First-order parity selection in an infinite well

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.

A first-order electric dipole transition connects opposite parities, but this does not prohibit all transitions between states of the same parity. In a symmetric infinite square well, use the centered position operator $X=x-L/2$. Each factor of $X$ reverses parity, so a second-order transition with two factors can connect the ground state to an excited state of the same parity through opposite-parity intermediate states.

For two-photon absorption, the second-order amplitude contains sums of products $⟨f|X|m⟩⟨m|X|i⟩$ divided by detuning-dependent energy denominators. Equal initial and final parity is therefore allowed, provided appropriate intermediate states and drive frequencies exist. “Allowed” does not guarantee a nonzero amplitude at every parameter value: different intermediate contributions can cancel. The distinction separates a symmetry-forbidden first-order matrix element from an all-orders prohibition.