Selection Rules in Transition Rates

A selection rule is a symmetry statement that makes a transition matrix element vanish. In transition calculations, the rate is often proportional to a squared matrix element, so a symmetry zero suppresses the transition before any density-of-states or resonance factor matters.

For a weak perturbation $V(t)$, the first-order transition amplitude contains

$$\langle f|V_I(t)|i\rangle.$$

In the golden-rule regime, a typical rate has the form

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

If $V_{fi}=0$ by symmetry, the leading transition rate vanishes. A nonzero matrix element permitted by symmetry may still be small or vanish through radial cancellation; selection rules give necessary conditions, not guaranteed strengths.

Required background. Parity gives symmetry labels, and Fermi’s Golden Rule explains how a channel amplitude enters a rate.

Matrix-Element Zeros

Selection rules are not guesses about smallness. They are exact zeros under stated assumptions:

• the Hamiltonian has the relevant symmetry;
• the states are chosen with definite symmetry labels;
• the perturbing operator transforms in a definite way;
• the symmetry is not broken by the environment, boundary conditions, or approximations.

If any assumption fails, the selection rule may be weakened or removed.

Parity Example

Let $\Pi$ be the parity operator. Suppose

$$\Pi\lvert i\rangle = \pi_i\lvert i\rangle, \qquad \Pi\lvert f\rangle = \pi_f\lvert f\rangle,$$

where $\pi_i,\pi_f=\pm1$. Let $O$ be an operator with parity $\pi_O$:

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

Then

$$\langle f|O|i\rangle = \pi_f\pi_O\pi_i \langle f|O|i\rangle.$$

Therefore the matrix element can be nonzero only if

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

For an odd operator, $\pi_O=-1$, transitions connect states of opposite parity. For an even operator, transitions connect states of the same parity.

Electric Dipole Preview

In the electric dipole approximation, the perturbation is proportional to

$$\mathbf d\cdot\boldsymbol\epsilon,$$

where $\mathbf d$ is odd under parity. Thus electric dipole transitions require opposite parity between initial and final states:

$$\pi_f=-\pi_i.$$

For central-potential orbital states, the position operator transforms as a vector. The familiar orbital angular-momentum rule is

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

For a spherical operator component $d_q$, the magnetic quantum number changes by $q$:

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

In the interaction $-\mathbf d\cdot\mathbf E=-\sum_q(-1)^q d_qE_{-q}$, the component of the applied field is paired with $d_q$ through $E_{-q}$. State the spherical-basis and propagation conventions before identifying $q$ with a polarization helicity.

These are electric-dipole rules in the simplest nonrelativistic setting. Spin, fine structure, identical particles, and many-body structure add further labels.

Angular Momentum Logic

Angular momentum selection rules come from rotational symmetry. A perturbing operator that transforms as a spherical tensor of rank $k$ can connect angular-momentum states only when angular momenta can couple consistently:

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

The magnetic quantum numbers satisfy

$$m_f=m_i+q, \qquad q=-k,\ldots,k.$$

For a vector operator, $k=1$, so $q=0,\pm1$ and $j=0\to j=0$ is excluded by the triangle condition. Clebsch–Gordan coefficients determine which allowed couplings have nonzero angular factors. The systematic theorem behind this factorization is the Wigner–Eckart Theorem.

Rates Versus Amplitudes

A selection rule first constrains the amplitude. The transition rate then inherits that constraint:

$$V_{fi}=0 \quad \Longrightarrow \quad \Gamma_{i\to f}=0$$

at the order being computed. Higher-order processes may still occur. For example, a one-photon electric dipole transition may be forbidden while a two-photon process, magnetic dipole process, or symmetry-breaking perturbation gives a much smaller nonzero rate.

Selection rules therefore classify leading mechanisms. They do not imply that a transition is impossible under all physical perturbations.

For continuum processes, the symmetry-allowed matrix element must still be combined with the correctly normalized final-state measure; see Density of States in Transition Rates.

A Weakly Allowed Transition

Let $|i\rangle$ and $|f\rangle$ have the same parity, so $\langle f|d_q|i\rangle=0$. Suppose a weak static parity-breaking perturbation mixes an opposite-parity state $|o\rangle$ into the final state:

$$|\widetilde f\rangle=|f\rangle+\epsilon|o\rangle+O(\epsilon^2), \qquad \epsilon=\frac{\langle o|\delta H|f\rangle}{E_f-E_o}.$$

Assuming the initial state is unchanged at this order,

$$\langle\widetilde f|d_q|i\rangle =\epsilon^*\langle o|d_q|i\rangle+O(\epsilon^2).$$

The borrowed transition strength scales as $|\epsilon|^2$ times the allowed strength, at fixed frequency and final-state measure. Near an $f$–$o$ degeneracy, the mixing coefficient need not be small; diagonalize the pair before estimating the rate. If the initial state also mixes, its amplitude correction must be added coherently.

Exercises

Allowed angular labels, zero angular factor

Can a rank-one $q=0$ operator connect $j_i=j_f=1$, $m_i=m_f=0$? Compare the triangle and magnetic conditions with the actual angular coefficient.

Solution

The triangle and magnetic conditions allow these labels, but the coefficient is proportional to the Wigner symbol $\left(\begin{smallmatrix}1&1&1\\0&0&0\end{smallmatrix}\right)=0$. Interchanging the first and third columns leaves all entries unchanged while multiplying the symbol by $(-1)^{1+1+1}=-1$, so it must vanish. Necessary angular-label rules do not replace evaluation of the angular factor.

Two mixing pathways

Suppose the final-state mixing above contributes amplitude $\epsilon^*D_f$ and initial-state mixing contributes $\eta D_i$. Write the strength through quadratic order in the mixing coefficients. Can two nonzero contributions cancel?

Solution

The strength is $|\epsilon^*D_f+\eta D_i|^2$, including the interference term $2\operatorname{Re}(\epsilon\eta D_f^*D_i)$. Equal and opposite amplitudes give zero even though each mixing pathway is individually nonzero. Adding their squared magnitudes would discard coherence.

References

• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Wiley, 1977.
• A. R. Edmonds, Angular Momentum in Quantum Mechanics, Princeton University Press, 1996.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii, Quantum Theory of Angular Momentum, World Scientific, 1988.