Symmetry and Selection Rules Preview

A selection rule is a symmetry reason for a matrix element to vanish. The basic object is

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

where $\lvert i\rangle$ is the initial state, $\lvert f\rangle$ is the final state, and $T$ is the operator that tries to connect them. If symmetry says $M_{fi}=0$, then the transition, coupling, expectation value, or mixing process is forbidden in the model being used.

This page gives the continuous-symmetry logic before the full tensor-operator treatment. The detailed angular-momentum machinery belongs to Selection Rules, Irreducible Spherical Tensors, and the Wigner–Eckart Theorem.

State labels and operator transformations

Selection rules compare the transformation labels of the states with the transformation label of the operator. Those labels must be meaningful for the Hamiltonian and the approximation under discussion.

If a conserved observable $Q$ commutes with the Hamiltonian,

$$[H,Q]=0,$$

then stationary states can often be chosen in sectors of fixed $Q$. If several commuting labels are available, the transition problem can be organized by simultaneous eigenstates:

$$\lvert E,q_1,\ldots,q_r,\lambda\rangle.$$

The extra label $\lambda$ reminds us that symmetry labels need not uniquely identify a state. The selection rule constrains the symmetry sector, not every dynamical detail inside the sector. See Simultaneous Eigenstates and Good Quantum Numbers for the label criterion.

Finite Symmetry Test

Let $U$ be a unitary symmetry. Suppose the states have definite transformation phases,

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

and suppose the operator transforms by

$$UTU^\dagger = u_TT.$$

Insert $U^\dagger U=I$ around the operator:

$$\begin{aligned} \langle f|T|i\rangle &= \langle f|U^\dagger UTU^\dagger U|i\rangle \\ &= u_f^*u_Tu_i \langle f|T|i\rangle. \end{aligned}$$

Therefore a nonzero matrix element requires

$$u_f^*u_Tu_i=1.$$

If the product is not $1$, the only consistent value is $M_{fi}=0$. This is the compact form of many parity, charge, and discrete-symmetry selection rules. It uses unitary $U$; for an antiunitary symmetry, conjugated amplitudes require a different argument.

Infinitesimal Version

For a continuous one-parameter symmetry, write

$$U(\alpha) = \exp\!\left(-\frac{i\alpha Q}{\hbar}\right),$$

where $Q$ is the generator. Suppose

$$Q\lvert i\rangle=q_i\lvert i\rangle, \qquad Q\lvert f\rangle=q_f\lvert f\rangle,$$

and suppose the operator has a definite charge under the adjoint action of $Q$:

$$[Q,T]=q_TT.$$

Taking a matrix element of this commutator gives

$$\begin{aligned} \langle f|[Q,T]|i\rangle &= (q_f-q_i)\langle f|T|i\rangle \\ &= q_T\langle f|T|i\rangle. \end{aligned}$$

Thus

$$(q_f-q_i-q_T)M_{fi}=0.$$

A nonzero matrix element can occur only if

$$q_f=q_i+q_T.$$

The operator must carry exactly the amount of quantum number needed to connect the two sectors. If $T$ is neutral, then $q_f=q_i$.

Parity as the Clean Example

Parity is discrete rather than continuous, but it is the cleanest first example. Let the states have definite parity

$$\Pi\lvert i\rangle = \pi_i\lvert i\rangle, \qquad \Pi\lvert f\rangle = \pi_f\lvert f\rangle, \qquad \pi_i,\pi_f=\pm1.$$

If

$$\Pi T\Pi^{-1} = \eta_TT, \qquad \eta_T=\pm1,$$

then the finite test gives

$$\pi_f\eta_T\pi_i=1.$$

An even operator, $\eta_T=+1$, connects only states of the same parity. An odd operator, $\eta_T=-1$, connects only states of opposite parity. The electric dipole operator is odd, which is why electric-dipole transitions between states of the same parity vanish in a parity-symmetric model.

Charge-Like Labels

For a conserved additive label, the infinitesimal test says that an operator must supply the missing charge. If

$$[Q,T]=q_TT,$$

then

$$\Delta q \equiv q_f-q_i = q_T$$

is required for a nonzero matrix element.

Common examples include particle-number-changing operators, spin ladder operators, oscillator ladder operators, and angular-momentum tensor components. For a neutral operator,

$$[Q,T]=0,$$

the matrix element cannot connect states with different $Q$ eigenvalues. This is the operator form of charge conservation.

Magnetic Quantum Numbers

Take $Q=J_z$. For a spherical tensor component $T_q^{(k)}$,

$$[J_z,T_q^{(k)}] = \hbar q\,T_q^{(k)}.$$

If

$$J_z\lvert j_i,m_i\rangle = \hbar m_i\lvert j_i,m_i\rangle, \qquad J_z\lvert j_f,m_f\rangle = \hbar m_f\lvert j_f,m_f\rangle,$$

then the charge-like rule gives

$$m_f=m_i+q.$$

This is the simplest angular-momentum selection rule. It follows from rotations about the chosen quantization axis alone. Full rotational covariance gives additional constraints, including the triangle condition

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

but that stronger statement uses the irreducible tensor structure developed later in the tensor-operator chapter.

Electric Dipole Preview

The electric dipole operator is a vector under rotations and odd under parity. In spherical components it has rank $k=1$ and component labels

$$q=-1,0,1.$$

The magnetic rule gives

$$\Delta m=q,$$

where the chosen polarization selects which component appears. Rotation also imposes a rank-$1$ triangle rule. For orbital angular momentum this permits

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

subject to the triangle inequality, which also forbids $\ell_i=\ell_f=0$. Since the dipole operator is odd and central-potential orbital states have parity $(-1)^\ell$, parity removes the $\Delta\ell=0$ option. The familiar orbital electric-dipole rule is therefore

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

The complete derivation, including polarization conventions and total-angular-momentum labels, is developed in Dipole Transitions.

Exact, Approximate, and Broken Rules

An exact symmetry gives an exact zero for the corresponding matrix element. If the symmetry is approximate, the same matrix element is usually suppressed rather than exactly zero. If the perturbation, boundary condition, external field, environment, or measurement setup breaks the symmetry, the rule must be rederived for the new Hamiltonian and operator.

This is why “forbidden” rarely means impossible in the laboratory. It means forbidden by a specified operator under specified symmetry assumptions. An electric-dipole-forbidden transition may occur through a magnetic dipole, electric quadrupole, or two-photon process. Mixing can relax a rule only if it mixes the labels that protected the zero: ordinary spin–orbit coupling can relax a spin selection rule but preserves spatial parity, so it cannot by itself permit an electric-dipole transition between states of equal exact parity.

Necessary, Not Sufficient

Selection rules are necessary tests for nonzero matrix elements. Passing a selection rule does not guarantee that the matrix element is nonzero or large.

Even if the symmetry labels match, the amplitude can still vanish because of:

• orthogonality in additional quantum numbers;
• a radial integral that happens to be zero;
• destructive interference among components;
• a missing physical coupling in the model;

A nonzero matrix element still does not guarantee an observable transition rate: energy conservation and available final states impose additional conditions.

Symmetry says which matrix elements are impossible under the assumptions. Dynamics decides the actual value of those not ruled out.

Degeneracy and Basis Choices

The phase test is simplest when the states are already eigenstates of the symmetry. In a degenerate subspace, an arbitrary basis vector may not have definite symmetry labels even when the subspace as a whole is symmetry invariant.

The practical instruction is: diagonalize the commuting symmetry labels before applying the rule. If the symmetry sectors are distinct, operators with definite transformation law have block-selection properties. If several copies of the same representation occur, symmetry alone may not distinguish the copies, and additional labels or dynamics are needed.

This is why selection-rule statements should name the basis and coupling scheme. After spin–orbit coupling becomes important, for example, $j,m_j$ may be the good angular labels, while $m_\ell,m_s$ are no longer exact labels of the Hamiltonian.

Exercises

1. A conserved charge $Q$ has eigenvalues $q_i=2$ and $q_f=5$ on the initial and final states. If $[Q,T]=3T$, is $\langle f|T|i\rangle$ forbidden by this charge?

Solution

The charge rule requires

$$q_f=q_i+q_T.$$

Here $q_T=3$, so

$$q_i+q_T=2+3=5.$$

This matches $q_f$. The matrix element is not forbidden by this charge. It may still vanish for another symmetry or for dynamical reasons.

2. Suppose $\pi_i=+1$, $\pi_f=-1$, and $T$ is parity even. What does parity say about $M_{fi}$?

Solution

For a parity-even operator, $\eta_T=+1$. The nonzero condition is

$$\pi_f\eta_T\pi_i=1.$$

Substituting the values gives

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

The condition fails, so parity forces

$$M_{fi}=0.$$

3. A rank-$1$ spherical tensor component has $q=0$. What magnetic quantum-number change can it produce?

Solution

The magnetic rule is

$$m_f=m_i+q.$$

For $q=0$,

$$\Delta m=m_f-m_i=0.$$

This is often associated with the component parallel to the chosen quantization axis.

4. Why does an exact electric-dipole selection rule not mean that the transition can never be observed?

Solution

An electric-dipole rule applies to the electric-dipole operator under a specified symmetry model. A transition forbidden at electric-dipole order may be allowed by a different operator, such as a magnetic dipole or electric quadrupole operator. It may also become weakly allowed when a perturbation mixes the particular protecting labels. For example, an electric field can mix opposite parity states; parity-preserving spin–orbit coupling cannot do that, though it can mix total-spin labels in many-electron states. The rule still matters because it explains why the leading amplitude vanishes and why the remaining process is usually weaker.

References

• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Vols. I–II, Wiley, 1977.
• A. R. Edmonds, Angular Momentum in Quantum Mechanics, Princeton University Press, 1957.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020, chapters 3–5.
• M. Tinkham, Group Theory and Quantum Mechanics, Dover, 2003 reprint of the 1964 edition.
• E. P. Wigner, Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra, Academic Press, 1959.