Selection Rules

A selection rule is a symmetry statement that makes a matrix element vanish. In transition problems the relevant matrix element often has the form

$$M_{fi} = \langle f|O|i\rangle,$$

where $O$ is the perturbing operator, coupling operator, or observable being tested. If symmetry forces $M_{fi}=0$, the corresponding transition, coupling, or mixing is forbidden at that order and under those assumptions.

Selection rules are not folklore about what is “unlikely.” They are exact consequences of a specified symmetry model.

Assumptions

A selection rule requires assumptions. At minimum, one must know:

• which symmetry is being used;
• how the initial and final states transform;
• how the operator transforms;
• whether the Hamiltonian and environment preserve the symmetry;
• which perturbative order or physical mechanism is being considered.

If the symmetry is broken, the states are not symmetry eigenstates, or a different operator is used, the rule can change. This is why a line called “forbidden” in one approximation may appear weakly in a more complete experiment.

Basic Symmetry Test

Let $U$ be a unitary symmetry. Suppose

$$U|i\rangle = u_i|i\rangle, \qquad U|f\rangle = u_f|f\rangle,$$

and suppose the operator transforms as

$$UOU^\dagger = u_O O.$$

Then

$$\begin{aligned} \langle f|O|i\rangle &= \langle f|U^\dagger UOU^\dagger U|i\rangle \\ &= u_f^*u_Ou_i\, \langle f|O|i\rangle. \end{aligned}$$

Therefore a nonzero matrix element requires

$$u_f^*u_Ou_i=1.$$

If this condition fails, the matrix element must vanish. Parity rules, charge-conservation rules, and many discrete symmetry rules are versions of this simple argument.

Parity Rules

The focused derivation and common parity examples are collected in Parity Selection Rules. The compact version is as follows.

For parity, the eigenvalues are

$$\pi_i,\pi_f=\pm1.$$

If the operator has parity $\pi_O$, then

$$\Pi O\Pi^{-1} = \pi_O O.$$

The nonzero condition becomes

$$\pi_f\pi_O\pi_i=1.$$

Thus:

• an even operator connects states of the same parity;
• an odd operator connects states of opposite parity.

For example, the electric dipole operator is odd under parity. In a parity-symmetric atom or molecule, an electric-dipole matrix element between two states of the same parity vanishes.

Charge-Like Rules

For a continuous $U(1)$ symmetry with conserved generator $Q$, states may be labeled by charge:

$$Q|i\rangle = q_i|i\rangle, \qquad Q|f\rangle = q_f|f\rangle.$$

If an operator carries charge $q_O$ in the sense that it changes the charge of a state by $q_O$, then a nonzero matrix element requires

$$q_f=q_i+q_O.$$

If $O$ is neutral, then $q_f=q_i$. This is the selection-rule form of charge conservation. In nonrelativistic quantum mechanics the same logic appears for particle number, magnetic quantum numbers, oscillator number under special ladder operators, and other additive labels when the operator has a definite charge under the symmetry.

Angular Momentum Rules

Angular momentum selection rules use the fact that operators can transform as irreducible spherical tensors. A rank-$k$ tensor component $T_q^{(k)}$ carries magnetic label $q$.

The Wigner–Eckart theorem gives the rotational selection rules

$$m_f=m_i+q,$$

and

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

The first rule is a magnetic quantum-number rule. The second is a triangle rule: the angular momentum $j_i$ of the initial state and the rank $k$ of the operator must be able to couple to the final angular momentum $j_f$.

For a vector operator, $k=1$, so

$$\Delta m=q=0,\pm1,$$

and

$$j_f=j_i-1,\ j_i,\ j_i+1$$

where the values are allowed by nonnegative angular momentum and the triangle condition.

Parity Plus Rotation

Many familiar rules combine more than one symmetry. Electric dipole transitions in central-potential orbital states are the standard example.

The position operator is a vector, so angular momentum allows

$$\ell_f=\ell_i-1,\ell_i,\ell_i+1.$$

But position is odd under parity, and orbital parity is

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

The parity rule requires opposite parity:

$$(-1)^{\ell_f} = -(-1)^{\ell_i}.$$

Combining both statements gives

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

The $\Delta\ell=0$ option is allowed by rotations for a vector operator but forbidden by parity in this setting.

Spin Rules

Spin selection rules depend on the operator. If an operator acts only on spatial variables and commutes with total spin, then spin labels are preserved:

$$\Delta s=0, \qquad \Delta m_s=0$$

in the basis where those labels are meaningful.

In atomic spectroscopy, electric dipole transitions often satisfy an approximate $\Delta S=0$ rule in an $LS$-coupling description because the leading electric dipole operator does not act on spin. But this is not a universal law. Spin–orbit coupling, magnetic dipole operators, hyperfine interactions, configuration mixing, and strong external fields can weaken or change spin selection rules.

The safe statement is: a spin rule follows from the spin transformation properties of the states and the operator in the Hamiltonian regime being used.

Forbidden Versus Weakly Allowed

“Forbidden” usually means forbidden in a specified approximation. A transition can become weakly allowed when:

• a different multipole operator contributes;
• a higher-order perturbative process is included;
• spin–orbit or hyperfine mixing changes the state labels;
• an external field breaks the symmetry;
• the environment or boundary conditions no longer respect the symmetry;
• the initial or final state is only approximately a symmetry eigenstate.

For example, an electric-dipole forbidden transition may occur through magnetic dipole or electric quadrupole multipoles, two-photon processes, or symmetry-breaking mechanisms. The selection rule still matters: it tells which leading mechanism vanishes and why the observed process is weaker.

Rates Versus Rules

Selection rules constrain amplitudes. Rates require additional physics. In a golden-rule calculation,

$$\Gamma_{i\to f} = \frac{2\pi}{\hbar} |M_{fi}|^2 \rho(E_f).$$

If symmetry gives $M_{fi}=0$, the leading rate vanishes. If $M_{fi}$ is allowed, the actual rate still depends on the reduced matrix element, density of final states, linewidths, populations, polarization, and experimental geometry.

The transition-rate machinery belongs to Selection Rules in Transition Rates. The electric-dipole specialization is derived in Dipole Transitions, while Atomic Selection Rules applies the hierarchy to fine and hyperfine labels, polarization, metastability, and perturbative mixing. This page explains the general symmetry zeros.

Common Mistakes

• Calling a transition forbidden without naming the symmetry and operator.
• Treating “allowed” as meaning “large.”
• Applying parity rules when the system is not parity symmetric.
• Forgetting that angular momentum rules and parity rules are separate constraints.
• Using weak-field quantum numbers in a strong-field regime where the coupling scheme has changed.
• Treating spin selection rules as exact when spin-dependent interactions mix the states.
• Forgetting that a higher-order or different-multipole process can bypass a leading selection rule.

Cross-Links

• Why Symmetry Matters
• Symmetry and Selection Rules Preview
• Parity
• Parity Selection Rules
• Multipole Operators
• Applications to Atomic Spectra
• Atomic Selection Rules
• Molecular Symmetry
• Explicit Symmetry Breaking
• Approximate Symmetry
• Irreducible Spherical Tensors
• Wigner–Eckart Theorem
• Dipole Transitions
• Selection Rule Problems
• Clebsch–Gordan Coefficients
• Hydrogen Atom Angular Structure
• From Angular Momentum to Helicity
• From Selection Rules to Ward Identities
• Selection Rules in Transition Rates
• Selection Rules in Spectroscopy
• Spectroscopy
• 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.
• D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii, Quantum Theory of Angular Momentum, World Scientific, 1988.
• C. J. Foot, Atomic Physics, Oxford University Press, 2005.

Exercises

1. Two states have parity $\pi_i=+1$ and $\pi_f=+1$. Can an odd operator connect them?

Solution

For an odd operator, $\pi_O=-1$. The nonzero condition is

$$\pi_f\pi_O\pi_i=1.$$

Here

$$(+1)(-1)(+1)=-1,$$

so the matrix element must vanish.

2. A rank-$1$ tensor component has $q=-1$. What magnetic quantum-number change does it imply?

Solution

The angular momentum rule is

$$m_f=m_i+q.$$

With $q=-1$,

$$\Delta m=m_f-m_i=-1.$$

3. Why does the electric dipole orbital rule exclude $\Delta\ell=0$ even though a vector operator can satisfy the triangle rule with $\ell_f=\ell_i$?

Solution

The vector nature of the dipole operator gives the triangle rule

$$\ell_f=\ell_i-1,\ell_i,\ell_i+1$$

where allowed. But the electric dipole operator is odd under parity. Orbital states have parity $(-1)^\ell$, so a dipole matrix element requires opposite parity between the initial and final orbital states. The case $\ell_f=\ell_i$ has the same parity and is therefore forbidden. The remaining options are

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

4. Give one mechanism that can make a forbidden transition weakly allowed.

Solution

One example is state mixing. If spin–orbit coupling mixes a small amount of an allowed symmetry character into an otherwise forbidden state, the leading forbidden matrix element can acquire a small nonzero contribution. Other mechanisms include magnetic dipole transitions, electric quadrupole transitions, two-photon processes, external-field symmetry breaking, and hyperfine mixing.