= Higher-order parity-allowed transitions in an infinite well
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 $\langle f|X|m\rangle\langle m|X|i\rangle$ 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.
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