Parity and electric-dipole transitions

For a spinless spatial wavefunction take $(P\psi)(\mathbf r)=\psi(-\mathbf r)$. With $P^2=I$, parity eigenvalues are $+1$ and $-1$. An inversion-symmetric Hamiltonian commutes with parity, so energy eigenstates can be organized into parity sectors. A nondegenerate eigenstate has definite parity. Within a degenerate eigenspace one may choose a parity basis, but an arbitrary superposition in that space need not itself have definite parity. Commutation does not make every written vector a parity eigenvector.

Position and momentum are polar vectors and reverse under inversion. Their cross product, orbital angular momentum, is axial and remains unchanged. Spin transforms as an axial vector too; it is not reversed by parity. This is a common point of confusion because both parity and time reversal change familiar motions, but their operator actions differ. In central potentials the spatial spherical harmonic contributes parity $(-1)^\ell$, whereas the radial part stays unchanged because radius is nonnegative and invariant.

In the electric-dipole approximation the interaction is minus the dipole moment dotted into the applied electric field. The approximation treats the field as nearly uniform across the bound state, requiring the state size to be small compared with the radiation wavelength. For a single charge measured relative to a fixed origin, the dipole is charge times position and is odd under inversion. Insert a pair of parity operators into its matrix element between definite-parity states to expose a sign constraint.

If the two state parities are equal, the matrix element equals its own negative and must vanish. If they are opposite, parity places no further restriction on that matrix element. The surviving integral can still vanish for angular or radial reasons. In a one-electron central-field model, parity requires an odd change of orbital label; combining that with the rank-one triangle condition gives a change of exactly one. Neither ingredient alone yields the full familiar electric-dipole rule.

A static external electric field can mix opposite-parity states, because its dipole interaction breaks the inversion symmetry of the fixed-system Hamiltonian. A previously vanishing amplitude can then acquire a small value. Weak interactions also violate parity, but ordinary electromagnetic spectroscopy often starts from a parity-conserving approximation and estimates such corrections separately. Calling a line forbidden therefore requires stating the operator and the symmetry assumptions, rather than assigning an absolute label to a pair of levels.

Beyond the long-wavelength dipole approximation, magnetic-dipole and electric-quadrupole operators have different parity and angular transformation properties. A line absent at electric-dipole order may remain visible through these weaker mechanisms or through multiphoton processes. Even in the ideal dipole model, population and frequency matching still matter: an allowed matrix element does not supply occupied initial states or a resonant field. Symmetry saves an integral calculation, but it does not replace the dynamical and experimental conditions for observing a transition.