= Quadratic Stark effect
{title2=$\delta E\propto |\mathbf E|^2$}
The quadratic <Stark effect> is an <energy level> shift whose leading term is quadratic in an applied <electric field>. For a nondegenerate <energy eigenstate> of definite parity, the diagonal <matrix element> of an odd electric-dipole perturbation vanishes. <Second-order nondegenerate perturbation theory> gives
$$
\delta E_n^{(2)}=\sum_{k\ne n}\frac{|\langle k|V|n\rangle|^2}{E_n^{(0)}-E_k^{(0)}}.
$$
Every term is negative for the ground state. If field reversal is implemented by the <parity operator>, an analytic nondegenerate energy branch is even in the field. This symmetry explains the absence of odd powers; it does not apply without qualification to a degeneracy split linearly by the field.
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