Parity Selection Rules

A parity selection rule is a statement that a matrix element vanishes because spatial inversion assigns incompatible signs to the initial state, final state, and operator.

The basic matrix element is

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

If the states have definite parity and the operator has definite parity, then $M_{fi}$ can be nonzero only when the product of the three parity signs is even:

$$\pi_f\,\eta_O\,\pi_i=1.$$

The parity operator is introduced in Parity. Here we apply it to transition amplitudes, including the distinction between opposite-parity mixing and ordinary configuration mixing.

The algebra requires actual parity eigenstates and an operator of definite parity. A parity-invariant Hamiltonian allows stationary states to be chosen with this property; it does not force an arbitrary superposition in a degenerate space to have definite parity. External electric fields, asymmetric boundaries, and weak interactions can mix opposite parities.

State and Operator Parity

Let $\Pi$ denote the parity operator. A parity eigenstate satisfies

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

where

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

An operator has definite parity $\eta_O$ if

$$\Pi O\Pi^{-1} = \eta_O O, \qquad \eta_O=\pm1.$$

The operator is parity even when $\eta_O=+1$ and parity odd when $\eta_O=-1$.

Derivation

Insert the identity $\Pi^{-1}\Pi$ on both sides of the operator:

$$\begin{aligned} \langle f|O|i\rangle &= \langle f|\Pi^{-1} \left( \Pi O\Pi^{-1} \right) \Pi|i\rangle \\ &= \pi_f\,\eta_O\,\pi_i \langle f|O|i\rangle. \end{aligned}$$

Therefore

$$\left( 1-\pi_f\eta_O\pi_i \right) \langle f|O|i\rangle=0.$$

A nonzero matrix element requires

$$\pi_f\eta_O\pi_i=1.$$

Equivalently:

Operator parity	Nonzero matrix element requires
even, $\eta_O=+1$	$\pi_f=\pi_i$
odd, $\eta_O=-1$	$\pi_f=-\pi_i$

In words: even operators connect states of the same parity, while odd operators connect states of opposite parity.

Integral Version

In one dimension, a parity eigenfunction obeys

$$\psi_n(-x) = \pi_n\psi_n(x).$$

If $O(x)$ is multiplication by a function with parity $\eta_O$,

$$O(-x) = \eta_OO(x),$$

then the integrand in

$$\langle f|O|i\rangle = \int_{-\infty}^{\infty} \psi_f^*(x)O(x)\psi_i(x)\,dx$$

has parity $\pi_f\eta_O\pi_i$. If that product is $-1$, the integrand is odd and the integral over a symmetric domain vanishes.

The operator derivation is more general. It still works when the Hilbert space has spin, degeneracies, internal labels, or nontrivial coordinate systems.

Common Operator Parities

Under spatial inversion,

$$\mathbf R\mapsto-\mathbf R, \qquad \mathbf P\mapsto-\mathbf P.$$

Thus polar vectors built from position or momentum are parity odd. Examples include:

• position $\mathbf R$;
• momentum $\mathbf P$;
• electric dipole moment $\mathbf d=\sum_a q_a\mathbf r_a$.

Axial vectors are parity even. Examples include:

• orbital angular momentum $\mathbf L=\mathbf R\times\mathbf P$;
• spin angular momentum $\mathbf S$ in the usual nonrelativistic setting;
• magnetic moment operators proportional to angular momentum.

Some useful scalar and tensor examples are:

Operator	Parity
$\mathbf R^2$, $\mathbf P^2$, $\mathbf L^2$	even
$\mathbf L\cdot\mathbf S$	even
electric dipole $\mathbf d$	odd
magnetic dipole $\boldsymbol\mu$	even
electric quadrupole $Q_{ij}\sim 3R_iR_j-\delta_{ij}\mathbf R^2$	even

The parity label is independent of the rotational rank. A vector under rotations can be parity odd, like $\mathbf R$, or parity even, like $\mathbf L$.

One-Dimensional Example

In an inversion-symmetric one-dimensional potential, stationary states can be chosen even or odd. The position operator $X$ is odd:

$$\Pi X\Pi^{-1}=-X.$$

Therefore

$$\langle \text{even}|X|\text{even}\rangle=0, \qquad \langle \text{odd}|X|\text{odd}\rangle=0.$$

But $X$ can connect even and odd states. By contrast, $X^2$ is even, so

$$\langle \text{even}|X^2|\text{odd}\rangle=0.$$

For the centered harmonic oscillator,

$$\pi_n=(-1)^n.$$

Thus $X$ connects only oscillator states with opposite parity, while $X^2$ connects only states with the same parity. The ladder-operator calculation further refines this to $\Delta n=\pm1$ for $X$ and $\Delta n=0,\pm2$ for $X^2$.

Orbital Parity

For orbital angular momentum eigenfunctions,

$$Y_\ell^m(-\hat{\mathbf r}) = (-1)^\ell Y_\ell^m(\hat{\mathbf r}).$$

Thus a scalar central-potential state with orbital angular momentum $\ell$ has orbital parity

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

This immediately gives the parity part of many atomic selection rules. If an operator is odd, it connects orbital states with opposite parity:

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

Equivalently, $\ell_f-\ell_i$ must be odd. If an operator is even, $\ell_f-\ell_i$ must be even.

Spin does not by itself fix parity. When spin and orbital angular momentum are coupled, the total $j$ label is not enough to determine parity; one must still know the orbital or intrinsic parity content.

Electric Dipole Rule

The electric dipole operator is parity odd:

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

Therefore electric-dipole transitions require opposite parity:

$$\pi_f=-\pi_i.$$

For central-potential orbital states, this means $\ell_f-\ell_i$ is odd. Combining this with the rotational rank-$1$ rule gives the familiar orbital electric-dipole result

$$\Delta\ell=\pm1,$$

where the angular-momentum labels are valid and the usual boundary cases are respected. The electric-dipole specialization, including polarization and magnetic quantum numbers, is treated in Dipole Transitions.

Magnetic Dipole and Electric Quadrupole

Magnetic dipole operators are axial vectors and are parity even:

$$\eta_{M1}=+1.$$

Thus magnetic-dipole matrix elements connect states of the same parity, subject to their rotational rank-$1$ angular rules. The $E\lambda/M\lambda$ rank-and-parity table is collected in Multipole Operators.

Electric quadrupole operators are built from two powers of position, with the trace removed. They are parity even:

$$\eta_{E2}=+1.$$

Thus electric-quadrupole matrix elements also connect states of the same parity, subject to rank-$2$ angular rules. For scalar central-potential orbital states, parity permits only even changes in $\ell$. Combining with the rank-$2$ triangle condition gives the usual possibilities

$$\Delta\ell=0,\pm2,$$

with the full triangle condition still essential: an initial $\ell_i=0$ requires $\ell_f=2$, not $0$. A particular magnetic component can impose additional zeros.

Broken and Approximate Parity

If parity is not an exact symmetry, states need not have definite parity. A weakly mixed state may have the schematic form

$$|\widetilde i\rangle = |i,+\rangle + \epsilon |i,-\rangle, \qquad |\epsilon|\ll1.$$

An otherwise forbidden matrix element can then become nonzero at order $\epsilon$. For an odd operator $O_-$ between two mostly even states,

$$\langle \widetilde f|O_-|\widetilde i\rangle = \epsilon_i \langle f,+|O_-|i,-\rangle + \epsilon_f^* \langle f,-|O_-|i,+\rangle + O(\epsilon^2).$$

A parity-odd perturbation, such as a static electric field, can supply these opposite-parity admixtures. By contrast, ordinary electrostatic configuration interaction and spin–orbit coupling preserve parity: they mix only configurations with the same total parity and cannot by themselves remove this zero. The corresponding rate is generally quadratic in the small admixture, unless a separate nonzero amplitude interferes with it.

Common Mistakes

• Assigning parity labels to states that are actually mixtures of opposite parity. The matrix-element proof needs parity eigenstates, whether or not they are energy eigenstates.
• Knowing the state parity but forgetting to determine the operator parity.
• Assuming every vector operator is parity odd; axial vectors are parity even.
• Assuming same $j$ means same parity. Total angular momentum does not determine parity by itself.
• Calling a transition impossible rather than forbidden at a specified order or by a specified operator.
• Combining parity and angular-momentum rules without checking both separately.

Exercises

1. Two states have the same parity. Can an odd operator connect them?

Solution

No. If $\pi_f=\pi_i$, then for an odd operator $\eta_O=-1$,

$$\pi_f\eta_O\pi_i = - \pi_f\pi_i = -1.$$

The nonzero condition $\pi_f\eta_O\pi_i=1$ fails, so the matrix element vanishes.

2. In a centered harmonic oscillator, which matrix elements can be nonzero by parity: $\langle n'|X|n\rangle$ and $\langle n'|X^2|n\rangle$?

Solution

The oscillator parity is $\pi_n=(-1)^n$. Since $X$ is odd, $X$ requires opposite parity:

$$(-1)^{n'}=-(-1)^n.$$

Thus $n'-n$ must be odd. Since $X^2$ is even, $X^2$ requires the same parity:

$$(-1)^{n'}=(-1)^n.$$

Thus $n'-n$ must be even. The actual oscillator ladder calculation gives the stronger rules $\Delta n=\pm1$ for $X$ and $\Delta n=0,\pm2$ for $X^2$.

3. Combine parity with the rank-$1$ orbital rule for an electric dipole operator in a central potential.

Solution

The rank-$1$ rotational rule permits

$$\ell_f=\ell_i,\ell_i\pm1$$

where valid. The electric dipole operator is parity odd, so it requires opposite parity:

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

Therefore $\ell_f-\ell_i$ must be odd. The $\ell_f=\ell_i$ option is removed, leaving

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

4. Suppose two mostly even states contain small odd admixtures:

$$|\widetilde a\rangle=|a,+\rangle+\epsilon_a|a,-\rangle, \qquad |\widetilde b\rangle=|b,+\rangle+\epsilon_b|b,-\rangle.$$

Estimate the leading parity-allowed contribution to $\langle\widetilde b|O_-|\widetilde a\rangle$ for an odd operator $O_-$.

Solution

Expand to first order:

$$\langle\widetilde b|O_-|\widetilde a\rangle = \langle b,+|O_-|a,+\rangle + \epsilon_a \langle b,+|O_-|a,-\rangle + \epsilon_b^* \langle b,-|O_-|a,+\rangle + O(\epsilon^2).$$

The first term vanishes because an odd operator cannot connect two even states. The leading contribution is therefore

$$\epsilon_a \langle b,+|O_-|a,-\rangle + \epsilon_b^* \langle b,-|O_-|a,+\rangle.$$

References

• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Wiley, 1977.
• A. R. Edmonds, Angular Momentum in Quantum Mechanics, Princeton University Press, 1957.
• L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, 3rd ed., Pergamon Press, 1977.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• E. P. Wigner, Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra, Academic Press, 1959.