Stark shifts and atomic polarizability

Write the perturbation as $V=-d_z\mathcal E$, where $\mathcal E$ is a uniform static electric field and $d_z$ is the atomic dipole operator along it. For one electron relative to a positive nucleus, $d_z=-ez$, so the perturbation is $+ez\mathcal E$. Under parity, position and dipole change sign. A nondegenerate parity eigenstate therefore has zero diagonal dipole expectation, making its first-order energy shift vanish in the absence of parity-violating effects.

Off-diagonal dipole elements between opposite-parity states need not vanish. They change the wavefunction at first order even when the energy does not move until second order. The induced expectation of the dipole consequently grows linearly with a weak field. This separates a response property from a permanent moment. It also explains why a photograph of an unperturbed symmetric electron density cannot establish that its polarizability is zero: response involves accessible excited states and their energy gaps.

For a nondegenerate ground state every excited-state denominator is negative, so its static scalar polarizability is positive and its leading shift is downward. An excited state has states both above and below it; contributions can oppose each other, and its polarizability need not be positive. The scalar notation assumes a specified substate response or an isotropic level. For angular momentum at least one, tensor contributions can make shifts depend on the magnetic projection and field geometry.

Differentiating the energy with respect to field gives the induced dipole: $\langle d_z\rangle=-\partial E/\partial\mathcal E=\alpha_a\mathcal E$ in the quadratic regime. The factor one half in the energy accounts for building up an induced dipole. Using the permanent-dipole expression with the final induced dipole would double the shift. The second-order expansion requires each relevant mixing energy to be small compared with its energy gap; a small field in laboratory units alone is not a sufficient criterion.

Consider two opposite-parity states with equal unperturbed energy and real off-diagonal dipole magnitude d. Diagonalizing their two-dimensional perturbation gives shifts $\pm d\mathcal E$ for a signed field, with branches exchanging under field reversal. The new eigenvectors are equal mixtures, so the original parity states are no longer field eigenstates. If the pair instead has a nonzero gap, sufficiently weak fields recover quadratic shifts. Hydrogen’s ideal Coulomb n manifolds supply a famous linear-Stark approximation, but fine, Lamb, and hyperfine splittings set its weak-field limits.

A static electric field also turns strict Coulomb bound states into metastable resonances because the potential permits escape far along one direction. Perturbative real shifts describe long-lived states when ionization is negligible on the measurement timescale. They cannot predict arbitrary strong-field stability. In a measured transition, the relevant coefficient is the upper-minus-lower polarizability; common shifts cancel. The parabola lab displays this quadratic dependence algebraically and does not mix orbitals, compute a polarizability, or model escape.