Electric-dipole selection rule

Because the position operator has odd parity, an electric-dipole matrix element vanishes between states of the same parity. For orbital-angular-momentum eigenstates, the full rule is $\Delta \ell =\pm 1$ and $\Delta m=0,\pm 1$, with the field polarization selecting the allowed change in $m$.

For $0<x<w$, normalized states ${\psi }_n=\sqrt{2/w}\sin [(n+1)\pi x/w]$ have centre-reflection eigenvalue $(-1{)}^n$. Since $x-w/2$ is odd, off-diagonal electric-dipole matrix elements vanish between equal-parity states. Ground state n=0 cannot reach n=2 at first order; an initial n=1 can reach even labels including n=0. This does not impose an all-orders prohibition: two dipole interactions can restore the original parity. The general nonzero elements follow from position matrix elements in a symmetric infinite square well after shifting the origin.

For an inversion-symmetric system, electric-dipole transitions require opposite parities. Transitions between d-shell terms of the same gerade parity can acquire weak vibronic or symmetry-breaking intensity.