm1.$$

This is why the familiar orbital electric-dipole rule is not merely an angular-momentum triangle rule. It is the combination of rank-$1$ rotation behavior and odd parity.

Hydrogenic Orbital Rules

For spinless hydrogenic orbital states

$$\psi_{n\ell m} = R_{n\ell}(r)Y_\ell^m(\theta,\phi),$$

the leading electric-dipole rules are

$$\Delta\ell=\pm1, \qquad \Delta m=0,\pm1,$$

with the value of $\Delta m$ selected by the polarization component $q$.

There is no symmetry rule requiring a particular $\Delta n$. The radial overlap and energy conservation decide which transitions are strong, weak, resonant, or possible for a given photon frequency.

Total Angular Momentum Rules

When spin and spin–orbit structure are included, states are often labeled by total angular momentum $j,m_j$ rather than by $\ell,m_\ell$ alone. The dipole operator is still a rank-$1$ tensor under total rotations, so the angular rules become

$$\Delta j=0,\pm1, \qquad j_i=0\not\to j_f=0,$$

and

$$\Delta m_j=q.$$

Parity must still change for an electric-dipole transition. Additional approximate rules can appear in a coupling scheme. For example, in $LS$ coupling the leading electric dipole operator acts on spatial coordinates and is often described as approximately preserving total spin:

$$\Delta S=0.$$

That spin rule is not as fundamental as parity and angular momentum. Spin–orbit mixing, configuration interaction, and relativistic corrections can make nominally spin-forbidden transitions weakly allowed.

Forbidden Does Not Mean Impossible

An electric-dipole forbidden transition is forbidden only for the electric-dipole operator under the stated symmetry assumptions. The transition may still occur through:

• magnetic dipole coupling;
• electric quadrupole coupling;
• two-photon processes;
• external-field mixing;
• hyperfine or spin–orbit mixing;
• symmetry breaking by the environment.

These mechanisms are usually weaker, which is exactly why the electric-dipole selection rules remain practically important.

Canonical Split

Use this page for the symmetry derivation of electric-dipole selection rules.

Use Dipole Approximation for the long-wavelength expansion, transition-density interpretation, center-of-mass phase, origin bookkeeping, and quantitative E1 validity tests.

Use Selection Rules in Transition Rates for how a matrix-element zero enters transition probabilities and golden-rule rates.

Use Transition Rates in Light–Matter Interaction for semiclassical absorption, stimulated emission, quantized-mode occupation factors, and the free-space spontaneous-emission rate.

Use Parity Selection Rules for the general even/odd operator proof behind the dipole parity rule.

Use Multipole Operators for how $E1$ compares with $M1$, $E2$, and higher multipoles by rank and parity.

Use Applications to Atomic Spectra for how these rules organize atomic line assignments.

Use Atomic Selection Rules for fine and hyperfine labels, polarization-resolved components, higher multipoles, metastable states, and intensity borrowing.

Use Applications to Molecular Rotations for the corresponding rigid-rotor selection-rule derivation.

Use Spectroscopy and later AMO material for line positions, line strengths, experimental geometry, linewidths, and real spectra.

Common Mistakes

• Deriving $\Delta\ell=\pm1$ from rotations alone. Parity is also needed.
• Forgetting that the polarization component fixes $\Delta m=q$.
• Treating $\Delta n$ as a symmetry selection rule. It is not.
• Applying spinless orbital rules unchanged after fine structure changes the good angular labels.
• Treating electric-dipole forbidden as absolutely impossible.
• Ignoring the convention dependence of $\sigma^\pm$ labels while using circular polarization language.

Cross-Links

• Dipole Approximation
• Selection Rules
• Parity Selection Rules
• Multipole Operators
• Applications to Atomic Spectra
• Atomic Selection Rules
• Applications to Molecular Rotations
• Irreducible Spherical Tensors
• Wigner–Eckart Theorem
• Parity
• Selection Rule Problems
• Hydrogen Atom Angular Structure
• Spin–Orbit Coupling
• From Angular Momentum to Helicity
• From Selection Rules to Ward Identities
• Selection Rules in Transition Rates
• Transition Rates in Light–Matter Interaction
• Spectroscopy
• Spectra to Atomic Physics
• Wigner–Eckart Formula Card

References

• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Wiley, 1977.
• A. R. Edmonds, Angular Momentum in Quantum Mechanics, Princeton University Press, 1957.
• C. J. Foot, Atomic Physics, Oxford University Press, 2005.
• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.

Exercises

1. Use parity to show that an electric-dipole matrix element between two $s$ orbitals vanishes.

Solution

An $s$ orbital has $\ell=0$, so its parity is

$$(-1)^0=+1.$$

The electric dipole operator is odd under parity. A nonzero matrix element requires opposite initial and final parity. Two $s$ orbitals both have even parity, so

$$\langle n'sm'|\mathbf d|nsm\rangle=0$$

in a parity-symmetric central potential.

2. For a $q=+1$ dipole component acting on a state with $m=-1$, what is the final magnetic quantum number?

Solution

The magnetic rule is

$$m_f=m_i+q.$$

With $m_i=-1$ and $q=+1$,

$$m_f=0.$$

3. Why is $j=0\to j'=0$ forbidden for electric dipole transitions?

Solution

The electric dipole operator is a rank-$1$ tensor. The angular momentum triangle rule is

$$|j_i-1|\le j_f\le j_i+1.$$

If $j_i=0$, this gives

$$1\le j_f\le1,$$

so $j_f=1$. A final state with $j_f=0$ is not allowed by angular momentum conservation for a rank-$1$ operator.

4. Explain why $\Delta n$ is not listed among the basic electric-dipole symmetry rules.

Solution

The principal quantum number $n$ is not a simple rotation or parity label. Electric-dipole symmetry rules constrain angular momentum and parity. Whether two radial states with different or equal $n$ have a nonzero radial overlap is a dynamical question, not a pure symmetry rule. Energy conservation and the photon frequency also matter.